Related papers: Solving the 3D High-Frequency Helmholtz Equation u…
We present a fast and approximate multifrontal solver for large-scale sparse linear systems arising from finite-difference, finite-volume or finite-element discretization of high-frequency wave equations. The proposed solver leverages the…
We study the convergence of multigrid schemes for the Helmholtz equation, focusing in particular on the choice of the coarse scale operators. Let G_c denote the number of points per wavelength at the coarse level. If the coarse scale…
This paper addresses several aspects of the linear Hybridizable Discontinuous Galerkin Method (HDG) for the Helmholtz equation with impedance boundary condition at high frequency. First, error estimates with explicit dependence on the wave…
We present a hybrid numerical-asymptotic (HNA) boundary element method (BEM) for high frequency scattering by two-dimensional screens and apertures, whose computational cost to achieve any prescribed accuracy remains bounded with increasing…
We propose a low-rank method for solving the Helmholtz equation. Our approach is based on the WaveHoltz method, which computes Helmholtz solutions by applying a time-domain filter to the solution of a related wave equation. The wave…
Inverse scattering involving microwave and ultrasound waves require numerical solution of nonlinear optimization problem. To alleviate the computational burden of a full three-dimensional (3-D) inverse problem, it is a common practice to…
Finite element methods are effective for Helmholtz problems involving complex geometries and heterogeneous media. However, the resulting linear systems are often large, indefinite, and challenging for iterative solvers, particularly at high…
The boundary integral method is an efficient approach for solving time-harmonic obstacle scattering problems by a bounded scatterer. This paper presents the directional preconditioner for the iterative solution of linear systems of the…
We introduce a new numerical method for solving time-harmonic acoustic scattering problems. The main focus is on plane waves scattered by smoothly varying material inhomogeneities. The proposed method works for any frequency $\omega$, but…
A discretization scheme for variable coefficient Helmholtz problems on two-dimensional domains is presented. The scheme is based on high-order spectral approximations and is designed for problems with smooth solutions. The resulting system…
The use of integral equation methods for the efficient numerical solution of PDE boundary value problems requires two main tools: quadrature rules for the evaluation of layer potential integral operators with singular kernels, and fast…
In this paper, we propose a deep learning-enhanced multigrid solver for high-frequency and heterogeneous Helmholtz equations. By applying spectral analysis, we categorize the iteration error into characteristic and non-characteristic…
This paper presents a fast iterative solver for Lippmann-Schwinger equation for high-frequency waves scattered by a smooth medium with a compactly supported inhomogeneity. The solver is based on the sparsifying preconditioner and a domain…
Solving large-scale Generalized Eigenvalue Problems (GEPs) is a fundamental yet computationally prohibitive task in science and engineering. As a promising direction, contour integral (CI) methods, such as the CIRR algorithm, offer an…
In this research, to solve the large indefinite least squares problem, we firstly transform its normal equation into a sparse block three-by-three linear systems, then use GMRES method with an accelerated preconditioner to solve it. The…
Implicit Neural Representations (INRs) have recently gained attention as a powerful approach for continuously representing signals such as images, videos, and 3D shapes using multilayer perceptrons (MLPs). However, MLPs are known to exhibit…
We propose a method for quantifying uncertainty in high-dimensional PDE systems with random parameters, where the number of solution evaluations is small. Parametric PDE solutions are often approximated using a spectral decomposition based…
We propose in this paper a globally numerical method to solve a phaseless coefficient inverse problem: how to reconstruct the spatially distributed refractive index of scatterers from the intensity (modulus square) of the full complex…
We establish a new iterative method for solving a class of large and sparse linear systems of equations with three-by-three block coefficient matrices having saddle point structure. Convergence properties of the proposed method are studied…
We show that the properties of the lower part of the spectrum of the Helmholtz equation for an heterogeneous system in a finite region in $d$ dimensions, where the solutions to the homogeneous problems are known, can be systematically…