Related papers: A Two-Level Direct Solver for the Hierarchical Poi…
Elliptic partial differential equations must be solved numerically for many problems in numerical relativity, such as initial data for every simulation of merging black holes and neutron stars. Existing elliptic solvers can take multiple…
We present a parallel hierarchical solver for general sparse linear systems on distributed-memory machines. For large-scale problems, this fully algebraic algorithm is faster and more memory-efficient than sparse direct solvers because it…
An efficient direct solver for solving the Lippmann-Schwinger integral equation modeling acoustic scattering in the plane is presented. For a problem with $N$ degrees of freedom, the solver constructs an approximate inverse in…
Matrix-free geometric multigrid solvers for elliptic PDEs that have been discretised with Higher-order Discontinuous Galerkin (DG) methods are ideally suited to exploit state-of-the-art computer architectures. Higher polynomial degrees…
The paper presents high-order accurate, energy-, and entropy-stable discretizations constructed from summation-by-parts (SBP) operators. Notably, the discretizations assemble global SBP operators and use continuous solutions, unlike…
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,…
In recent years, GPU-accelerated optimization solvers based on second-order methods (e.g., interior-point methods) have gained momentum with the advent of mature and efficient GPU-accelerated direct sparse linear solvers, such as cuDSS.…
We present a model and variance reduction method for the fast and reliable computation of statistical outputs of stochastic elliptic partial differential equations. Our method consists of three main ingredients: (1) the hybridizable…
We revisit the Hierarchical Poincar\'{e}-Steklov (HPS) method for the Poisson equation using standard Q1 finite elements, building on the original in work on HPS of Martinsson from 2013. While corner degrees of freedom were implicitly…
High order methods have shown great potential to overcome performance issues of simulations of partial differential equations (PDEs) on modern hardware, still many users stick to low-order, matrix-based simulations, in particular in porous…
High fidelity scientific simulations modeling physical phenomena typically require solving large linear systems of equations which result from discretization of a partial differential equation (PDE) by some numerical method. This step often…
Global spectral methods offer the potential to compute solutions of partial differential equations numerically to very high accuracy. In this work, we develop a novel global spectral method for linear partial differential equations on cubes…
This paper introduces a new approximation scheme for solving high-dimensional semilinear partial differential equations (PDEs) and backward stochastic differential equations (BSDEs). First, we decompose a target semilinear PDE (BSDE) into…
In this article, we present a parallel recursive algorithm based on multi-level domain decomposition that can be used as a precondtioner to a Krylov subspace method to solve sparse linear systems of equations arising from the discretization…
This paper presents an efficient parallel direct algorithm with near-optimal complexity for the compact fourth and sixth-order approximation of the three-dimensional Helmholtz equations [1] with the problem coefficient depending on only one…
In this paper, we use an implicit two-derivative deferred correction time discretization approach and combine it with a spatial discretization of the discontinuous Galerkin spectral element method to solve (non-)linear PDEs. The resulting…
This paper is focused on performing a new method for solving linear and nonlinear higher-order boundary value problems (HBVPs). This direct numerical method based on spectral method. The trial function of this method is the Monic Chebyshev…
In this paper, we propose a mesh-free numerical method for solving elliptic PDEs on unknown manifolds, identified with randomly sampled point cloud data. The PDE solver is formulated as a spectral method where the test function space is the…
Relying on the classical connection between Backward Stochastic Differential Equations (BSDEs) and non-linear parabolic partial differential equations (PDEs), we propose a new probabilistic learning scheme for solving high-dimensional…
Direct factorization methods for the solution of large, sparse linear systems that arise from PDE discretizations are robust, but typically show poor time and memory scalability for large systems. In this paper, we describe an efficient…