A splitting-based KPIK method for eddy current optimal control problems in an all-at-once approach

Abstract

In this paper, we explore efficient methods for discretized linear systems that arise from eddy current optimal control problems utilizing an all-at-once approach. We propose a novel low-rank matrix equation method based on a special splitting of the coefficient matrix and the Krylov-plus-inverted-Krylov (KPIK) algorithm. First, we reformulate the resulting discretized linear system into a matrix equation format. Then, by employing the KPIK algorithm, we derive a low-rank approximation solution. This new approach is referred to as the splitting-based Krylov-plus-inverted-Krylov (SKPIK) method. The SKPIK method exhibits the potential for efficiently tackle large and sparse discretized systems, while also significantly reducing both storage requirements and computational time. Next, theoretical results regarding the existence of low-rank solutions are provided. Furthermore, numerical experiments are conducted to demonstrate the effectiveness of the proposed low-rank matrix equation method in comparison to several established classical efficient techniques.