AI summary · arXiv: 2103.08632v1 · math.NA
Solving Backward Doubly Stochastic Differential Equations through Splitting Schemes
The essentials
- Authors propose a splitting scheme for solving backward doubly stochastic differential equations
- Main idea is to decompose the equation into a backward stochastic differential equation and a stochastic differential equation
- Approximate the two equations using first-order finite difference schemes
- Conduct numerical experiments to demonstrate convergence rate and accuracy of the proposed scheme
- Backward doubly stochastic differential equations involve both forward and backward components, making them difficult to solve directly
- By splitting the equation, existing numerical methods can be applied to each component individually
- Preliminary concepts related to Brownian motions are introduced and collections of sigma-algebras associated with these motions are defined
- Conditions on step size and constants are established for convergence of the scheme
- Proposed method provides a novel approach for solving backward doubly stochastic differential equations through a splitting scheme
- Shows promise in accurately approximating solutions and provides valuable insights into tackling these complex equations numerically
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
-
DelayDiffEq: Generating Delay Differential Equation Solvers via Recursive Embedding of Ordinary Differential Equation SolversDavid Widmann, Chris Rackauckas · 2022 · Numerical Analysis · AI summary available
DelayDiffEq develops numerical solvers for delay differential equations, whereas this paper constructs a numerical scheme for backward doubly stochastic differential equations.
Abstract
Traditional solvers for delay differential equations (DDEs) are designed around only a single method and do not effectively use the infrastructure of their more-developed ordinary differential equation (ODE) counterparts. In this work we present DelayDiffEq, a Julia package for numerically solving delay differential equations (DDEs) which leverages the multitude of numerical algorithms in OrdinaryDiffEq for solving both stiff and non-stiff ODEs, and manages to solve challenging stiff DDEs. We describe how compiling the ODE integrator within itself, and accounting for discontinuity propagation, leads to a design that is effective for DDEs while using all of the ODE internals. We highlight som…
-
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
The overview reviews deep-learning and Monte Carlo approximations for high-dimensional PDEs, while this paper approximates backward doubly stochastic differential equations with a splitting scheme.
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
The mpEDMD algorithm computes spectral information for measure-preserving dynamical systems, rather than solving backward doubly stochastic differential equations.
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…
-
A robust solution strategy for the Cahn-Larché equationsErlend Storvik, Jakub Wiktor Both, Jan Martin Nordbotten et al. · 2022 · Numerical Analysis · AI summary available
The Cahn-Larché study proposes a time discretization for a coupled phase-field and elasticity system, unlike this paper's scheme for backward doubly stochastic differential equations.
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…
-
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
The anisotropic diffusion study uses operator splitting and finite differences for a diffusion equation, while this paper splits a backward doubly stochastic differential equation into two stochastic equations.
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 …
-
Tensor-Product Split-Simplex Summation-By-Parts OperatorsZelalem Arega Worku, Jason E. Hicken, David W. Zingg · 2024 · Numerical Analysis · AI summary available
The study constructs summation-by-parts operators for stable discretizations, a numerical discretization tool distinct from this paper's first-order finite-difference splitting scheme.
Abstract
We present an approach to construct efficient sparse summation-by-parts (SBP) operators on triangles and tetrahedra with a tensor-product structure. The operators are constructed by splitting the simplices into quadrilateral or hexahedral subdomains, mapping tensor-product SBP operators onto the subdomains, and assembling back using a continuous-Galerkin-type procedure. These tensor-product split-simplex operators do not have repeated degrees of freedom at the interior interfaces between the split subdomains. Furthermore, they satisfy the SBP property by construction, leading to stable discretizations. The accuracy and sparsity of the operators substantially enhance the efficiency of SBP dis…
-
Isomeric trees and the order of Runge--Kutta methodsJohn C. Butcher, Helmut Podhaisky · 2021 · Numerical Analysis · AI summary available
The Runge–Kutta study derives order conditions and methods for ordinary differential equations, while this paper establishes a first-order scheme for backward doubly stochastic differential equations.
Abstract
The conditions for a Runge--Kutta method to be of order $p$ with $p\ge 5$ for a scalar non-autonomous problem are a proper subset of the order conditions for a vector problem. Nevertheless, Runge--Kutta methods that were derived historically only for scalar problems happened to be of the same order for vector problems. We relate the order conditions for scalar problems to factorisations of the Runge--Kutta trees into "atomic stumps" and enumerate those conditions up to $p=20$. Using a special search procedure over unsatisfied order conditions, new Runge--Kutta methods of "ambiguous orders" five and six are constructed. These are used to verify the validity of the results.
-
An embedded corrector problem for homogenization. Part II: Algorithms and discretizationEric Cancès, Virginie Ehrlacher, Frederic Legoll et al. · 2018 · Numerical Analysis · Not summarized yet
The homogenization study develops algorithms and discretization for an embedded corrector problem, a numerical PDE task distinct from this paper's stochastic equation splitting scheme.
Abstract
This contribution is the numerically oriented companion article of the work [E. Canc\`es, V. Ehrlacher, F. Legoll, B. Stamm and S. Xiang, arxiv preprint 1807.05131]. We focus here on the numerical resolution of the embedded corrector problem introduced in [E. Canc\`es, V. Ehrlacher, F. Legoll and B. Stamm, CRAS 2015; E. Canc\`es, V. Ehrlacher, F. Legoll, B. Stamm and S. Xiang, arxiv preprint 1807.05131] in the context of homogenization of diffusion equations. Our approach consists in considering a corrector-type problem, posed on the whole space, but with a diffusion matrix which is constant outside some bounded domain. In [E. Canc\`es, V. Ehrlacher, F. Legoll, B. Stamm and S. Xiang, arxiv p…
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.