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Solving the normal equations corresponding to large sparse linear least-squares problems is an important and challenging problem. For very large problems, an iterative solver is needed and, in general, a preconditioner is required to…
Even in cases where quantum linear solvers provide significant speedup compared to their classical counterparts, their performance depends on some of the same parameters. In particular, the condition number of the matrix which is to be…
We study for the first time Schwarz domain decomposition methods for the solution of the Navier equations modeling the propagation of elastic waves. These equations in the time harmonic regime are difficult to solve by iterative methods,…
We study a variant of the Schwarz-preconditioned HMC algorithm. In contrast to the original proposal of L\"uscher, we apply the domain decomposition in one lattice direction only. This is sufficient to reduce the condition number of the…
The overlap operator is a lattice discretization of the Dirac operator of quantum chromodynamics, the fundamental physical theory of the strong interaction between the quarks. As opposed to other discretizations it preserves the important…
The discretization of certain integral equations, e.g., the first-kind Fredholm equation of Laplace's equation, leads to symmetric positive-definite linear systems, where the coefficient matrix is dense and often ill-conditioned. We…
Numerical algorithms for solving problems of mathematical physics on modern parallel computers employ various domain decomposition techniques. Domain decomposition schemes are developed here to solve numerically initial/boundary value…
In this paper, we propose a parallel space-time domain decomposition method for solving an unsteady source identification problem governed by the linear convection-diffusion equation. Traditional approaches require to solve repeatedly a…
Domain decomposition methods are among the most efficient for solving sparse linear systems of equations. Their effectiveness relies on a judiciously chosen coarse space. Originally introduced and theoretically proved to be efficient for…
With recent advancements in computer hardware and software platforms, there has been a surge of interest in solving partial differential equations with deep learning-based methods, and the integration with domain decomposition strategies…
We discuss the implementation and optimization challenges for a Wilson-Dirac solver with Clover term on QPACE, a parallel machine based on Cell processors and a torus network. We choose the mixed-precision Schwarz preconditioned FGCR…
A domain decomposition method for the solution of general variable-coefficient elliptic partial differential equations on regular domains is introduced. The method is based on tessellating the domain into overlapping thin slabs or shells,…
We present a parallelizable SSOR preconditioning scheme for Krylov subspace iterative solvers which proves to be efficient in lattice QCD applications involving Wilson fermions. Our preconditioner is based on a locally lexicographic…
The additive Schwarz method is usually presented as a preconditioner for a PDE linearization based on overlapping subsets of nodes from a global discretization. It has previously been shown how to apply Schwarz preconditioning to a…
Domain decomposition methods are used for approximate solving boundary problems for partial differential equations on parallel computing systems. Specific features of unsteady problems are taken into account in the most complete way in…
This paper is concerned with the numerical solution of porous-media flow and transport problems , i. e. heterogeneous, advection-diffusion problems. Its aim is to investigate numerical schemes for these problems in which different time…
Solving time-harmonic wave propagation problems by iterative methods is a difficult task, and over the last two decades, an important research effort has gone into developing preconditioners for the simplest representative of such wave…
We propose a method to reduce the computational effort to solve a partial differential equation on a given domain. The main idea is to split the domain of interest in two subdomains, and to use different approximation methods in each of the…
Optimization with time-dependent partial differential equations (PDEs) as constraints {appears} in many science and engineering applications. The associated first-order necessary optimality system consists of one forward and one backward…
In this paper we present an algebraic dimension-oblivious two-level domain decomposition solver for discretizations of elliptic partial differential equations. The proposed parallel solver is based on a space-filling curve partitioning…