Iterative solvers for Tikhonov regularization of dense inverse problems

According to the special demands arising from the development of science and technology, in the last decades appeared a special class of problems that are inverse to the classical direct ones. Such an inverse problem is concerned with the opposite way, usually followed by a direct one: finding the cause of a given effect or finding the law of evolution given the cause and effect. Very frequently, such inverse problems are modelled by Fredholm first-kind integral equations that give rise after discretization to (very) ill-conditioned linear systems, in classical or least squares formulation. Then, an efficient numerical solution can be obtained by using the Tikhonov regularization technique. In this respect, in the present paper, we propose three Kovarik-like algorithms for numerical solution of the regularized problem. We prove convergence for all three methods and present numerical experiments on a mathematical model of an inverse problem concerned with the determination of charge distribution generating a given electric field.

[1]  Ulrich Rüde,et al.  An iterative algorithm for approximate orthogonalisation of symmetric matrices , 2004, Int. J. Comput. Math..

[2]  Constantin Popa Modified Kovarik Algorithm For Approximate Orthogonalization Of Arbitrary Matrices , 2003, Int. J. Comput. Math..

[3]  Per Christian Hansen,et al.  Regularization methods for large-scale problems , 1993 .

[4]  Constantin Popa Extension of an approximate orthogonalization algorithm to arbitrary rectangular matrices , 2001 .

[5]  J. Douglas Faires,et al.  Numerical Analysis , 1981 .

[6]  Gene H. Golub,et al.  Matrix computations , 1983 .

[7]  D. Calvetti,et al.  Tikhonov Regularization of Large Linear Problems , 2003 .

[8]  Constantin Popa A method for improving orthogonality of rows and columns of matrices , 2001, Int. J. Comput. Math..

[9]  Yimin Wei,et al.  On mixed and componentwise condition numbers for Moore-Penrose inverse and linear least squares problems , 2006, Math. Comput..

[10]  Per Christian Hansen,et al.  Rank-Deficient and Discrete Ill-Posed Problems , 1996 .

[11]  J. Miller Numerical Analysis , 1966, Nature.

[12]  C. Popa On numerical solution of arbitrary symmetric linear systems by approximate orthogonalization , 2008, Math. Comput. Simul..

[13]  Yimin Wei,et al.  Perturbation Identities for Regularized Tikhonov Inverses and Weighted Pseudoinverses , 2000 .

[14]  Constantin Popa,et al.  Projections and preconditioning for inconsistent least-squares problems , 2001 .

[15]  Constantin Popa ON A MODIFIED KOVARIK ALGORITHM FOR SYMMETRIC MATRICES , 2003 .

[16]  A. Zients Andy , 2003 .

[17]  C. Popa A fast Kaczmarz-Kovarik algorithm for consistent least-squares problems , 2001 .

[18]  Dana Petcu,et al.  A new version of Kovarik’s approximate orthogonalization algorithm without matrix inversion , 2005, Int. J. Comput. Math..

[19]  Z. Kovarik Some Iterative Methods for Improving Orthonormality , 1970 .