Abstract
We present a new rank-adaptive tensor method to compute the numerical solution of high-dimensional nonlinear PDEs. The method combines functional tensor train (FTT) series expansions, operator splitting time integration, and a new rank-adaptive algorithm based on a thresholding criterion that limits the component of the PDE velocity vector normal to the FTT tensor manifold. This yields a scheme that can add or remove tensor modes adaptively from the PDE solution as time integration proceeds. The new method is designed to improve computational efficiency, accuracy and robustness in numerical integration of high-dimensional problems. In particular, it overcomes well-known computational challenges associated with dynamic tensor integration, including low-rank modeling errors and the need to invert covariance matrices of tensor cores at each time step. Numerical applications are presented and discussed for linear and nonlinear advection problems in two dimensions, and for a four-dimensional Fokker–Planck equation.
Highlights
High-dimensional partial differential equations (PDEs) arise in many areas of engineering, physical sciences and mathematics
The numerical method presented in this work combines all these features, i.e., functional tensor train (FTT) series expansions, operator splitting time integration, and a new rank-adaptive algorithm to add and remove tensor modes from the PDE solution based on a thresholding criterion that limits the component of the velocity vector normal to the FTT tensor manifold
We presented a new rank-adaptive tensor method to integrate high-dimensional nonlinear PDEs
Summary
High-dimensional partial differential equations (PDEs) arise in many areas of engineering, physical sciences and mathematics. High-dimensional PDEs have become central to many new areas of application such as optimal mass transport [27,59], random dynamical systems [57,58], mean field games [19,52], and functionaldifferential equations [55,56]. Computing the numerical solution to high-dimensional PDEs is an extremely challenging problem which has attracted substantial research efforts in recent years. Techniques such as sparse collocation methods [4,9,12,25,41], high-dimensional model representations [3,10,37], deep neural networks [46,47,60], and numerical tensor
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