Golub\u2013Kahan vs. Monte Carlo: a comparison of bidiagonlization and a randomized SVD method for the solution of linear discrete ill-posed problems

Xianglan Bai,Alessandro Buccini,Lothar Reichel

doi:10.1007/s10543-021-00857-0

Xianglan Bai, Alessandro Buccini + Show 1 more

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https://doi.org/10.1007/s10543-021-00857-0

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Abstract

Randomized methods can be competitive for the solution of problems with a large matrix of low rank. They also have been applied successfully to the solution of large-scale linear discrete ill-posed problems by Tikhonov regularization (Xiang and Zou in Inverse Probl 29:085008, 2013). This entails the computation of an approximation of a partial singular value decomposition of a large matrix A that is of numerical low rank. The present paper compares a randomized method to a Krylov subspace method based on Golub–Kahan bidiagonalization with respect to accuracy and computing time and discusses characteristics of linear discrete ill-posed problems that make them well suited for solution by a randomized method.

Highlights

We are concerned with the solution of linear least-squares problems min Ax − b, (1)x∈Rn where A ∈ Rm×n is a large matrix, whose singular values “cluster” at the origin, and b ∈ Rm
This section reviews solution methods for the Tikhonov minimization problem described in [2,22]. They are based on reducing the matrix A to a small bidiagonal matrix by the application of 1 ≤ min{m, n} steps of Golub–Kahan bidiagonalization to A
The matrix A is not explicitly formed; instead we evaluate matrix–vector products with A and AT by using the fast Fourier transform (FFT) algorithm

Summary

Introduction

We are concerned with the solution of linear least-squares problems min Ax − b , (1)x∈Rn where A ∈ Rm×n is a large matrix, whose singular values “cluster” at the origin, and b ∈ Rm. Least-squares problem with a matrix of this kind commonly are referred to as linear discrete ill-posed problems. They arise, for instance, from the discretization of Fredholm integral equations of the first kind; see, e.g., [10,19]. Applications of this kind of least-squares problems include image reconstruction and remote sensing. Throughout this paper · denotes the Euclidean vector norm or the spectral matrix norm. Both the situations when m ≥ n and when m < n will be considered

Methods

Conclusion