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.
In this paper, we focus on solving a sequence of linear systems with an identical (or similar) coefficient matrix. For this type of problems, we investigate the subspace correction and deflation methods, which use an auxiliary matrix (subspace) to accelerate the convergence of the iterative method. In practical simulations, these acceleration methods typically work well when the range of the auxiliary matrix contains eigenspaces corresponding to small eigenvalues of the coefficient matrix. We have developed a new algebraic auxiliary matrix construction method based on error vector sampling, in…
In this paper, we propose and analyze the numerical algorithms for fast solution of periodic elliptic problems in random media in $\mathbb{R}^d$, $d=2,3$. We consider the stochastic realizations using checkerboard configuration of the equation coefficients built on a large $L \times L \times L$ lattice, where $L$ is the size of representative volume elements. The Kronecker tensor product scheme is introduced for fast generation of the stiffness matrix for FDM discretization on a tensor grid. We describe tensor techniques for the construction of the low Kronecker rank spectrally equivalent prec…
We introduce tensor numerical techniques for solving optimal control problems constrained by elliptic operators in $\mathbb{R}^d$, $d=2,3$, with variable coefficients, which can be represented in a low rank separable form. We construct a preconditioned iterative method with an adaptive rank truncation for solving the equation for the control function, governed by a sum of the elliptic operator and its inverse $M=A + A^{-1}$, both discretized over large $n^{\otimes d}$, $d=2,3$, spatial grids. Two basic solution schemes are proposed and analyzed. In the first approach, one solves iteratively th…
