Related papers: Coarse spaces for non-symmetric two-level precondi…
Realistic physical phenomena exhibit random fluctuations across many scales in the input and output processes. Models of these phenomena require stochastic PDEs. For three-dimensional coupled (vector-valued) stochastic PDEs (SPDEs), for…
Linear systems with large differences between coefficients ("discontinuous coefficients") arise in many cases in which partial differential equations(PDEs) model physical phenomena involving heterogeneous media. The standard approach to…
We present a novel uncertainty quantification approach for high-dimensional stochastic partial differential equations that reduces the computational cost of polynomial chaos methods by decomposing the computational domain into…
This work introduces nodal auxiliary space preconditioners for discretizations of mixed-dimensional partial differential equations. We first consider the continuous setting and generalize the regular decomposition to this setting. With the…
Randomized methods are becoming increasingly popular in numerical linear algebra. However, few attempts have been made to use them in developing preconditioners. Our interest lies in solving large-scale sparse symmetric positive definite…
This paper introduces a novel approach to analyzing overlapping Schwarz methods for N\'{e}d\'{e}lec and Raviart--Thomas vector field problems. The theory is based on new regular stable decompositions for vector fields that are robust to the…
We present a meshless Schwarz-type non-overlapping domain decomposition method based on artificial neural networks for solving forward and inverse problems involving partial differential equations (PDEs). To ensure the consistency of…
The research of two-level overlapping Schwarz (TL-OS) method based on constrained energy minimizing coarse space is still in its infancy, and there exist some defects, e.g. mainly for second order elliptic problem and too heavy…
Domain decomposition methods are essential in solving applied problems on parallel computer systems. For boundary value problems for evolutionary equations the implicit schemes are in common use to solve problems at a new time level…
The goal of this work is to present a fast and viable approach for the numerical solution of the high-contrast state problems arising in topology optimization. The optimization process is iterative, and the gradients are obtained by an…
This work develops optimal preconditioners for the discrete H(curl) and H(div) problems on two-dimensional surfaces by nodal auxiliary space preconditioning [R. Hiptmair, J. Xu: SIAM J. Numer. Anal. \textbf{45}, 2483-2509 (2007)]. In…
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…
The multigroup neutron transport equations has been widely used to study the interactions of neutrons with their background materials in nuclear reactors. High-resolution simulations of the multigroup neutron transport equations using…
The finite cell method is a highly flexible discretization technique for numerical analysis on domains with complex geometries. By using a non-boundary conforming computational domain that can be easily meshed, automatized computations on a…
Interior point methods solve small to medium sized problems to high accuracy in a reasonable amount of time. However, for larger problems as well as stochastic problems, one needs to use first-order methods such as stochastic gradient…
This paper proposes a domain decomposition subspace neural network method for efficiently solving linear and nonlinear partial differential equations. By combining the principles of domain decomposition and subspace neural networks, the…
In this work, we consider alternative discretizations for PDEs which use expansions involving integral operators to approximate spatial derivatives. These constructions use explicit information within the integral terms, but treat boundary…
This paper presents and studies an approach for constructing auxiliary space preconditioners for finite element problems using a constrained nonconforming reformulation, that is based on a proposed modified version of the mortar method. The…
A combination of block-Jacobi and deflation preconditioning is used to solve a high-order discontinuous collocation-based discretization of the Schur complement of the Poisson-Neumann system as arises in the operator splitting of the…
GenEO (`Generalised Eigenvalue problems on the Overlap') is a method for computing an operator-dependent spectral coarse space to be combined with local solves on subdomains to form a robust parallel domain decomposition preconditioner for…