Related papers: A Lightweight, Geometrically Flexible Fast Algorit…
One of the major issues in the computational mechanics is to take into account the geometrical complexity. To overcome this difficulty and to avoid the expensive mesh generation, geometrically unfitted methods, i.e. the numerical methods…
We describe three approaches for computing a gravity signal from a density anomaly. The first approach consists of the classical "summation" technique, whilst the remaining two methods solve the Poisson problem for the gravitational…
We present a spectrally accurate fast algorithm for evaluating the solution to the scalar wave equation in free space driven by a large collection of point sources in a bounded domain. With $M$ sources temporally discretized by $N_t$ time…
We propose an efficient algorithm for the evaluation of the potential and its gradient of gravitational/electrostatic $N$-body systems, which we call particle mesh multipole method (PMMM or PM$^3$). PMMM can be understood both as an…
Density-potential functional theory (DPFT) is an alternative formulation of orbital-free density functional theory that may be suitable for modeling the electronic structure of large systems. To date, DPFT has been applied mainly to quantum…
Quantum computing holds the promise of solving computational mechanics problems in polylogarithmic time, meaning computational time scales as $\mathscr{O}((\log N)^c)$, where $N$ is the problem size and $c$ a constant. We propose a quantum…
We present a method of CutFEM type for the Poisson problem with either Dirichlet or Neumann boundary conditions. The computational mesh is obtained from a background (typically uniform Cartesian) mesh by retaining only the elements…
Bertaut's equivalent electric density idea (E. F. Bertaut, Journal de Physique {\bf 39}, 1331 (1978)) is applied to the case of two dimensional periodic continuous charge density distributions. The following derivation differs from what was…
Consider the Poisson equation with the Dirichlet boundary condition on a three-dimensional polyhedral domain. For singular solutions from the non-smoothness of the domain boundary, we propose new anisotropic tetrahedral mesh refinement…
This work proposes a fast iterative method for local steric Poisson--Boltzmann (PB) theories, in which the electrostatic potential is governed by the Poisson's equation and ionic concentrations satisfy equilibrium conditions. To present the…
We present a simple, yet general, end-to-end deep neural network representation of the potential energy surface for atomic and molecular systems. This methodology, which we call Deep Potential, is "first-principle" based, in the sense that…
This paper concerns the construction and analysis of a numerical scheme for a mixed discrete-continuous fragmentation equation. A finite volume scheme is developed, based on a conservative formulation of a truncated version of the…
This paper presents a quantum algorithm for the solution of prototypical second-order linear elliptic partial differential equations discretized by $d$-linear finite elements on Cartesian grids of a bounded $d$-dimensional domain. An…
We present an exponentially convergent semi-implicit meshless algorithm for the solution of Navier-Stokes equations in complex domains. The algorithm discretizes partial derivatives at scattered points using radial basis functions as…
Solving a Poisson equation is generally reduced to solving a linear system with a coefficient matrix $A$ of entries $a_{ij}$, $i,j=1,2,...,n$, from the discretized Poisson equation. Although the variational quantum algorithms are promising…
This work aims to introduce the framework of polynomial optimization theory to solve fractional polynomial problems (FPPs). Unlike other widely used optimization frameworks, the proposed one applies to a larger class of FPPs, not…
Density-functional theory (DFT) has revolutionized computer simulations in chemistry and material science. A faithful implementation of the theory requires self-consistent calculations. However, this effort involves repeatedly diagonalizing…
The finite element method is used to approximately solve boundary value problems for differential equations. The method discretises the parameter space and finds an approximate solution by solving a large system of linear equations. Here we…
We present a novel numerical method for solving the elliptic partial differential equation problem for the electrostatic potential with piecewise constant conductivity. We employ an integral equation approach for which we derive a system of…
We develop a universally applicable embedded boundary finite difference method, which results in a symmetric positive definite linear system and does not suffer from small cell stiffness. Our discretization is efficient for the wave, heat…