Regularization of Inverse ill-posed problems with General Penalizing Terms

G. MAZZIERI; RUBEN D. SPIES; K. TEMPERINI

San Andrés

Conferencia; International Conference on Applied Mathematics and Informatics; 2010

Departamento de Matemáticas, Facultad de Ciencias Naturales y Exactas, Universidad del Valle. Cali, Colombia

In a quite general framework an inverse problem can be formulated as the need for determining x in an equation of the form (0.1) Tx = y; where T is a linear bounded operator between two infinite dimensional Hilbert spaces X and Y , R(T), the range of T, is non-closed and y is the data, supposed to be known, perhaps with a certain degree of error. It is well known that under these hypotheses the problem (0.1) is ill-posed in the sense of Hadamard. In this case the ill-posedness is a result of the unboundedness of the Moore-Penrose generalized inverse of T, T+, which has as an undesired consequence the fact small errors or noise in the data y can result in arbitrarily large errors in the corresponding approximated solutions, turning unstable all standard numerical approximation methods. The so called "regularization methods" are mathematical tools designed to restore stability to the inversion process and consist essentially of parametric families of continuous linear operators approximating Ty ([2]). There exist several ways of regularizing an ill-posed inverse problem. The Tikhonov-Phillips method is widely used for this due to the simplicity of its formulation as an optimization problem. In fact, the regularized solution of problem (0.1) obtained by applying Tikhonov-Phillips method is the minimizer x_alpha of the functional J_alpha(x) := ||Tx - y||^2 +alpha|| x||^2 ; where alpha is a positive constant known as the regularization parameter and alpha||x||^2 is the penalizing term. The use of different penalizers in the functionals associated to the corresponding optimization problems has originated a few other methods which can be considered as "variants" of the traditional Tikhonov-Phillips method. Such is the case for instance of the Tikhonov-Phillips method of order one, the total variation regularization method ([1]), etc. The purpose of this work is twofold. First we study the problem of determining general suffcient conditions on the penalizers in generalized Tikhonov- Phillips functionals which guarantee existence and uniqueness of minimizers, in such a way that finding such minimizers constitutes a regularization method, that is, in such a way that these minimizers approximate, as the regularization parameter tend to 0+, a least squares solution of the problem. Secondly we also study the problem of characterizing those limiting least square solutions in terms of properties of the penalizers and find conditions which guarantee that the regularization method thus defined is stable under different types of perturbations. Finally, several examples with different penalizers are presented and a few numerical results in an image restoration problem are shown which better illustrate the results.x in an equation of the form (0.1) Tx = y; where T is a linear bounded operator between two infinite dimensional Hilbert spaces X and Y , R(T), the range of T, is non-closed and y is the data, supposed to be known, perhaps with a certain degree of error. It is well known that under these hypotheses the problem (0.1) is ill-posed in the sense of Hadamard. In this case the ill-posedness is a result of the unboundedness of the Moore-Penrose generalized inverse of T, T+, which has as an undesired consequence the fact small errors or noise in the data y can result in arbitrarily large errors in the corresponding approximated solutions, turning unstable all standard numerical approximation methods. The so called "regularization methods" are mathematical tools designed to restore stability to the inversion process and consist essentially of parametric families of continuous linear operators approximating Ty ([2]). There exist several ways of regularizing an ill-posed inverse problem. The Tikhonov-Phillips method is widely used for this due to the simplicity of its formulation as an optimization problem. In fact, the regularized solution of problem (0.1) obtained by applying Tikhonov-Phillips method is the minimizer x_alpha of the functional J_alpha(x) := ||Tx - y||^2 +alpha|| x||^2 ; where alpha is a positive constant known as the regularization parameter and alpha||x||^2 is the penalizing term. The use of different penalizers in the functionals associated to the corresponding optimization problems has originated a few other methods which can be considered as "variants" of the traditional Tikhonov-Phillips method. Such is the case for instance of the Tikhonov-Phillips method of order one, the total variation regularization method ([1]), etc. The purpose of this work is twofold. First we study the problem of determining general suffcient conditions on the penalizers in generalized Tikhonov- Phillips functionals which guarantee existence and uniqueness of minimizers, in such a way that finding such minimizers constitutes a regularization method, that is, in such a way that these minimizers approximate, as the regularization parameter tend to 0+, a least squares solution of the problem. Secondly we also study the problem of characterizing those limiting least square solutions in terms of properties of the penalizers and find conditions which guarantee that the regularization method thus defined is stable under different types of perturbations. Finally, several examples with different penalizers are presented and a few numerical results in an image restoration problem are shown which better illustrate the results.