Related papers: EXPODE -- Advanced Exponential Time Integration To…
Discretizing continuous-time linear systems typically requires numerical integration. This document presents a convenient method for discretizing the dynamics, input, and process noise state-space matrices of a continuous-time linear system…
Chain of thought is a natural inference-time method for increasing the computational power of transformer-based large language models (LLMs), but comes at the cost of sequential decoding. Are there more efficient alternatives to expand a…
We propose a computationally efficient and systematically convergent approach for elastodynamics simulations. We recast the second-order dynamical equation of elastodynamics into an equivalent first-order system of coupled equations, so as…
In this work, we present a novel class of parallelizable high-order time integration schemes for the approximate solution of additive ODEs. The methods achieve high order through a combination of a suitable quadrature formula involving…
With growing demand for time-domain simulations of correlated many-body systems, the development of efficient and stable integration schemes for the time-dependent Schr\"odinger equation is of keen interest in modern electronic structure…
We propose an effective and flexible way to implement 2D and 3D elastoplastic problems in MATLAB using fully vectorized codes. Our technique is applied to a broad class of the problems including perfect plasticity or plasticity with…
This course, intended for undergraduates familiar with elementary calculus and linear algebra, introduces the extension of differential calculus to functions on more general vector spaces, such as functions that take as input a matrix and…
Time integration of ODEs or time-dependent PDEs with required resolution of the fastest time scales of the system, can be very costly if the system exhibits multiple time scales of different magnitudes. If the different time scales are…
Building on the recently developed theory of magnitude, we introduce the optimization algorithm EXPLO2 and carefully benchmark it. EXPLO2 advances the state of the art for optimizing high-dimensional ($D \gtrapprox 40$) multimodal functions…
We present $TimeEvolver$, a program for computing time evolution in a generic quantum system. It relies on well-known Krylov subspace techniques to tackle the problem of multiplying the exponential of a large sparse matrix $i H$, where $H$…
An algorithm for numerically computing the exponential of a matrix is presented. We have derived a polynomial expansion of $e^x$ by computing it as an initial value problem using a symbolic programming language. This algorithm is shown to…
Suzuki-Trotter decompositions of exponential operators like $\exp(Ht)$ are required in almost every branch of numerical physics. Often the exponent under consideration has to be split into more than two operators, for instance as local…
This paper introduces Exp-ParaDiag, a novel time-parallel method that combines the strength of exponential integrators into the ParaDiag framework. We develop and analyze Exp-ParaDiag based on first and second order accurate exponential…
The logic embedding tool provides a procedural encoding for non-classical reasoning problems into classical higher-order logic. It is extensible and can support an increasing number of different non-classical logics as reasoning targets.…
We discuss a numerical package, named ORTHOCUB, for the computation of linear functionals of both integral and differential type on multivariate polynomial spaces. The weighted sums corresponding to such integral and differential cubatures…
Parareal is a well-known parallel-in-time algorithm that combines a coarse and fine propagator within a parallel iteration. It allows for large-scale parallelism that leads to significantly reduced computational time compared to serial…
To approximate solutions of a linear differential equation, we project, via trigonometric interpolation, its solution space onto a finite-dimensional space of trigonometric polynomials and construct a matrix representation of the…
In this paper, we investigate the application of exponential integrators to advection-dominated problems. We focus on Krylov subspace and Leja interpolation methods to compute the action of exponential and related matrix functions.…
In this paper, we present new types of exponential integrators for Stochastic Differential Equations (SDEs) that take the advantage of the exact solution of (generalised) geometric Brownian motion. We examine both Euler and Milstein…
By exploiting the error functions of explicit symplectic integrators for solving separable Hamiltonians, I show that it is possible to develop explicit, time-reversible symplectic integrators for solving non-separable Hamiltonians of the…