Regularized Linear Inversion with Randomized Singular Value Decomposition

TL;DR

This paper introduces efficient linear inverse problem solvers using randomized singular value decomposition combined with classical regularization techniques, ensuring solutions preserve specific structural properties.

Contribution

It presents a novel approach integrating RSVD with regularization methods that maintains the solution's structure and provides theoretical error estimates.

Findings

The method achieves high accuracy in numerical experiments.

It preserves the structure of regularized solutions.

The approach is computationally efficient.

Abstract

In this work, we develop efficient solvers for linear inverse problems based on randomized singular value decomposition (RSVD). This is achieved by combining RSVD with classical regularization methods, e.g., truncated singular value decomposition, Tikhonov regularization, and general Tikhonov regularization with a smoothness penalty. One distinct feature of the proposed approach is that it explicitly preserves the structure of the regularized solution in the sense that it always lies in the range of a certain adjoint operator. We provide error estimates between the approximation and the exact solution under canonical source condition, and interpret the approach in the lens of convex duality. Extensive numerical experiments are provided to illustrate the efficiency and accuracy of the approach.