Randomized Core Reduction for Discrete Ill-Posed Problem
For practitioners solving large-scale ill-posed inverse problems, this offers a potentially faster TLS approximation, but the contribution is incremental given existing randomized methods.
The paper proposes a randomized algorithm to approximate the total least squares solution for large-scale discrete ill-posed problems, using multiplicative randomization and subspace iteration to obtain an approximate core problem. Numerical examples demonstrate the method's effectiveness, but no concrete performance numbers are provided.
In this paper, we apply randomized algorithms to approximate the total least squares (TLS) solution of the problem $Ax\approx b$ in the large-scale discrete ill-posed problems. A regularization technique, based on the multiplicative randomization and the subspace iteration, is proposed to obtain the approximate core problem.In the error analysis, we provide upper bounds %in terms of the $(k\!\!+\!\!1)$-th singular value of $A$ for the errors of the solution and the residual of the randomized core reduction. Illustrative numerical examples and comparisons are presented.