相关论文: A family of high-order accurate contour integral m…

In this paper, we introduce a general constructive method to compute solutions of initial value problems of semilinear parabolic partial differential equations on hyper-rectangular domains via semigroup theory and computer-assisted proofs.…

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…

A high-order quadrature algorithm is presented for computing integrals over curved surfaces and volumes whose geometry is implicitly defined by the level sets of (one or more) multivariate polynomials. The algorithm recasts the implicitly…

A new family of methods involving complex coefficients for the numerical integration of differential equations is presented and analyzed. They are constructed as linear combinations of symmetric-conjugate compositions obtained from a basic…

This paper investigates a class of non-autonomous highly oscillatory ordinary differential equations characterized by a linear component inversely proportional to a small parameter $\varepsilon$, with purely imaginary eigenvalues, and an…

This paper is devoted to the construction of exponential integrators of first and second order for the time discretization of constrained parabolic systems. For this extend, we combine well-known exponential integrators for unconstrained…

We propose a finite-dimensional control-based method to approximate solution operators for evolutional partial differential equations (PDEs), particularly in high-dimensions. By employing a general reduced-order model, such as a deep neural…

Exponential integrators are time stepping schemes which exactly solve the linear part of a semilinear ODE system. This class of schemes requires the approxima- tion of a matrix exponential in every step, and one successful modern method is…

Dozens of exponential integration formulas have been proposed for the high-accuracy solution of stiff PDEs such as the Allen-Cahn, Korteweg-de Vries and Ginzburg-Landau equations. We report the results of extensive comparisons in MATLAB and…

We propose a family of reliable symplectic integrators adapted to the Discrete Non-Linear Schr\"odinger equation; based on an idea of Yoshida (H. Yoshida, Construction of higher order symplectic integrators, Physics Letters A, 150, 5,6,7,…

The concept of effective order is a popular methodology in the deterministic literature for the construction of efficient and accurate integrators for differential equations over long times. The idea is to enhance the accuracy of a…

This paper presents a high-order accurate numerical quadrature algorithm for evaluating integrals over curved surfaces and regions defined implicitly via a level set of a given function restricted to a hyperrectangle. The domain is divided…

We develop a general framework for numerically solving differential equations while preserving invariants. As in standard projection methods, we project an arbitrary base integrator onto an invariant-preserving manifold, however, our method…

Stochastic interpolants offer a robust framework for continuously transforming samples between arbitrary data distributions, holding significant promise for generative modeling. Despite their potential, rigorous finite-time convergence…

Exponential integrators are special time discretization methods where the traditional linear system solves used by implicit schemes are replaced with computing the action of matrix exponential-like functions on a vector. A very general…

Seismic imaging is a major challenge in geophysics with broad applications. It involves solving wave propagation equations with absorbing boundary conditions (ABC) multiple times. This drives the need for accurate and efficient numerical…

A variable stepsize exponential multistep integrator, with contour integral approximation of the operator-valued exponential functions, is proposed for solving semilinear parabolic equations with nonsmooth initial data. By this approach,…

In the last two decades, significant effort has been put in understanding and designing so-called structure-preserving numerical methods for the simulation of mechanical systems. Geometric integrators attempt to preserve the geometry…

In this paper, we begin the exploration of vertex-ordering problems through the lens of exponential-time approximation algorithms. In particular, we ask the following question: Can we simultaneously beat the running times of the fastest…

This paper addresses the problem of efficiently computing higher-order variational integrators in simulation and trajectory optimization of mechanical systems as those often found in robotic applications. We develop $O(n)$ algorithms to…