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

Věnceslav Chumchal, Pavel Marton, Martin Plešinger, Martina Šimůnková

TL;DR

The paper addresses efficient solution of discretized Poisson equations $- abla^2\upsilon=\eta$ on ferroelectric domain patterns by exploiting Kronecker-structured Laplacians. The authors develop and compare several preconditioners for preconditioned conjugate gradients, notably a Moore–Penrose pseudoinverse-based preconditioner that delivers rapid convergence across 2D and 3D problems with dense right-hand sides. They provide a detailed cost analysis and practical implementation notes, including the interpretation of the method as a direct solver with iterative refinement and its Matlab/Ferrodo2 realization. The results demonstrate robust, scalable performance for large-scale, Kronecker-structured Poisson problems relevant to ferroelectric material simulations, with clear implications for fast electrostatic computations in domain pattern studies.

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.

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

TL;DR

The paper addresses efficient solution of discretized Poisson equations on ferroelectric domain patterns by exploiting Kronecker-structured Laplacians. The authors develop and compare several preconditioners for preconditioned conjugate gradients, notably a Moore–Penrose pseudoinverse-based preconditioner that delivers rapid convergence across 2D and 3D problems with dense right-hand sides. They provide a detailed cost analysis and practical implementation notes, including the interpretation of the method as a direct solver with iterative refinement and its Matlab/Ferrodo2 realization. The results demonstrate robust, scalable performance for large-scale, Kronecker-structured Poisson problems relevant to ferroelectric material simulations, with clear implications for fast electrostatic computations in domain pattern studies.

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.
Paper Structure (25 sections, 4 theorems, 56 equations, 10 figures, 2 tables, 7 algorithms)

This paper contains 25 sections, 4 theorems, 56 equations, 10 figures, 2 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Problem description and its discretization
  3. Notation and structure of the work
  4. Basic tensor concepts and notation
  5. Vectorization
  6. Matrix-tensor product (MT)
  7. Description of the algebraic system
  8. Discretization of 1D Laplace operator
  9. Spectral properties of $L_n$ matrices
  10. Discretization of 2D & 3D Laplacians and their spectra
  11. Modifications of the right-hand side enforced by BCs
  12. Orthogonality to null-space
  13. Dirichlet and Neumann BCs update
  14. Preconditioned method of conjugate gradients
  15. PCG algorithm
  16. ...and 10 more sections

Key Result

Lemma 3.1

Let $n\in\mathbb{N}$, $n\geq 3$. Consider $\ell\in\mathbb{N}$ such that either $n = 2\ell-1$ or $n = 2\ell$. Then represent the eigenvalues and the eigenvectors of $L_{{n},{\mathsf{P}}}$ (except of the last pair in the odd $n$ case).

Figures (10)

  • Figure 1: Four right-hand sides of the regular two-periodic problem (see Problem \ref{['pb:1']}).
  • Figure 2: Right-hand side of the problem with mixed boundary conditions (see Problem \ref{['pb:2']}).
  • Figure 3: Four right-hand sides of problem with randomly distributed charge (see Problem \ref{['pb:3']}). Notice that strips are of different width --- negative charge is more localized. The last three images show the first frontal slices ($k=0$) of three-way tensors.
  • Figure 4: Convergence of unpreconditioned CG and Jacobi-like stationary iterative method for different values of $\omega$. Time axis is in logarithmic scale; markers at convergence curves for Jacobi denotes each $250$th iterations (see Experiment \ref{['exp:1']}).
  • Figure 5: Behavior of unpreconditioned CG observed by several quantities (see Experiment \ref{['exp:1']}).
  • ...and 5 more figures

Theorems & Definitions (7)

  • Lemma 3.1: Spectral properties of $L_{{n},{\mathsf{P}}}$
  • Lemma 3.2: Spectral properties of $L_{{n},{\mathsf{D}}}$
  • Lemma 3.3: Spectral properties of $L_{{n},{\mathsf{N}}}$
  • Lemma 3.4: Spectral properties of $L_{{n},{\mathsf{DN}}}$ and $L_{{n},{\mathsf{ND}}}$
  • Remark 4.1: Orthogonality to null-space
  • Remark 4.2: Stopping criterion
  • Remark 4.3