Square regularization matrices for large linear discrete ill-posed problems. (English) Zbl 1289.65096
The authors are concerned with large-scale discrete ill-posed problems. Tikhonov regularization based on the range restricted Arnoldi process and range restricted GMRES require a square regularization matrix. When the solution is smooth, common choices of regularization operators are the identity matrix and scaled rectangular finite difference matrices. In this paper they discuss how to define square smoothing operators with a good approximation of a prescribed null space and such that the matrix-vector product with the pseudoinverse can be computed efficiently. An effective and simple strategy is obtained imposing appropriate boundary conditions to finite difference approximations of a derivative. Numerical experiments are presented to confirm the performance of the approach.
Reviewer: Fiorella Sgallari (Bologna)
MSC:
| 65F22 | Ill-posedness and regularization problems in numerical linear algebra |
| 65F10 | Iterative numerical methods for linear systems |
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
| 65F20 | Numerical solutions to overdetermined systems, pseudoinverses |
Keywords:
ill-posed problem; regularization operator; Tikhonov regularization; Arnoldi process; range restricted GMRES; finite difference matrices; smoothing operators; pseudoinverse; numerical experimentsReferences:
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