Related papers: Optimised finite difference computation from symbo…
We introduce an efficient variational hybrid quantum-classical algorithm designed for solving Caputo time-fractional partial differential equations. Our method employs an iterable cost function incorporating a linear combination of overlap…
High level domain specific languages for the finite element method underpin high productivity programming environments for simulations based on partial differential equations (PDE) while employing automatic code generation to achieve high…
In this article, we propose a Milstein finite difference scheme for a stochastic partial differential equation (SPDE) describing a large particle system. We show, by means of Fourier analysis, that the discretisation on an unbounded domain…
Numerical algorithms for solving problems of mathematical physics on modern parallel computers employ various domain decomposition techniques. Domain decomposition schemes are developed here to solve numerically initial/boundary value…
Multiscale and multiphysics problems need novel numerical methods in order for them to be solved correctly and predictively. To that end, we develop a wavelet based technique to solve a coupled system of nonlinear partial differential…
The paper introduces a new finite element numerical method for the solution of partial differential equations on evolving domains. The approach uses a completely Eulerian description of the domain motion. The physical domain is embedded in…
In this article, we introduce a finite element method designed for the robust computation of approximate signed distance functions to arbitrary boundaries in two and three dimensions. Our method employs a novel prediction-correction…
Derivative boundary conditions introduce challenges for mesh-free discretizations of PDEs on surfaces, especially when the domain is represented by randomly sampled point clouds. The recently developed two-step tangent-space RBF-generated…
This work develops a class of probabilistic algorithms for the numerical solution of nonlinear, time-dependent partial differential equations (PDEs). Current state-of-the-art PDE solvers treat the space- and time-dimensions separately,…
The paper presents a variational quantum algorithm to solve initial-boundary value problems described by second-order partial differential equations. The approach uses hybrid classical/quantum hardware that is well suited for quantum…
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…
The development of data-driven approaches for solving differential equations has been followed by a plethora of applications in science and engineering across a multitude of disciplines and remains a central focus of active scientific…
This paper deals with fractional differential equations, with dependence on a Caputo fractional derivative of real order. The goal is to show, based on concrete examples and experimental data from several experiments, that fractional…
Symbolic encoding has been used in multi-operator learning as a way to embed additional information for distinct time-series data. For spatiotemporal systems described by time-dependent partial differential equations, the equation itself…
This work introduces and analyzes a finite element scheme for evolution problems involving fractional-in-time and in-space differentiation operators up to order two. The left-sided fractional-order derivative in time we consider is employed…
Ordinary Differential Equations (ODE) are used throughout science where the capture of rates of change in states is sought. While both pieces of commercial and open software exist to study such systems, their efficient and accurate usage…
Partial differential equations have a wide range of applications in modeling multiple physical, biological, or social phenomena. Therefore, we need to approximate the solutions of these equations in computationally feasible terms. Nowadays,…
We present a novel computational paradigm for process design in manufacturing processes that incorporates simulation responses to optimize manufacturing process parameters in high-dimensional temporal and spatial design spaces. We developed…
The long term aim is to use modern dynamical systems theory to derive discretisations of noisy, dissipative partial differential equations. As a first step we here consider a small domain and apply stochastic centre manifold techniques to…
Next-generation exascale machines with extreme levels of parallelism will provide massive computing resources for large scale numerical simulations of complex physical systems at unprecedented parameter ranges. However, novel numerical…