Related papers: A high-order fast direct solver for surface PDEs
The demand for substantial increases in the spatial resolution of global weather- and climate- prediction models makes it necessary to use numerically efficient and highly scalable algorithms to solve the equations of large scale…
Many important problems in science and engineering require solving the so-called parametric partial differential equations (PDEs), i.e., PDEs with different physical parameters, boundary conditions, shapes of computation domains, etc.…
The present work proposes a well-balanced finite volume-type numerical method for the solution of non-conservative hyperbolic partial differential equations (PDEs) with source terms. The method is characterized, first, by the use of a…
We present a higher-order finite volume method for solving elliptic PDEs with jump conditions on interfaces embedded in a 2D Cartesian grid. Second, fourth, and sixth order accuracy is demonstrated on a variety of tests including problems…
This paper develops and analyzes an efficient numerical method for solving elliptic partial differential equations, where the diffusion coefficients are random perturbations of deterministic diffusion coefficients. The method is based upon…
This paper presents an implicit solution formula for the Hamilton-Jacobi partial differential equation (HJ PDE). The formula is derived using the method of characteristics and is shown to coincide with the Hopf and Lax formulas in the case…
We present a novel differentiable grid-based representation for efficiently solving differential equations (DEs). Widely used architectures for neural solvers, such as sinusoidal neural networks, are coordinate-based MLPs that are both…
Three-dimensional target identification using scattering techniques requires high accuracy solutions and very fast computations for real-time predictions in some critical applications. We first train a deep neural operator~(DeepONet) to…
In this paper, we develop fast procedures for solving linear systems arising from discretization of ordinary and partial differential equations with Caputo fractional derivative w.r.t time variable. First, we consider a finite difference…
In this work, we consider the approximation of Hilbert space-valued meromorphic functions that arise as solution maps of parametric PDEs whose operator is the shift of an operator with normal and compact resolvent, e.g. the Helmholtz…
This paper discusses a fast direct solver using boundary integral equations for Helmholtz transmission problems involving multiple inclusions in two dimensions. Efficiently addressing scattering problems in the presence of numerous…
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…
The hybrid high-order method is a modern numerical framework for the approximation of elliptic PDEs. We present here an extension of the hybrid high-order method to meshes possessing curved edges/faces. Such an extension allows us to…
Multiscale and multiphysics problems need novel numerical methods in order for them to be solved correctly and predictively. To that end, we develop a wavelet based technique to solve a coupled system of nonlinear partial differential…
We provide improved convergence rates for various \emph{non-smooth} optimization problems via higher-order accelerated methods. In the case of $\ell_\infty$ regression, we achieves an $O(\epsilon^{-4/5})$ iteration complexity, breaking the…
We propose machine learning methods for solving fully nonlinear partial differential equations (PDEs) with convex Hamiltonian. Our algorithms are conducted in two steps. First the PDE is rewritten in its dual stochastic control…
Higher-order accurate solution to electromagnetic scattering problems are obtained at reduced computational cost in a {\it p}-variable finite volume time domain method. Spatial operators of lower, including first-order accuracy, are…
A deep learning approach to numerically approximate the solution to the Eikonal equation is introduced. The proposed method is built on the fast marching scheme which comprises of two components: a local numerical solver and an update…
The paper studies a method for solving elliptic partial differential equations posed on hypersurfaces in $\mathbb{R}^N$, $N=2,3$. The method allows a surface to be given implicitly as a zero level of a level set function. A surface equation…
We develop a spectral low-mode reduced solver for second-order elliptic boundary value problems with spatially varying diffusion coefficients. The approach projects standard finite difference or finite element discretization onto a global…