A Bidiagonalization-Regularization Procedure for Large Scale Discretizations of Ill-Posed Problems

In this paper, we consider ill-posed problems which discretize to linear least squares problems with matrices K of high dimensions. The algorithm proposed uses K only as an operator and does not need to explicitly store or modify it. A method related to one of Lanczos is used to project the problem onto a subspace for which K is bidiagonal. It is then an easy matter to solve the projected problem by standard regularization techniques. These ideas are illustrated with some integral equations of the first kind with convolution kernels, and sample numerical results are given.

[1]  C. Lanczos.  An iteration method for the solution of the eigenvalue problem of linear differential and integral operators , 1950 .

[2]  C. Lanczos.  Solution of Systems of Linear Equations by Minimized Iterations1 , 1952 .

[3]  David L. Phillips,et al.  A Technique for the Numerical Solution of Certain Integral Equations of the First Kind , 1962, JACM.

[4]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[5]  C. Paige.  Computational variants of the Lanczos method for the eigenproblem , 1972 .

[6]  G. Stewart.  Error and Perturbation Bounds for Subspaces Associated with Certain Eigenvalue Problems , 1973 .

[7]  C. Paige.  Bidiagonalization of Matrices and Solution of Linear Equations , 1974 .

[8]  Ii John Franklin Palmer.  Conjugate-direction methods and parallel-computing. , 1974 .

[9]  William Kahan,et al.  HOW FAR SHOULD YOU GO WITH THE LANCZOS PROCESS , 1976 .

[10]  O. Axelsson.  Solution of linear systems of equations: Iterative methods , 1977 .

[11]  L. Eldén.  Algorithms for the regularization of ill-conditioned least squares problems , 1977 .

[12]  W. Gander.  On the linear least squares problem with a quadratic constraint , 1978 .

[13]  F. Gustavson,et al.  Fast algorithms for rational Hermite approximation and solution of Toeplitz systems , 1979 .

[14]  B. Parlett,et al.  The Lanczos algorithm with selective orthogonalization , 1979 .

[15]  A Fast Algorithm to Solve Fredholm Integral Equations of the First Kind with Stationary Kernels , 1979 .

[16]  G. Wahba.  Ill Posed Problems: Numerical and Statistical Methods for Mildly, Moderately and Severely Ill Posed Problems with Noisy Data. , 1980 .

[17]  C. K. Yuen,et al.  Digital Filters , 1979, IEEE Transactions on Systems, Man, and Cybernetics.

[18]  Franklin T. Luk,et al.  A Block Lanczos Method for Computing the Singular Values and Corresponding Singular Vectors of a Matrix , 1981, TOMS.