Related papers: A Framework for Solving Parabolic Partial Differen…
In this paper we consider multi-dimensional partial differential equations of parabolic type involving divergence form operators that possess a discontinuous coefficient matrix along some smooth interface. The solution of the equation is…
In this paper, we study the convergence behavior of the diffuse domain method (DDM) for solving a class of second-order parabolic partial differential equations with Neumann boundary condition posed on general irregular domains. The DDM…
An adaptive method for parabolic partial differential equations that combines sparse wavelet expansions in time with adaptive low-rank approximations in the spatial variables is constructed and analyzed. The method is shown to converge and…
High-dimensional partial differential equations (PDEs) are ubiquitous in economics, science and engineering. However, their numerical treatment poses formidable challenges since traditional grid-based methods tend to be frustrated by the…
A common strategy in the numerical solution of partial differential equations is to define a uniform discretization of a tensor-product multi-dimensional logical domain, which is mapped to a physical domain through a given coordinate…
We present an abstract framework for treating the theory of well-posedness of solutions to abstract parabolic partial differential equations on evolving Hilbert spaces. This theory is applicable to variational formulations of PDEs on…
We present a new technique to apply finite element methods to partial differential equations over curved domains. A change of variables along a coordinate transformation satisfying only low regularity assumptions can translate a Poisson…
The numerical solution methods for partial differential equation (PDE) solution allow obtaining a discrete field that converges towards the solution if the method is applied to the correct problem. Nevertheless, the numerical methods…
In this paper, for solving a class of linear parabolic equations in rectangular domains, we have proposed an efficient Parareal exponential integrator finite element method. The proposed method first uses the finite element approximation…
Recent advances in deep learning makes solving parabolic partial differential equations (PDEs) in high dimensional spaces possible via forward-backward stochastic differential equation (FBSDE) formulations. The implementation of most…
Hamilton-Jacobi (HJ) partial differential equations (PDEs) have diverse applications spanning physics, optimal control, game theory, and imaging sciences. This research introduces a first-order optimization-based technique for HJ PDEs,…
We present a new method based on functional tensor decomposition and dynamic tensor approximation to compute the solution of a high-dimensional time-dependent nonlinear partial differential equation (PDE). The idea of dynamic approximation…
We introduce a generic numerical schemes for fully nonlinear parabolic PDEs on the full domain, where the nonlinearity is convex on the Hessian of the solution. The main idea behind this paper is reduction of a fully nonlinear problem to a…
In this paper, we propose an efficient exponential integrator finite element method for solving a class of semilinear parabolic equations in rectangular domains. The proposed method first performs the spatial discretization of the model…
We present a semi-real-time algorithm for minimal-time optimal path planning based on optimal control theory, dynamic programming, and Hamilton-Jacobi (HJ) equations. Partial differential equation (PDE) based optimal path planning methods…
Parabolic partial differential equations (PDEs) appear in many disciplines to model the evolution of various mathematical objects, such as probability flows, value functions in control theory, and derivative prices in finance. It is often…
We propose machine learning methods for solving fully nonlinear partial differential equations (PDEs) with convex Hamiltonian. Our algorithms are conducted in two steps. First the PDE is rewritten in its dual stochastic control…
Partial differential equations are fundamental tools in mathematics,sciences and engineering. This book is mainly an exposition of the various algebraic techniques of solving partial differential equations for exact solutions developed by…
In this paper we introduce a new hierarchical model reduction framework for the Fokker-Planck equation. We reduce the dimension of the equation by a truncated basis expansion in the velocity variable, obtaining a hyperbolic system of…
The problem of solving partial differential equations (PDEs) on manifolds can be considered to be one of the most general problem formulations encountered in computational multi-physics. The required covariant forms of balance laws as well…