The reduction of a large‐scale symmetric linear discrete ill‐posed problem with multiple right‐hand sides to a smaller problem with a symmetric block tridiagonal matrix can easily be carried out by the application of a small number of steps of the symmetric block Lanczos method. We show that the subdiagonal blocks of the reduced problem converge to zero fairly rapidly with increasing block number. This quick convergence indicates that there is little advantage in expressing the solutions of discrete ill‐posed problems in terms of eigenvectors of the coefficient matrix when compared with using a basis of block Lanczos vectors, which are simpler and cheaper to compute. Similarly, for nonsymmetric linear discrete ill‐posed problems with multiple right‐hand sides, we show that the solution subspace defined by a few steps of the block Golub–Kahan bidiagonalization method usually can be applied instead of the solution subspace determined by the singular value decomposition of the coefficient matrix without significant, if any, reduction of the quality of the computed solution.
The analysis of linear illposed problems often is carried out in function spaces using tools from functional analysis. However, the numerical solution of these problems typically is computed by first discretizing the problem and then applying tools from finitedimensional linear algebra. The present paper explores the feasibility of applying the Chebfun package to solve illposed problems with a regularizefirst approach numerically. This allows a user to work with functions instead of vectors and with integral operators instead of matrices. The solution process therefore is much closer to the analysis of illposed problems than standard linear algebrabased solution methods. Furthermore, the difficult process of explicitly choosing a suitable discretization is not required.
more » « less NSFPAR ID:
 10371471
 Publisher / Repository:
 Springer Science + Business Media
 Date Published:
 Journal Name:
 Numerical Algorithms
 Volume:
 92
 Issue:
 4
 ISSN:
 10171398
 Page Range / eLocation ID:
 p. 23412364
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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