Spectral Regularization Methods for an Abstract Ill-Posed Elliptic Problem

Nadjib Boussetila,Salim Hamida,Faouzia Rebbani

doi:10.1155/2013/947379

Nadjib Boussetila, Salim Hamida + Show 1 more

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https://doi.org/10.1155/2013/947379

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Abstract

We study an abstract elliptic Cauchy problem associated with an unbounded self-adjoint positive operator which has a continuous spectrum. It is well-known that such a problem is severely ill-posed; that is, the solution does not depend continuously on the Cauchy data. We propose two spectral regularization methods to construct an approximate stable solution to our original problem. Finally, some other convergence results including some explicit convergence rates are also established under a priori bound assumptions on the exact solution.

Highlights

IntroductionThroughout this paper H denotes a complex Hilbert space endowed with the inner product (⋅, ⋅), and the norm ‖⋅‖, L(H) stands for the Banach algebra of bounded linear operators on H
Throughout this paper H denotes a complex Hilbert space endowed with the inner product (⋅, ⋅), and the norm ‖⋅‖, L(H) stands for the Banach algebra of bounded linear operators on H.Let A be a linear unbounded operator with dense domain D(A)
This work is mainly devoted to theoretical aspects of the spectral regularization methods to problem (1) in the abstract setting, by considering more general self-adjoint operators when A is positive and induces the elliptic case, that is, has the following properties: for any λ ∈ (−∞, 0], the resolvent R(λ; A) = (A − λI)−1 exists and satisfies the estimates

Summary

Introduction

Throughout this paper H denotes a complex Hilbert space endowed with the inner product (⋅, ⋅), and the norm ‖⋅‖, L(H) stands for the Banach algebra of bounded linear operators on H. It is well known that this operator is self-adjoint with continuous spectrum σ (Aγ) = σ (A0) + γ = [0, +∞[ + γ = [γ, +∞[ . Some regularization methods for the Cauchy problem for elliptic equations have been proposed by many authors. This work is mainly devoted to theoretical aspects of the spectral regularization methods to problem (1) in the abstract setting, by considering more general self-adjoint operators when A is positive and induces the elliptic case, that is, has the following properties: for any λ ∈ (−∞, 0], the resolvent R(λ; A) = (A − λI)−1 exists and satisfies the estimates. In the case when A is a linear positive self-adjoint operator with compact inverse, problem (1) has been treated by a different method and there is a large literature in this direction. In the present paper we shall use two spectral regularization methods to construct a stable solution to our original illposed problem

Preliminaries and Basic Results

Regularization and Error Estimates

Example

The Nonlocal Boundary Value Problem Method and Some Extensions