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Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general…
We present an efficient MPI-parallel geometric multigrid library for quadtree (2D) or octree (3D) grids with adaptive refinement. Cartesian 2D/3D and cylindrical 2D geometries are supported, with second-order discretizations for the…
We present a multi-block finite-difference solver for massively parallel Direct Numerical Simulations (DNS) of incompressible flows. The algorithm combines the versatility of a multi-block solver with the method of eigenfunctions…
We present an accurate and efficient method to calculate the gravitational potential of an isolated system in three-dimensional Cartesian and cylindrical coordinates subject to vacuum (open) boundary conditions. Our method consists of two…
In [5] Cauchon introduced the so-called deleting derivations algorithm. This algorithm was first used in noncommutative algebra to prove catenarity in generic quantum matrices, and then to show that torus-invariant primes in these algebras…
This work is motivated by the frequent occurrence of boundary value problems with various boundary conditions in the modeling of some problems in engineering and physical science. Here we propose a new technique to force the positive…
We consider the numerical solution of the scattering of time-harmonic plane waves from an infinite periodic array of reflection or transmission obstacles in a homogeneous background medium, in two dimensions. Boundary integral formulations…
In this paper we prove discrete to continuum convergence rates for Poisson Learning, a graph-based semi-supervised learning algorithm that is based on solving the graph Poisson equation with a source term consisting of a linear combination…
We introduce and derive the Fourier-enhanced 3D electrostatic field solver of the gyrokinetic full-f PIC code PICLS. The solver makes use of a Fourier representation in one periodic direction of the domain to make the solving of the system…
Floorplanning is the first stage of VLSI physical design. An effective floorplanning engine definitely has positive impact on chip design speed, quality and performance. In this paper, we present a novel mathematical model to characterize…
Quantum linear system (QLS) algorithms offer the potential to solve large-scale linear systems exponentially faster than classical methods. However, applying QLS algorithms to real-world problems remains challenging due to issues such as…
We present a nonvariational setting for the Neumann problem for the Poisson equation for solutions that are H\"{o}lder continuous and that may have infinite Dirichlet integral. We introduce a distributional normal derivative on the boundary…
This paper considers the existence of weak and strong solutions to the Poisson equation on a surface with a boundary condition in co-normal direction. We apply the Lax-Milgram theorem and some properties of $H^1$-functions to show the…
We propose Neural Walk-on-Spheres (NWoS), a novel neural PDE solver for the efficient solution of high-dimensional Poisson equations. Leveraging stochastic representations and Walk-on-Spheres methods, we develop novel losses for neural…
Research on smooth vector graphics is separated into two independent research threads: one on interpolation-based gradient meshes and the other on diffusion-based curve formulations. With this paper, we propose a mathematical formulation…
Recently, a nonlinear Poisson equation has been introduced to model nonlinear and nonlocal hyperpolarization effects in electrostatic solute-solvent interaction for biomolecular solvation analysis. Due to a strong nonlinearity associated…
For a connected weighted hypergraph, we give a randomized almost-linear-time solver for the Poisson problem for the cut-based hypergraph Laplacian in the natural input size $P=\sum_{e\in E}|e|$, the sum of hyperedge sizes. For every fixed…
This paper is the second part of a threefold article, aimed at solving numerically the Poisson problem in three-dimensional prismatic or axisymmetric domains. In the first part of this series, the Fourier Singular Complement Method was…
We describe extensions to the siesta density functional theory (dft) code [30], for the simulation of isolated molecules and their absorption spectra. The extensions allow for: - Use of a multi-grid solver for the Poisson equation on a…
In this paper we prove that any solution of the $m$-polyharmonic Poisson equation in a Reifenberg-flat domain with homogeneous Dirichlet boundary condition, is $\mathscr{C}^{m-1,\alpha}$ regular up to the boundary. To achieve this result we…