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An extrapolated TSVD method for linear discrete ill-posed problems with Kronecker structure. (English) Zbl 1210.65091
Summary: This paper describes a new numerical method for the solution of large linear discrete ill-posed problems, whose matrix is a Kronecker product. Problems of this kind arise, for instance, from the discretization of Fredholm integral equations of the first kind in two space-dimensions with a separable kernel. The available data (right-hand side) of many linear discrete ill-posed problems that arise in applications is contaminated by measurement errors. Straightforward solution of these problems generally is not meaningful because of severe error propagation. We discuss how to combine the truncated singular value decomposition (TSVD) with reduced rank vector extrapolation to determine computed approximate solutions that are fairly insensitive to the error in the data. Exploitation of the structure of the problem keeps the computational effort quite small.

MSC:
65F22 Ill-posedness and regularization problems in numerical linear algebra
65F20 Numerical solutions to overdetermined systems, pseudoinverses
45B05 Fredholm integral equations
65R20 Numerical methods for integral equations
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[1] Brezinski, C.; Rodriguez, G.; Seatzu, S., Error estimates for linear systems with applications to regularization, Numer. algorithms, 49, 85-104, (2008) · Zbl 1162.65018
[2] Brezinski, C.; Rodriguez, G.; Seatzu, S., Error estimates for the regularization of least squares problems, Numer. algorithms, 51, 61-76, (2009) · Zbl 1166.65331
[3] Datta, B.N., Linear and numerical linear algebra in control theory: some research problems, Linear algebra appl., 197-198, 755-790, (1994) · Zbl 0798.15015
[4] Datta, B.N., Stability and inertia, Linear algebra appl., 302-303, 563-600, (1999) · Zbl 0972.15009
[5] Datta, B.N.; Datta, K., The matrix equation \(\mathit{XA} = A^T X\) and an associated algorithm for solving the inertia and stability problems, Linear algebra appl., 97, 103-119, (1987)
[6] Eddy, R.P., Extrapolation to the limit of a vector sequence, (), 387-396
[7] Golub, G.H.; Van Loan, C.F., Matrix computations, (1996), Johns Hopkins University Press Baltimore · Zbl 0865.65009
[8] Hansen, P.C., Rank-deficient and discrete ill-posed problems, (1998), SIAM Philadelphia
[9] Hansen, P.C., Regularization tools version 4.0 for MATLAB 7.3, Numer. algorithms, 46, 189-194, (2007) · Zbl 1128.65029
[10] Horn, R.A.; Johnson, C.R., Topics in matrix analysis, (1991), Cambridge University Press Cambridge, England · Zbl 0729.15001
[11] Jbilou, K.; Reichel, L.; Sadok, H., Vector extrapolation enhanced TSVD for linear discrete ill-posed problems, Numer. algorithms, 51, 195-208, (2009) · Zbl 1167.65019
[12] Jbilou, K.; Sadok, H., Analysis of some vector extrapolation methods for linear systems, Numer. math., 70, 73-89, (1995) · Zbl 0836.65044
[13] Jbilou, K.; Sadok, H., Vector extrapolation methods, applications and numerical comparison, J. comput. appl. math., 122, 149-165, (2000) · Zbl 0974.65034
[14] Mesina, M., Convergence acceleration for the iterative solution of \(x = \mathit{Ax} + f\), Comput. methods appl. mech. engrg., 10, 165-173, (1977) · Zbl 0344.65019
[15] Reichel, L.; Sadok, H., A new L-curve for ill-posed problems, J. comput. appl. math., 219, 493-508, (2008) · Zbl 1145.65035
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