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Recent advances in quantum computing and their increased availability has led to a growing interest in possible applications. Among those is the solution of partial differential equations (PDEs) for, e.g., material or flow simulation.…
Although deep models have been widely explored in solving partial differential equations (PDEs), previous works are primarily limited to data only with up to tens of thousands of mesh points, far from the million-point scale required by…
We implemented the pressure-implicit with splitting of operators (PISO) and semi-implicit method for pressure-linked equations (SIMPLE) solvers of the Navier-Stokes equations on Fermi-class graphics processing units (GPUs) using the CUDA…
The problem of solving a system of polynomial equations is one of the most fundamental problems in applied mathematics. Among them, the problem of solving a system of binomial equations form a important subclass for which specialized…
We solve Poisson's equation by combining a spectral-element method with a mapped infinite-element method. We focus on problems in geostatics and geodynamics, where Earth's gravitational field is determined by Poisson's equation inside the…
Kinetic plasma simulations solve the Vlasov-Poisson or Vlasov-Maxwell equations to evolve scalar-variable distribution functions in position-velocity phase space and vector-variable electromagnetic fields in physical space. The…
The graphics processing unit (GPU) has emerged as a powerful and cost effective processor for general performance computing. GPUs are capable of an order of magnitude more floating-point operations per second as compared to modern central…
This paper proposes a novel Machine Learning-based approach to solve a Poisson problem with mixed boundary conditions. Leveraging Graph Neural Networks, we develop a model able to process unstructured grids with the advantage of enforcing…
In this paper, we develop a low-order three-dimensional finite-element solver for fast multiple-case crust deformation analysis on GPU-based systems. Based on a high-performance solver designed for massively parallel CPU based systems, we…
A fast Poisson solver software package PoisFFT is presented. It is available as a free software licensed under the GNU GPL license version 3. The package uses the fast Fourier transform to directly solve the Poisson equation on a uniform…
The kernel-free boundary integral (KFBI) method has successfully solved partial differential equations (PDEs) on irregular domains. Diverging from traditional boundary integral methods, the computation of boundary integrals in KFBI is…
We discuss the development, verification, and performance of a GPU accelerated discontinuous Galerkin method for the solutions of two dimensional nonlinear shallow water equations. The shallow water equations are hyperbolic partial…
In this manuscript, we propose matrix- and tensor-oriented methods for the numerical solution of the multidimensional evolutionary space-fractional complex Ginzburg--Landau equation. After a suitable spatial semidiscretization, the…
Obtaining a thermodynamically accurate phase diagram through numerical calculations is a computationally expensive problem that is crucially important to understanding the complex phenomena of solid state physics, such as superconductivity.…
We present a novel approach for enhancing the resolution and geometric fidelity of 3D Gaussian Splatting (3DGS) beyond native training resolution. Current 3DGS methods are fundamentally limited by their input resolution, producing…
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…
Effective Hamiltonian calculations for large quantum systems can be both analytically intractable and numerically expensive using standard techniques. In this manuscript, we present numerical techniques inspired by Nonperturbative…
We investigate the potential of Graphics Processing Units (GPUs) to solve large-scale nonlinear programs with a dynamic structure. Using ExaModels, a GPU-accelerated automatic differentiation tool, and the interior-point solver MadNLP, we…
We study exact sparse linear regression with an $\ell_0-\ell_2$ penalty and develop a branch-and-bound (BnB) algorithm explicitly designed for GPU execution. Starting from a perspective reformulation, we derive an interval relaxation that…
We discuss how to implement the linear finite element method for solving the Poisson equation. We begin with the data structure to represent the triangulation and boundary conditions, introduce the sparse matrix, and then discuss the…