Name
Abstract | ||
|---|---|---|
Regularization approaches based on spectral filtering can be highly effective in solving ill-posed inverse problems. These methods, however, require computing the singular value decomposition (SVD) and choosing appropriate regularization parameters. These tasks can be prohibitively expensive for large-scale problems. In this paper, we present a framework that uses operator approximations to efficiently obtain good regularization parameters without an SVD of the original operator. Instead, we approximate the original operator with a nearby structured or separable one whose SVD is easily computable. Highly effective methods can then be used to efficiently compute good regularization parameters for the nearby problem. Then, we solve the original problem iteratively using the regularization determined for the approximate problem. A variety of regularization approaches can be incorporated into this framework, but we focus here on the recently developed windowed regularization, a generalization of Tikhonov regularization in which different regularization parameters are used in different regions of the spectrum. We derive bounds on the perturbation to the computed solution and residual resulting from using the regularization determined for the approximate operator. We demonstrate the effectiveness of our method in computations using operator approximations such as sums of Kronecker products, block circulant with circulant blocks matrices, and Krylov subspace approximations. |
Year | DOI | Venue |
|---|---|---|
2015 | 10.1137/130945363 | SIAM JOURNAL ON SCIENTIFIC COMPUTING |
Keywords | Field | DocType |
singular value decomposition,Tikhonov regularization,inverse problem,ill-posed,deconvolution,Kronecker approximation,spectral filter,Golub-Kahan,parameter selection,hybrid iterative methods | Tikhonov regularization,Singular value decomposition,Well-posed problem,Mathematical optimization,Mathematical analysis,Backus–Gilbert method,Regularization (mathematics),Inverse problem,Operator (computer programming),Mathematics,Regularization perspectives on support vector machines | Journal |
Volume | Issue | ISSN |
37 | 2 | 1064-8275 |
Citations | PageRank | References |
3 | 0.47 | 0 |
Authors | ||
3 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Julianne Chung | 1 | 46 | 7.68 |
| Misha E. Kilmer | 2 | 320 | 39.27 |
| O'Leary, Dianne P. | 3 | 1064 | 222.93 |