SVD Approximations of Large Scale Inverse Problems Open Access

Meng, Chang (2016)

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This thesis studies efficient methods for computing approximations of the singular value decomposition (SVD) for large matrices that arise in ill-posed inverse problems, with a focus on image deblurring. These methods are: the Lanczos method, the randomized method, and the Kronecker product approximation method. After introducing the SVD and describing the approximation methods, we show test results involving the accuracy and speed comparisons of these methods, and provide some deblurring examples.

Table of Contents

1 Introduction. 1

2 Singular Value Decomposition. 7

2.1 Definition and Properties. 7

2.2 Computational Difficulties. 9

2.3 Regularization Methods. 13

2.3.1 Truncated SVD. 13

2.3.2 Tikhonov Regularization. 14

3 SVD Approximations. 19

3.1 Lanczos Method. 19

3.2 Randomized Method. 20

3.3 Kronecker Product Approximation Method. 22

4 Numerical Experiments. 25

4.1 Accuracy Test. 25

4.2 Speed Test. 30

4.3 Deblurring Examples. 33

5 Conclusion and Discussion 40

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