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Hybrid Enriched Bidiagonalization forDiscrete Ill-Posed Problems
Per Christian HansenTechnical University of Denmark
Joint work with
Kuniyoshi Abe - Gifu Shotoku Gakuen University
Yiqiu Dong - Technical University of Denmark
With thanks toHenrik Garde and Nao Kuroiwa
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Japan, Spring 20202/27 P. C. Hansen – Hybrid Enriched Bidiagonalization for Discrete Ill-Posed Problems
Setting the Stage – Overview of Talk
Forward problem:b = A x
Inverse problem:solve Ax = b
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Japan, Spring 20203/27 P. C. Hansen – Hybrid Enriched Bidiagonalization for Discrete Ill-Posed Problems
Inverse Problems Ill-Conditioning
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Japan, Spring 20204/27 P. C. Hansen – Hybrid Enriched Bidiagonalization for Discrete Ill-Posed Problems
Regularization Algorithms
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Japan, Spring 20205/27 P. C. Hansen – Hybrid Enriched Bidiagonalization for Discrete Ill-Posed Problems
Krylov Subspaces and Semi-Convergence
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Japan, Spring 20206/27 P. C. Hansen – Hybrid Enriched Bidiagonalization for Discrete Ill-Posed Problems
Illustration of Semi-Convergence
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Japan, Spring 20207/27 P. C. Hansen – Hybrid Enriched Bidiagonalization for Discrete Ill-Posed Problems
Hybrid Methods
Projectedproblem
Regularizedprojectedproblem
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Japan, Spring 20208/27 P. C. Hansen – Hybrid Enriched Bidiagonalization for Discrete Ill-Posed Problems
Augmented Krylov Subspace
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Japan, Spring 20209/27 P. C. Hansen – Hybrid Enriched Bidiagonalization for Discrete Ill-Posed Problems
Example: Augmented GMRES
GMRES Augmented GMRES
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Japan, Spring 202010/27 P. C. Hansen – Hybrid Enriched Bidiagonalization for Discrete Ill-Posed Problems
Overview of Methods
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Japan, Spring 202011/27 P. C. Hansen – Hybrid Enriched Bidiagonalization for Discrete Ill-Posed Problems
Enriched CGNR
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Japan, Spring 202012/27 P. C. Hansen – Hybrid Enriched Bidiagonalization for Discrete Ill-Posed Problems
Towards our Algorithm
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Japan, Spring 202013/27 P. C. Hansen – Hybrid Enriched Bidiagonalization for Discrete Ill-Posed Problems
Setting the Stage for Our Algorithm
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Japan, Spring 202014/27 P. C. Hansen – Hybrid Enriched Bidiagonalization for Discrete Ill-Posed Problems
More Details
Projectedproblem
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Japan, Spring 202015/27 P. C. Hansen – Hybrid Enriched Bidiagonalization for Discrete Ill-Posed Problems
Basic Enriched Bidiagonalization
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Japan, Spring 202016/27 P. C. Hansen – Hybrid Enriched Bidiagonalization for Discrete Ill-Posed Problems
Efficient and Stable Implementation
Algorithm HYBR (Chung, Nagy, O’Leary 2008) also uses full reorthogonalization.
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Japan, Spring 202017/27 P. C. Hansen – Hybrid Enriched Bidiagonalization for Discrete Ill-Posed Problems
And Now: a Hybrid Algorithm
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Japan, Spring 202018/27 P. C. Hansen – Hybrid Enriched Bidiagonalization for Discrete Ill-Posed Problems
HEB: Hybrid Enriched Bidiagonalization
Regularized projected problem
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Japan, Spring 202019/27 P. C. Hansen – Hybrid Enriched Bidiagonalization for Discrete Ill-Posed Problems
Terrible Computational Details of Step 6
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Japan, Spring 202020/27 P. C. Hansen – Hybrid Enriched Bidiagonalization for Discrete Ill-Posed Problems
Numerical Examples
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Japan, Spring 202021/27 P. C. Hansen – Hybrid Enriched Bidiagonalization for Discrete Ill-Posed Problems
Large Component in Augment. Subspace
Results next page
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Japan, Spring 202022/27 P. C. Hansen – Hybrid Enriched Bidiagonalization for Discrete Ill-Posed Problems
Large Component in Augment. Subspace
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Japan, Spring 202023/27 P. C. Hansen – Hybrid Enriched Bidiagonalization for Discrete Ill-Posed Problems
1D Deconvolution and “Inpainting”
Results next page
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Japan, Spring 202024/27 P. C. Hansen – Hybrid Enriched Bidiagonalization for Discrete Ill-Posed Problems
1D Deconvolution and “Inpainting”
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Japan, Spring 202025/27 P. C. Hansen – Hybrid Enriched Bidiagonalization for Discrete Ill-Posed Problems
2D Image Deblurring and Inpainting
Results next page
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Japan, Spring 202026/27 P. C. Hansen – Hybrid Enriched Bidiagonalization for Discrete Ill-Posed Problems
2D Image Deblurring and Inpainting
LSQR
HEB
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Japan, Spring 202027/27 P. C. Hansen – Hybrid Enriched Bidiagonalization for Discrete Ill-Posed Problems
Conclusions We augment the bidiagonalization algorithm underlying LSQR.
Our algorithm uses an enriched subspace:the Krylov subspace plus a low-dimensional linear subspace.
We add standard-form Tikhonov regularization, thus arriving at a hybrid enriched bidiagonalization algorithm.
We choose the regularization parameter adaptively in each iteration, e.g., by means of GCV.
Possible extension: use general-form Tikhonov regularization.