Related papers: FiniteFieldSolve: Exactly Solving Large Linear Sys…
In April 2025 GMV announced a competition for finding the best method to solve a particular polynomial system over a finite field. In this paper we provide a method for solving the given equation system significantly faster than what is…
We introduce DDE-Solver, a Maple package designed for solving Discrete Differential Equations (DDEs). These equations are functional equations relating algebraically a formal power series F(t, u) with polynomial coefficients in a…
This paper presents a universal numerical scheme tailored for tackling linear integral, integro-differential, and both initial and boundary value problems of ordinary differential equations. The numerical scheme is readily adapted for…
Discretizations of infinite-dimensional variational inequalities lead to linear and nonlinear complementarity problems with many degrees of freedom. To solve these problems in a parallel computing environment, we propose two active-set…
This document summarizes the main concepts of the finite element (FE) theory and constitutive relations as implemented in the open-source code phase-field multiphysics materials simulator PHIMATS https://github.com/ahcomat/PHIMATS. PHIMATS…
This survey describes a class of methods known as "fast direct solvers". These algorithms address the problem of solving a system of linear equations $\boldsymbol{Ax}=\boldsymbol{b}$ arising from the discretization of either an elliptic PDE…
Within the finite element community, discontinuous Galerkin (DG) and mixed finite element methods have become increasingly popular in simulating geophysical flows. However, robust and efficient solvers for the resulting saddle-point and…
In this paper, we introduce a new finite expression method (FEX) to solve high-dimensional partial integro-differential equations (PIDEs). This approach builds upon the original FEX and its inherent advantages with new advances: 1) A novel…
Finite model finders give users the ability to specify properties of a system in mathematical logic and then automatically find concrete examples, called solutions, that satisfy the properties. These solutions are often viewed as a key…
Leveraging Trace Theory, we investigate the efficient parallelization of direct solvers for large linear equation systems. Our focus lies on a multi-frontal algorithm, and we present a methodology for achieving near-optimal scheduling on…
It is shown that large classes of nonlinear systems of PDEs, with possibly associated initial and/or boundary value problems, can be solved by the method of order completion. The solutions obtained can be assimilated with Hausdorff…
We present algorithms to solve coupled systems of linear differential equations, arising in the calculation of massive Feynman diagrams with local operator insertions at 3-loop order, which do {\it not} request special choices of bases.…
Until now, Computer Scientists have concerned themselves with identifying efficient algorithms for solving the general case of some problem -- that is finding one which performs well when the size of the input tends to infinity. In this…
The finite element method is a well-established method for the numerical solution of partial differential equations (PDEs), both linear and nonlinear. However, the repeated reassemblage of finite element matrices for nonlinear PDEs is…
To improve the computational efficiencies of the real-space orbital-free density functional theory, this work develops a new single-grid solver by directly providing the closed-form solution to the inner iteration and using an improved…
Scaling hyperparameter optimisation to very large datasets remains an open problem in the Gaussian process community. This paper focuses on iterative methods, which use linear system solvers, like conjugate gradients, alternating…
Designing efficient and accurate numerical solvers for high-dimensional partial differential equations (PDEs) remains a challenging and important topic in computational science and engineering, mainly due to the "curse of dimensionality" in…
A high-order accurate adjoint-based optimization framework is presented for unsteady multiphysics problems. The fully discrete adjoint solver relies on the high-order, linearly stable, partitioned solver introduced in [1], where different…
This work presents a matrix-free finite element solver for finite-strain elasticity adopting an $hp$-multigrid preconditioner. Compared to classical algorithms relying on a global sparse matrix, matrix-free solution strategies significantly…
This paper presents a large-scale parallel solver, specifically designed to tackle the challenges of solving high-dimensional and high-contrast linear systems in heat transfer topology optimization. The solver incorporates an interpolation…