AI summary · arXiv: 2208.12879v1 · math.NA
DelayDiffEq: Generating Delay Differential Equation Solvers via Recursive Embedding of Ordinary Differential Equation Solvers
The essentials
- Delay Differential Equations (DDEs) involve time delays in system evolution
- Traditional DDE solvers lack effective utilization of ODE infrastructure
- DelayDiffEq is a Julia package that solves DDEs using numerical algorithms from OrdinaryDiffEq
- DelayDiffEq can handle challenging stiff DDEs by compiling the ODE integrator within itself and accounting for discontinuity propagation
- Demonstrates efficiency in solving difficult equations and handling problems with time-scale separation
- Enables generality and flexibility, can be repurposed for stochastic delay differential equations
- Offers improved performance and versatility compared to traditional methods
Key points written by AI: they may contain errors, check them against the paper.
Ask questions about this paper to our AI assistant
How to chat with this paper? Illustrated guide
You can also chat with multiple papers at once here.
Keep this conversation — a free account saves your questions and answers, your collections of papers and your alerts.
Create a free accountTools for this paper
Advanced tools Premium
For researchers who follow a field closely: citation alerts, journal-club slides, version comparison and Zotero export. Discover Premium
Papers around this one
See them on a mapRanked by how close their title and abstract are to this paper’s. Select some to read them together.
Closest papers
-
Solving Backward Doubly Stochastic Differential Equations through Splitting SchemesFeng Bao, Yanzhao Cao, He Zhang · 2021 · Numerical Analysis · AI summary available
Abstract
A splitting scheme for backward doubly stochastic differential equations is proposed. The main idea is to decompose a backward doubly stochastic differential equation into a backward stochastic differential equation and a stochastic differential equation. The backward stochastic differential equation and the stochastic differential equation are then approximated by first order finite difference schemes, which results in a first order scheme for the backward doubly stochastic differential equation. Numerical experiments are conducted to illustrate the convergence rate of the proposed scheme.
-
An overview on deep learning-based approximation methods for partial differential equationsChristian Beck, Martin Hutzenthaler, Arnulf Jentzen et al. · 2020 · Discrete Contin. Dyn. Syst. Ser. B 28 (2023), no. 6, 3697-3746 · AI summary available
Abstract
It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). Recently, several deep learning-based approximation algorithms for attacking this problem have been proposed and tested numerically on a number of examples of high-dimensional PDEs. This has given rise to a lively field of research in which deep learning-based methods and related Monte Carlo methods are applied to the approximation of high-dimensional PDEs. In this article we offer an introduction to this field of research by revisiting selected mathematical results related to deep learning approximation methods for PDEs and reviewing the main ide…
-
The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical SystemsMatthew J. Colbrook · 2022 · Numerical Analysis · AI summary available
Abstract
Koopman operators globally linearize nonlinear dynamical systems and their spectral information is a powerful tool for the analysis and decomposition of nonlinear dynamical systems. However, Koopman operators are infinite-dimensional, and computing their spectral information is a considerable challenge. We introduce measure-preserving extended dynamic mode decomposition ($\texttt{mpEDMD}$), the first truncation method whose eigendecomposition converges to the spectral quantities of Koopman operators for general measure-preserving dynamical systems. $\texttt{mpEDMD}$ is a data-driven algorithm based on an orthogonal Procrustes problem that enforces measure-preserving truncations of Koopman op…
-
Compatible finite element methods for geophysical fluid dynamicsColin J. Cotter · 2023 · Numerical Analysis · AI summary available
Abstract
This article surveys research on the application of compatible finite element methods to large scale atmosphere and ocean simulation. Compatible finite element methods extend Arakawa's C-grid finite difference scheme to the finite element world. They are constructed from a discrete de Rham complex, which is a sequence of finite element spaces which are linked by the operators of differential calculus. The use of discrete de Rham complexes to solve partial differential equations is well established, but in this article we focus on the specifics of dynamical cores for simulating weather, oceans and climate. The most important consequence of the discrete de Rham complex is the Hodge-Helmholtz d…
-
On the implementation of the eXtended Finite Element Method (XFEM) for interface problemsThomas Carraro, Sven Wetterauer · 2015 · Numerical Analysis · AI summary available
Abstract
The eXtended Finite Element Method (XFEM) is used to solve interface problems with an unfitted mesh. We present an implementation of the XFEM in the FEM-library deal.II. The main parts of the implementation are (i) the appropriate quadrature rule; (ii) the shape functions for the extended part of the finite element formulation; (iii) the boundary and interface conditions. We show how to handle the XFEM formulation providing a code that demonstrates the solution of two exemplary interface problems for a strong and a weak discontinuity respectively. In the weak discontinuity case, the loss of conformity due to the blending effect and its remedy are discussed. Furthermore, the optimal convergen…
-
Provably stable numerical method for the anisotropic diffusion equation in toroidally confined magnetic fieldsDean Muir, Kenneth Duru, Matthew Hole et al. · 2023 · Numerical Analysis · AI summary available
Abstract
We present a novel numerical method for solving the anisotropic diffusion equation in toroidally confined magnetic fields which is efficient, accurate and provably stable. The continuous problem is written in terms of a derivative operator for the perpendicular transport and a linear operator, obtained through field line tracing, for the parallel transport. We derive energy estimates of the solution of the continuous initial boundary value problem. A discrete formulation is presented using operator splitting in time with the summation by parts finite difference approximation of spatial derivatives for the perpendicular diffusion operator. Weak penalty procedures are derived for implementing …
-
Gradual Variation Analysis for Groundwater FlowLi Chen · 2010 · Numerical Analysis · Not summarized yet
Abstract
Groundwater flow in Washington DC greatly influences the surface water quality in urban areas. The current methods of flow estimation, based on Darcy's Law and the groundwater flow equation, can be described by the diffusion equation (the transient flow) and the Laplace equation (the steady-state flow). The Laplace equation is a simplification of the diffusion equation under the condition that the aquifer has a recharging boundary. The practical way of calculation is to use numerical methods to solve these equations. The most popular system is called MODFLOW, which was developed by USGS. MODFLOW is based on the finite-difference method in rectangular Cartesian coordinates. MODFLOW can be vie…
-
A robust solution strategy for the Cahn-Larché equationsErlend Storvik, Jakub Wiktor Both, Jan Martin Nordbotten et al. · 2022 · Numerical Analysis · AI summary available
Abstract
In this paper we propose a solution strategy for the Cahn-Larch\'e equations, which is a model for linearized elasticity in a medium with two elastic phases that evolve subject to a Ginzburg-Landau type energy functional. The system can be seen as a combination of the Cahn-Hilliard regularized interface equation and linearized elasticity, and is non-linearly coupled, has a fourth order term that comes from the Cahn-Hilliard subsystem, and is non-convex and nonlinear in both the phase-field and displacement variables. We propose a novel semi-implicit discretization in time that uses a standard convex-concave splitting method of the nonlinear double-well potential, as well as special treatment…
Go further
How are these papers found?
We compare the title and abstract of this paper with those of every paper summarized on this site, and with the papers Semantic Scholar recommends, those it cites and those citing it (for a much-cited paper, its most recent citations and, from OpenAlex, the most cited ones). Among the prior work and the follow-ups, the papers that use each other in depth, a method it uses and the much-cited follow-ups come a little higher. The levels (very close, close, related) come from the similarity of the texts, not from a judgment of quality; * marks a paper without an abstract, compared by its title. The sentence under a paper is written by an AI from the two abstracts: check it in the papers. A paper marked not relevant by three readers leaves the list.
For now only among the papers summarized on this site: search again to add the prior work, the follow-ups and Semantic Scholar’s recommendations.
The previous summary was created more than a year ago and can be re-run (if necessary) by clicking on the Run button below.
Disclaimer: The AI-based summarization tool and virtual assistant provided on this website may not always provide accurate and complete summaries or responses. We encourage you to carefully review and evaluate the generated content to ensure its quality and relevance to your needs.