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Two regularization methods for a class of inverse boundary value problems of elliptic type

Abdallah Bouzitouna1, Nadjib Boussetila2* and Faouzia Rebbani1

Author Affiliations

1 Applied Mathematics Laboratory, University Badji Mokhtar Annaba, P.O. Box 12, Annaba, 23000, Algeria

2 Department of Mathematics, 8 Mai 1945 Guelma University, P.O. Box 401, Guelma, 24000, Algeria

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Boundary Value Problems 2013, 2013:178  doi:10.1186/1687-2770-2013-178


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/178


Received:13 March 2013
Accepted:18 July 2013
Published:2 August 2013

© 2013 Bouzitouna et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper deals with the problem of determining an unknown boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M1">View MathML</a> in the boundary value problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M4">View MathML</a>, with the aid of an extra measurement at an internal point. It is well known that such a problem is severely ill-posed, i.e., the solution does not depend continuously on the data. In order to overcome the instability of the ill-posed problem, we propose two regularization procedures: the first method is based on the spectral truncation, and the second is a version of the Kozlov-Maz’ya iteration method. Finally, some other convergence results including some explicit convergence rates are also established under a priori bound assumptions on the exact solution.

MSC: 35R25, 65J20, 35J25.

Keywords:
ill-posed problems; elliptic problems; cut-off spectral regularization; iterative regularization

1 Formulation of the problem

Throughout this paper, H denotes a complex separable Hilbert space endowed with the inner product <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M5">View MathML</a> and the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M7">View MathML</a> stands for the Banach algebra of bounded linear operators on H.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M8">View MathML</a> be a positive, self-adjoint operator with compact resolvent, so that A has an orthonormal basis of eigenvectors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M9">View MathML</a> with real eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M10">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M11">View MathML</a>

In this paper, we are interested in the following inverse boundary value problem: find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M12">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M13">View MathML</a>

(1.1)

where f is the unknown boundary condition to be determined from the interior data

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M14">View MathML</a>

(1.2)

This problem is an abstract version of an inverse boundary value problem, which generalizes inverse problems for second-order elliptic partial differential equations in a cylindrical domain, for example we mention the following problem.

Example 1.1 An example of (1.1) is the boundary value problem for the Laplace equation in the strip <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M15">View MathML</a>, where the operator A is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M16">View MathML</a>

which takes the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M17">View MathML</a>

To our knowledge, there are few papers devoted to this class of problems in the abstract setting, except for [1,2]. In [3], the author studied a similar problem posed on a bounded interval. In this study, the algebraic invertibility of the inverse problem was established. However, the regularization aspect was not investigated.

We note here that this inverse problem was studied by Levine and Vessella [2], where the authors considered the problem of recovering <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M1">View MathML</a> from the experimental data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M19">View MathML</a> associated to the internal measurements <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M20">View MathML</a>, in which the temperature is measured at various depths <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M21">View MathML</a> as approximate functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M22">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M23">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M24">View MathML</a> are positive weights with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M25">View MathML</a> and ε denotes the level noisy.

The regularizing strategy employed in [2] is essentially based on the Tikhonov regularization and the conditional stability estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M26">View MathML</a> for some a priori constant E.

In practice, the use of N-measurements or the average of a series of measurements is an expensive operation, and sometimes unrealizable. Moreover, the numerical implementation of the stabilized solutions by the Tikhonov regularization method for this class of problems will be a very complex task.

For these reasons, we propose in our study a practical regularizing strategy. We show that we can recover <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M1">View MathML</a> from the internal measurement <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M28">View MathML</a> under the conditional stability estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M29">View MathML</a> for some a priori constant E. Moreover, our investigation is supplemented by numerical simulations justifying the feasibility of our approach.

2 Preliminaries and basic results

In this section we present the notation and the functional setting which will be used in this paper and prepare some material which will be used in our analysis.

2.1 Notation

We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M30">View MathML</a> the set of all closed linear operators densely defined in H. The domain, range and kernel of a linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M31">View MathML</a> are denoted as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M32">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M33">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M34">View MathML</a>; the symbols <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M36">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M37">View MathML</a> are used for the resolvent set, spectrum and point spectrum of B, respectively. If V is a closed subspace of H, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M38">View MathML</a> the orthogonal projection from H to V.

For the ease of reading, we summarize some well-known facts in spectral theory.

2.2 Spectral theorem and properties

By the spectral theorem, for each strictly positive self-adjoint operator B,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M39">View MathML</a>

there is a unique right continuous family <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M40">View MathML</a> of orthogonal projection operators such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M41">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M42">View MathML</a>

Theorem 2.1 [[4], Theorem 6, XII.2.5, pp.1196-1198]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M43">View MathML</a>be the spectral resolution of the identity associated toB, and let Φ be a complex Borel function definedE-almost everywhere on the real axis. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M44">View MathML</a>is a closed operator with dense domain. Moreover,

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M45">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M48">View MathML</a>,

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M49">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M47">View MathML</a>,

(iv) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M51">View MathML</a>. In particular, if Φ is a real Borel function, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M44">View MathML</a>is self-adjoint.

We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M54">View MathML</a>, the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M55">View MathML</a>-semigroup generated by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M56">View MathML</a>. Some basic properties of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M57">View MathML</a> are listed in the following theorem.

Theorem 2.2 (see [5], Chapter 2, Theorem 6.13, p.74)

For this family of operators, we have:

1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M59">View MathML</a>;

2. the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M61">View MathML</a>, is analytic;

3. for every real<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M62">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M63">View MathML</a>, the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M64">View MathML</a>;

4. for every integer<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M65">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M66">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M67">View MathML</a>;

5. for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M62">View MathML</a>, we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M70">View MathML</a>.

Theorem 2.3For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M57">View MathML</a>is self-adjoint and one-to-one operator with dense range (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M74">View MathML</a>).

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M75">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M76">View MathML</a>. Then, by virtue of (iv) of Theorem 2.1, we can write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M77">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M79">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M80">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M81">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M82">View MathML</a>. Using analyticity, we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M83">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M54">View MathML</a>. Strong continuity at 0 now gives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M85">View MathML</a>. This shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M86">View MathML</a>.

Thanks to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M87">View MathML</a>

we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M88">View MathML</a> is dense in H. □

Remark 2.1 For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M89">View MathML</a>, this theorem ensures that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M90">View MathML</a> is self-adjoint and one-to-one operator with dense range <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M91">View MathML</a>. Then we can define its inverse <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M92">View MathML</a>, which is an unbounded self-adjoint strictly positive definite operator in H with dense domain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M93">View MathML</a>

Let us consider the following problem: for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M94">View MathML</a> find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M95">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M96">View MathML</a>

(2.1)

Theorem 2.4 [[6], Theorem 7.5, p.191]

For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M94">View MathML</a>, problem (2.1) has a unique solution, given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M98">View MathML</a>

(2.2)

Moreover, for all integer<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M100">View MathML</a>. If, in addition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M101">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M102">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M103">View MathML</a>

On the other hand, Theorem 2.4 provides smoothness results with respect to y: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M104">View MathML</a> whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M105">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M106">View MathML</a>. Under this same hypothesis, we also have smoothness in space: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M107">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M108">View MathML</a>.

Here we recall a crucial theorem in the analysis of the inverse problems.

Theorem 2.5 [[7], Generalized Picard theorem, p.502]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M109">View MathML</a>be a self-adjoint operator and the Hilbert spaceH, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M110">View MathML</a>be its spectral resolution of unity. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M111">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M112">View MathML</a>. We suppose that the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M113">View MathML</a>either is empty or contains isolated point only. Then the vectorial equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M114">View MathML</a>

is solvable if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M115">View MathML</a>

Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M116">View MathML</a>

On the basis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M117">View MathML</a>, we introduce the Hilbert scale <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M118">View MathML</a> (resp. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M119">View MathML</a>) induced by as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M121">View MathML</a>

2.3 Non-expansive operators

Definition 2.1 A linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M122">View MathML</a> is called non-expansive if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M123">View MathML</a>

Theorem 2.6 [[8], Theorem 2.2]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M122">View MathML</a>be a positive, self-adjoint operator with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M125">View MathML</a>. Putting<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M126">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M127">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M128">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M129">View MathML</a>

For more details concerning the theory of non-expansive operators, we refer to Krasnosel’skii et al. [[9], p.66].

Let use consider the operator equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M130">View MathML</a>

(2.3)

for non-expansive operators M.

Theorem 2.7LetMbe a linear self-adjoint, positive and non-expansive operator onH. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M131">View MathML</a>be such that equation (2.3) has a solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M132">View MathML</a>. If 1 is not an eigenvalue ofM, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M133">View MathML</a>is injective (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M134">View MathML</a>), then the successive approximations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M135">View MathML</a>

converge to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M132">View MathML</a>for any initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M137">View MathML</a>.

Proof From the hypothesis and by virtue of Theorem 2.6, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M138">View MathML</a>

(2.4)

By induction with respect to n, it is easily seen that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M139">View MathML</a> has the explicit form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M140">View MathML</a>

and (2.4) allows us to conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M141">View MathML</a>

(2.5)

 □

Remark 2.2 In many situations, some boundary value problems for partial differential equations which are ill-posed can be reduced to Fredholm operator equations of the first kind of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M142">View MathML</a>, where B is compact, positive, and self-adjoint operator in a Hilbert space H. This equation can be rewritten in the following way:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M143">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M144">View MathML</a>, and ω is a positive parameter satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M145">View MathML</a>. It is easily seen that the operator L is non-expansive and 1 is not an eigenvalue of L. It follows from Theorem 2.7 that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M146">View MathML</a> converges and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M147">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M148">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M149">View MathML</a>.

3 Ill-posedness and stabilization of the inverse boundary value problem

3.1 Cauchy problem with Dirichlet conditions

Consider the following well-posed boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M150">View MathML</a>

(3.1)

where ξ is an H-valued function.

Definition 3.1 [[10], p.250]

• A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M151">View MathML</a> is called a generalized solution to equation (3.1) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M152">View MathML</a>, and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M153">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M154">View MathML</a> and obeys equation (3.1) on the same interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M155">View MathML</a>.

• A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M151">View MathML</a> is called a classical solution to equation (3.1) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M157">View MathML</a>, and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M153">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M154">View MathML</a> and obeys equation (3.1) on the same interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M155">View MathML</a>.

Theorem 3.1Problem (3.1) admits a unique generalized (resp. classical) solution if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M94">View MathML</a> (resp. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M162">View MathML</a>).

Proof By using the Fourier expansion and the given Dirichlet boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M163">View MathML</a>

we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M164">View MathML</a>

(3.2)

This differential equation admits two linearly independent fundamental solutions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M165">View MathML</a>

Thus, its general solution can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M166">View MathML</a>

Applying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M167">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M168">View MathML</a> yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M169">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M170">View MathML</a>. Finally, the solution of (3.2) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M171">View MathML</a>

(3.3)

Remark 3.1 It is easy to check that the expression (3.3) solves the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M172">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M94">View MathML</a> (resp. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M162">View MathML</a>), by virtue of Theorem 2.4 and Remark 3.1, we easily check the inclusion <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M175">View MathML</a> (resp. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M176">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M177">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M153">View MathML</a>. □

3.2 Inverse boundary value problem

Our inverse problem is to determine <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M179">View MathML</a> from the supplementary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M180">View MathML</a>, then we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M181">View MathML</a>

(3.4)

We define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M182">View MathML</a>

(3.5)

The operator equation (3.5) is the main instrument in investigating problem (3.4). More precisely, we want to study the following properties:

1. Injectivity of K (identifiability);

2. Continuity of K and the existence of its inverse (stability);

3. The range of K.

It is easy to see that K is a linear compact self-adjoint operator with the singular values <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M183">View MathML</a>, and by virtue of Remark 2.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M184">View MathML</a>

Now, to conclude the solvability of problem (3.4) it is enough to apply Theorem 2.5.

Corollary 3.1The inverse problem (3.4) is uniquely solvable if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M185">View MathML</a>

(3.6)

In this case, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M186">View MathML</a>

(3.7)

In other words, the solution f of the inverse problem is obtained from the data g via the unbounded operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M187">View MathML</a> defined on functions g in the subspace

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M188">View MathML</a>

Corollary 3.2Problem (1.1)-(1.2) admits a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M189">View MathML</a>if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M190">View MathML</a>

In this case, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M191">View MathML</a>

(3.8)

From this representation, we see that:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M192">View MathML</a> is stable in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M193">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M194">View MathML</a>);

u is unstable in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M195">View MathML</a>. This follows from the high-frequency <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M196">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M197">View MathML</a>.

3.3 Regularization by truncation method and error estimates

A natural way to stabilize the problem is to eliminate all the components of large n from the solution and instead consider (3.7) only for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M198">View MathML</a>.

Definition 3.2 For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M199">View MathML</a>, the regularized solution of problem (1.1)-(1.2) is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M200">View MathML</a>

(3.9)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M201">View MathML</a>

(3.10)

Remark 3.2 If the parameter N is large, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M202">View MathML</a> is close to the exact solution f. On the other hand, if the parameter N is fixed, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M202">View MathML</a> is bounded. So, the positive integer N plays the role of regularization parameter.

Remark 3.3 In view of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M204">View MathML</a>

and if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M205">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M206">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M207">View MathML</a>

implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M208">View MathML</a>

Since the data g are based on (physical) observations and are not known with complete accuracy, we assume that g and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M209">View MathML</a> satisfy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M210">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M209">View MathML</a> denotes the measured data and δ denotes the level noisy.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M212">View MathML</a> denote the regularized solution of problem (1.1), (1.2) with measured data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M213">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M214">View MathML</a>

(3.11)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M215">View MathML</a>

(3.12)

As usual, in order to obtain convergence rate, we assume that there exists an a priori bound for problem (1.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M216">View MathML</a>

(3.13)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M217">View MathML</a> is a given constant.

Remark 3.4 For given two exact conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M218">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M219">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M220">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M221">View MathML</a> be the corresponding regularized solutions, respectively. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M222">View MathML</a>

(3.14)

The main theorem of this method is as follows.

Theorem 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M223">View MathML</a>be the regularized solution given by (3.11), and letfbe the exact solution given by (3.7). If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M224">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M225">View MathML</a>and if we choose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M226">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M227">View MathML</a>, then we have the error bound

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M228">View MathML</a>

(3.15)

Proof From direct computations, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M229">View MathML</a>

Using the triangle inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M230">View MathML</a>

we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M231">View MathML</a>

(3.16)

By choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M232">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M227">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M234">View MathML</a>

 □

Finally, from (3.4) and (3.15), we deduce the following corollary.

Corollary 3.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M235">View MathML</a>be the regularized solution given by (3.12), and letube the exact solution given by (3.8). If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M224">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M225">View MathML</a>and if we choose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M232">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M227">View MathML</a>, then we have the error bound

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M240">View MathML</a>

(3.17)

4 Regularization by the Kozlov-Maz’ya iteration method and error estimates

In [11,12] Kozlov and Maz’ya proposed an alternating iterative method to solve boundary value problems for general strongly elliptic and formally self-adjoint systems. After that, the idea of this method has been successfully used for solving various classes of ill-posed (elliptic, parabolic and hyperbolic) problems; see, e.g., [13-15].

In this section we extend this method to our ill-posed problem.

4.1 Description of the method

The iterative algorithm for solving the inverse problem (1.1)-(1.2) starts by letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M241">View MathML</a> be arbitrary. The first approximation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M242">View MathML</a> is the solution to the direct problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M243">View MathML</a>

(4.1)

If the pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M244">View MathML</a> has been constructed, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M245">View MathML</a>

(4.2)

where ω is such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M246">View MathML</a>

Finally, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M247">View MathML</a> by solving the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M248">View MathML</a>

(4.3)

We set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M249">View MathML</a>. If we iterate backwards in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M250">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M251">View MathML</a>

(4.4)

This implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M252">View MathML</a>

(4.5)

Proposition 4.1The operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M253">View MathML</a>is self-adjoint and non-expansive on H. Moreover, it has not 1 as eigenvalue.

Proof The self-adjointness follows from the definition of G (see Theorem 2.1). Since the inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M254">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M255">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M256">View MathML</a>, then 1 is not an eigenvalue of G. □

In general, the exact solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M257">View MathML</a> is required to satisfy the so-called source condition; otherwise, the convergence of the regularization method approximating the problem can be arbitrarily slow. Since our problem is exponentially ill-posed (the eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M258">View MathML</a> of K converge exponentially to 0), it is well known in this case [16,17] that the best choice to accelerate the convergence of the regularization method is to use logarithmic-type source conditions, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M259">View MathML</a>

(4.6)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M260">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M261">View MathML</a>.

Remark 4.1 [[16], p.34]

The logarithmic source condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M262">View MathML</a> is equivalent to the inclusion <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M263">View MathML</a>.

Proof The proof is based on the following equivalence:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M264">View MathML</a>

 □

Lemma 4.1 [[18], Appendix, Lemma A.1]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M261">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M266">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M267">View MathML</a>. Then the real-valued function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M268">View MathML</a>defined on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M269">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M270">View MathML</a>

(4.7)

Remark 4.2 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M271">View MathML</a>. Then the real-valued function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M272">View MathML</a> defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M269">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M274">View MathML</a>

(4.8)

Proof Using the mean value theorem, we can write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M275">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M276">View MathML</a>

 □

Let us consider the following real-valued functions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M277">View MathML</a>

Using the change of variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M278">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M279">View MathML</a>

Now we are in a position to state the main result of this method.

Theorem 4.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M205">View MathML</a>andωsatisfy<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M281">View MathML</a>, let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M282">View MathML</a>be an arbitrary element for the iterative procedure suggested above, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M283">View MathML</a>be thekth approximate solution. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M284">View MathML</a>

(4.9)

Moreover, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M285">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M261">View MathML</a>), i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M287">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M94">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M289">View MathML</a>, then the rate of convergence of the method is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M290">View MathML</a>

(4.10)

Proof By virtue of Proposition 4.1 and Theorem 2.7, it follows immediately

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M291">View MathML</a>

We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M292">View MathML</a>

and by virtue of Lemma 4.1 (estimate (4.7)), we conclude the desired estimate. □

Theorem 4.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M205">View MathML</a>andωsatisfy<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M294">View MathML</a>, let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M282">View MathML</a>be an arbitrary element for the iterative procedure suggested above, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M283">View MathML</a> (resp. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M297">View MathML</a>) be thekth approximate solution for the exact datag (resp. for the inexact data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M209">View MathML</a>) such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M299">View MathML</a>. Then, under condition (4.6), the following inequality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M300">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M301">View MathML</a>.

Proof Using (4.4) and the triangle inequality, we can write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M302">View MathML</a>

(4.11)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M303">View MathML</a>

(4.12)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M304">View MathML</a>

(4.13)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M305">View MathML</a>

By using inequality (4.8), the quantity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M306">View MathML</a> can be estimated as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M307">View MathML</a>

(4.14)

Combining (4.13) and (4.14) and taking the supremum with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M308">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M309">View MathML</a>, we obtain the desired bound.

Remark 4.3 Choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M310">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M311">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M312">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M313">View MathML</a>

 □

5 Numerical results

In this section we give a two-dimensional numerical test to show the feasibility and efficiency of the proposed methods. Numerical experiments were carried out using MATLAB.

We consider the following inverse problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M314">View MathML</a>

(5.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M315">View MathML</a> is the unknown source and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M316">View MathML</a> is the supplementary condition.

It is easy to check that the operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M317">View MathML</a>

is positive, self-adjoint with compact resolvent (A is diagonalizable).

The eigenpairs <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M318">View MathML</a> of A are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M319">View MathML</a>

In this case, formula (3.7) takes the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M320">View MathML</a>

(5.2)

Truncation method

We use trapezoid’s rule to approach the integral and do an approximate truncation for the series by choosing the sum of the front <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M321">View MathML</a> terms. After considering an equidistant grid <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M322">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M323">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M324">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M325">View MathML</a>

(5.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M326">View MathML</a>

(5.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M327">View MathML</a>

(5.5)

In the following, we consider an example which has an exact expression of solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M328">View MathML</a>.

Example

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M329">View MathML</a>, then the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M330">View MathML</a> is the exact solution of problem (5.1). Consequently, the data function is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M331">View MathML</a>.

Adding a random distributed perturbation (obtained by the Matlab command randn) to each data function, we obtain the vector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M209">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M333">View MathML</a>

where ε indicates the noise level of the measurement data and the function ‘<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M334">View MathML</a>’ generates arrays of random numbers whose elements are normally distributed with mean 0, variance <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M335">View MathML</a>, and standard deviation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M336">View MathML</a>. ‘<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M337">View MathML</a>’ returns an array of random entries that is the same size as g. The bound on the measurement error δ can be measured in the sense of Root Mean Square Error (RMSE) according to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M338">View MathML</a>

Using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M209">View MathML</a> as a data function, we obtain the computed approximation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M223">View MathML</a> by (5.5). The relative error <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M341">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M342">View MathML</a>

(5.6)

Kozlov-Maz’ya iteration method

By using the central difference with step length <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M343">View MathML</a> to approximate the first derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M344">View MathML</a> and the second derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M345">View MathML</a>, we can get the following semi-discrete problem (ordinary differential equation):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M346">View MathML</a>

(5.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M347">View MathML</a> is the discretization matrix stemming from the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M348">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M349">View MathML</a>

is a symmetric, positive definite matrix. We assume that it is fine enough so that the discretization errors are small compared to the uncertainty δ of the data; this means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M347">View MathML</a> is a good approximation of the differential operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M351">View MathML</a>, whose unboundedness is reflected in a large norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M347">View MathML</a> (see [[19], p.5]). The eigenpairs <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M353">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M347">View MathML</a> are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M355">View MathML</a>

The discrete iterative approximation of (4.12) takes the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M356">View MathML</a>

(5.8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M357">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M358">View MathML</a>.

Figures 1-4, Table 1 show the comparisons between the exact solution and its computed approximations for different values N, M and ε.

thumbnailFigure 1. TM with (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M359">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M360">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M361">View MathML</a>).

thumbnailFigure 2. TM with (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M359">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M360">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M364">View MathML</a>).

thumbnailFigure 3. TM with (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M365">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M360">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M361">View MathML</a>).

thumbnailFigure 4. TM with (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M365">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M360">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M364">View MathML</a>).

Table 1. Truncation method: Relative error<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M341">View MathML</a>

Figures 5-12, Table 2 show the comparisons between the exact solution and its computed approximations for different values N, k, ω and ε.

Table 2. Kozlov-Maz’ya method: Relative error<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M341">View MathML</a>

Conclusion

The numerical results (Figures 1-4) are quite satisfactory. Even with the noise level <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M383">View MathML</a>, the numerical solutions are still in good agreement with the exact solution. In addition, the numerical results (Figures 5-12) are better for (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M384">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M383">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M386">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/178/mathml/M387">View MathML</a>) and the other values are also acceptable.

In this study, a convergent and stable reconstruction of an unknown boundary condition has been obtained using two regularizing methods: truncation method and Kozlov-Maz’ya iteration method. Both theoretical and numerical studies have been provided.

Future work will involve the error effect arising in computing eigenfunctions and eigenvalues of the operator A on the truncation method. The question is how to obtain some optimal balance between the accuracy of eigensystem and the noise level of input data.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors have contributed equally. All authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank the editor and the anonymous referees for their valuable comments and helpful suggestions that improved the quality of our paper. This work is supported by the DGRST of Algeria (PNR Project 2011-code: 8\92 u23\92 997).

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