Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2501.03377

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

Věnceslav Chumchal·Pavel Marton·Martin Plešinger·Martina Šimůnková
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Věnceslav Chumchal, Pavel Marton, Martin Plešinger, Martina Šimůnková. 2025-01-06. Tensor-structured PCG for finite difference solver of domain patterns in ferroelectric material. https://arxiv.org/abs/2501.03377

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Fully spectral scheme for the linear BGK equation on the whole space

In this article, we design a fully spectral method in both space and velocity for a linear inhomogeneous kinetic equation with mass, momentum and energy conservation. We focus on the linear BGK equation with a confinement potential $Φ$, even if the method could be applied to different collision operators. It is based upon the projection on Hermite polynomials in velocity and orthonormal polynomials with respect to the weight $e^{-$Φ$}$ in space. The potential $Φ$ is assumed to be a polynomial. It is, to the author's knowledge, the first scheme which preserves hypocoercive behavior in addition to the conservation laws. These different properties are illustrated numerically on both quadratic and double well potential.

math.NA↗
arXiv2026

Inverse inequalities for kernel-based approximation on bounded domains and Riemannian manifolds

This paper establishes inverse inequalities for kernel-based approximation spaces defined on bounded Lipschitz domains in $\mathbb{R}^d$ and compact Riemannian manifolds. While inverse inequalities are well-studied for polynomial spaces, their extension to kernel-based trial spaces poses significant challenges. For bounded Lipschitz domains, we extend prior Bernstein inequalities, which only apply to a limited range of Sobolev orders, to the full range of lower and upper orders, and derive Nikolskii inequalities that bound $L_\infty$ norms by $L_2$ norms. For compact Riemannian manifolds, we focus on restricted kernels, which are defined as the restriction of positive definite kernels from the ambient Euclidean space to the manifold, and prove their counterparts.

math.NA↗
arXiv2026

Error Estimates for Hyperbolic Scaling Limits of Linear Kinetic Models on Networks

This paper studies linear discrete kinetic models on networks and their asymptotic behavior in the small Knudsen number limit. For coupling conditions at an n-edge junction under a symmetric formulation, we introduce a change of variables that reformulates the system into n independent initial-boundary value problems. The asymptotic expansions are then constructed and rigorously justified by deriving an error estimate based on the energy method.

math.NA↗