Tensor-structured PCG for finite difference solver of domain patterns in ferroelectric material

TL;DR

This paper develops an efficient tensor-structured preconditioned conjugate gradient method for solving Poisson's equation in ferroelectric materials, demonstrating its effectiveness through numerical experiments and real-world application.

Contribution

It introduces a novel preconditioner based on the Moore--Penrose pseudoinverse tailored for Kronecker-structured operators in finite difference problems.

Findings

The pseudoinverse-based preconditioner significantly accelerates convergence.

The method is computationally efficient and suitable for large-scale problems.

Numerical experiments confirm the effectiveness of the approach.

Abstract

This paper presents a case study of application of the preconditioned method of conjugate gradients (CG) on a problem with operator resembling the structure of sum of Kronecker products. In particular, we are solving the Poisson's equation on a sample of homogeneous isotropic ferroelectric material of cuboid shape, where the Laplacian is discretized by finite difference. We present several preconditioners that fits the Kronecker structure and thus can be efficiently implemented and applied. Preconditioner based on the Moore--Penrose pseudoinverse is extremely efficient for this particular problem, and also applicable (if we are able to store the dense right-hand side of our problem). We briefly analyze the computational cost of the method and individual preconditioners, and illustrate effectiveness of the chosen one by numerical experiments. Although we describe our method as preconditioned CG with pseudoinverse-based preconditioner, it can also be seen as pseudoinverse-based direct solver with iterative refinement by CG iteration. This work is motivated by real application, the method was already implemented in C/C++ code Ferrodo2 and first results were published in Physical Review B 107(9) (2023), paper id 094102.