arXiv subjects

Martin Plešinger

Publications and source records attributed to Martin Plešinger.

3 recordsLinked to original sources

arXiv2025

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

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.

math.NA↗

arXiv2022

Zig-zag charged domain walls in ferroelectric PbTiO$_3$

We report a theoretical investigation of a charged 180$^\circ$ domain wall in ferroelectric PbTiO$_3$, compensated by randomly distributed immobile charge defects. For this we utilize atomistic shell-model simulations and continuous phase-field simulations in the framework of the Ginzburg-Landau-Devonshire model. We predict that domain walls form a zig-zag pattern and we discuss its properties in a broad interval of compensation-region widths, ranging from a couple to over a hundred nanometers.

cond-mat.mtrl-sci↗

arXiv2017

Noise representation in residuals of LSQR, LSMR, and CRAIG regularization

Golub-Kahan iterative bidiagonalization represents the core algorithm in several regularization methods for solving large linear noise-polluted ill-posed problems. We consider a general noise setting and derive explicit relations between (noise contaminated) bidiagonalization vectors and the residuals of bidiagonalization-based regularization methods LSQR, LSMR, and CRAIG. For LSQR and LSMR residuals we prove that the coefficients of the linear combination of the computed bidiagonalization vectors reflect the amount of propagated noise in each of these vectors. For CRAIG the residual is only a multiple of a particular bidiagonalization vector. We show how its size indicates the regularization effect in each iteration by expressing the CRAIG solution as the exact solution to a modified compatible problem. Validity of the results for larger two-dimensional problems and influence of the loss of orthogonality is also discussed.

math.NA↗