Related papers: Fast global spectral methods for three-dimensional…
We present a general purpose method for solving partial differential equations on a closed surface, based on a technique for discretizing the surface introduced by Wenjun Ying and Wei-Cheng Wang [J. Comput. Phys. 252 (2013), pp. 606-624]…
Fast and accurate solution of time-dependent partial differential equations (PDEs) is of key interest in many research fields including physics, engineering, and biology. Generally, implicit schemes are preferred over the explicit ones for…
Quantum computers can produce a quantum encoding of the solution of a system of differential equations exponentially faster than a classical algorithm can produce an explicit description. However, while high-precision quantum algorithms for…
This study proposes an approach based on a perturbation technique to construct global solutions to dynamic stochastic general equilibrium models (DSGE). The main idea is to expand a solution in a series of powers of a small parameter…
We develop a spectral method for solving univariate singular integral equations over unions of intervals by utilizing Chebyshev and ultraspherical polynomials to reformulate the equations as almost-banded infinite-dimensional systems. This…
In this paper, we present a deep learning-based numerical method for approximating high dimensional stochastic partial differential equations (SPDEs). At each time step, our method relies on a predictor-corrector procedure. More precisely,…
We propose a new framework to design and analyze accelerated methods that solve general monotone equation (ME) problems $F(x)=0$. Traditional approaches include generalized steepest descent methods and inexact Newton-type methods. If $F$ is…
In [7], a new iterative method for solving linear system of equations was presented which can be considered as a modification of the Gauss-Seidel method. Then in [4] a different approach, say 2D-DSPM, and more effective one was introduced.…
We introduce a framework for solving a class of parabolic partial differential equations on triangle mesh surfaces, including the Hamilton-Jacobi equation and the Fokker-Planck equation. PDE in this class often have nonlinear or stiff terms…
This manuscript presents an efficient solver for the linear system that arises from the Hierarchical Poincar\'e-Steklov (HPS) discretization of three dimensional variable coefficient Helmholtz problems. Previous work on the HPS method has…
Elliptic partial differential equations arise in many fields of science and engineering such as steady state distribution of heat, fluid dynamics, structural/mechanical engineering, aerospace engineering and seismology etc. In three…
This paper introduces PDEformer, a neural solver for partial differential equations (PDEs) capable of simultaneously addressing various types of PDEs. We propose to represent the PDE in the form of a computational graph, facilitating the…
Nonlinear systems of partial differential equations (PDEs) may permit several distinct solutions. The typical current approach to finding distinct solutions is to start Newton's method with many different initial guesses, hoping to find…
We introduce a general framework for solving partial differential equations (PDEs) using generative diffusion models. In particular, we focus on the scenarios where we do not have the full knowledge of the scene necessary to apply classical…
We present an amelioration of current known algorithms for optimal spectral partitioning problems. The idea is to use the advantage of a representation using density functions while decreasing the computational time. This is done by…
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…
We consider regular polynomial interpolation algorithms on recursively defined sets of interpolation points which approximate global solutions of arbitrary well-posed systems of linear partial differential equations. Convergence of the…
We present an enhanced version of the row-based randomized block-Kaczmarz method to solve a linear system of equations. This improvement makes use of a regularization during block updates in the solution, and a dynamic proposal distribution…
We present an overarching framework for stable spectral methods on a triangle, defined by a multivariate W-system and based on orthogonal polynomials on the triangle. Motivated by the Koornwinder orthogonal polynomials on the triangle, we…
In this paper, we seek to construct nontrivial global solutions to some quasilinear wave equations in three space dimensions. We first present a conditional result on the construction of nontrivial global solutions to a general system of…