Related papers: Efficient Explicit Taylor ODE Integrators with Sym…

In this paper, we develop a regularized higher-order Taylor based method for solving composite (e.g., nonlinear least-squares) problems. At each iteration, we replace each smooth component of the objective function by a higher-order Taylor…

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…

NonlinearSchrodinger.jl is a Julia package with a simple interface for studying solutions of nonlinear Schr\"odinger equations (NLSEs). In approximately ten lines of code, one can perform a simulation of the cubic NLSE using one of 32…

It has been found that residual networks are an Euler discretization of solutions to Ordinary Differential Equations (ODEs). In this paper, we explore a deeper relationship between Transformer and numerical methods of ODEs. We show that a…

In this paper, we introduce an algorithm that provides approximate solutions to semi-linear ordinary differential equations with highly oscillatory solutions, which, after an appropriate change of variables, can be rewritten as…

Function approximation is a generic process in a variety of computational problems, from data interpolation to the solution of differential equations and inverse problems. In this work, a unified approach for such techniques is…

In this paper, we first present an explicit expression for the inverse\emph{} of a type of matrices. As special applications, the inverse of some matrices arising from implicit time integration techniques, such as the well-known implicit…

Stiff systems of ordinary differential equations (ODEs) and sparse training data are common in scientific problems. This paper describes efficient, implicit, vectorized methods for integrating stiff systems of ordinary differential…

Many complex systems can be accurately modeled as a set of coupled time-dependent partial differential equations (PDEs). However, solving such equations can be prohibitively expensive, easily taxing the world's largest supercomputers. One…

This paper investigates, a new class of fractional order Runge-Kutta (FORK) methods for numerical approximation to the solution of fractional differential equations (FDEs). By using the Caputo generalizedTaylor formula and the total…

We present a hybrid a-priori/a-posteriori goal oriented error estimator for a combination of dynamic iteration-based solution of ordinary differential equations discretized by finite elements. Our novel error estimator combines estimates…

We present two new algorithms for the computation of the q-integer linear decomposition of a multivariate polynomial. Such a decomposition is essential for the treatment of q-hypergeometric symbolic summation via creative telescoping and…

We present a boundary integral equation solver for computing Taylor relaxed states in non-axisymmetric solid and shell-like toroidal geometries. The computation of Taylor states in these geometries is a key element for the calculation of…

Scientific studies often require the precise calculation of derivatives. In many cases an analytical calculation is not feasible and one resorts to evaluating derivatives numerically. These are error-prone, especially for higher-order…

This paper proposes a higher-order multiscale computational method for nonlinear thermo-electric coupling problems of composite structures, which possess temperature-dependent material properties and nonlinear Joule heating. The innovative…

When a computer algebra system fails to solve an Ordinary Differential Equation, is this a limitation of its implementation, or a genuine computational barrier? Three traditions bear on the question. Modern computer algebra algorithms can…

Quantum algorithms offer an exponential advantage with respect to the number of dependent variables for solving certain nonlinear ordinary differential equations (ODEs). These algorithms typically begin by transforming the original…

When applied to stiff, linear differential equations with time-dependent forcing, Runge-Kutta methods can exhibit convergence rates lower than predicted by the classical order condition theory. Commonly, this order reduction phenomenon is…

In the paper we analyse the exact solutions to scalar PDEs obtained thanks to summable Taylor series provided by Adomian's decomposition method. We propose the modification of the method which makes the calculations of Taylor coefficients…

This work presents algorithms for the efficient implementation of discontinuous Galerkin methods with explicit time stepping for acoustic wave propagation on unstructured meshes of quadrilaterals or hexahedra. A crucial step towards…