Randomized and Inner-product Free Krylov Methods for Large-scale Inverse Problems

TL;DR

This paper introduces a new Krylov method for large-scale inverse problems that is both inner-product-free and minimizes a residual-norm closer to the true residual, improving efficiency and robustness.

Contribution

The paper proposes a novel Krylov algorithm combining an inner-product-free Hessenberg approach with randomized sketching, addressing limitations of existing methods in inverse problems.

Findings

Efficiently solves large-scale inverse problems without inner-products.

Achieves residual minimization closer to the true residual norm.

Demonstrates effectiveness through numerical experiments.

Abstract

Iterative Krylov projection methods have become widely used for solving large-scale linear inverse problems. However, methods based on orthogonality include the computation of inner-products, which become costly when the number of iterations is high; are a bottleneck for parallelization; and can cause the algorithms to break down in low precision due to information loss in the projections. Recent works on inner-product free Krylov iterative algorithms alleviate these concerns, but they are quasi-minimal residual rather than minimal residual methods. This is a potential concern for inverse problems where the residual norm provides critical information from the observations via the likelihood function, and we do not have any way of controlling how close the quasi-norm is from the norm we want to minimize. In this work, we introduce a new Krylov method that is both inner-product-free and minimizes a functional that is theoretically closer to the residual norm. The proposed scheme combines an inner-product free Hessenberg projection approach for generating a solution subspace with a randomized sketch-and-solve approach for solving the resulting strongly overdetermined projected least-squares problem. Numerical results show that the proposed algorithm can solve large-scale inverse problems efficiently and without requiring inner-products.