Advancement: Numerical tensor methods for high-dimensional nonlinear PDEs

Speaker Name
Alec Dektor
Speaker Title
Applied Mathematics Ph.D. Student
Speaker Organization
Applied Mathematics Ph.D.
Start Time
End Time
Virtual Event

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Description: High-dimensional partial differential equations (PDEs) arise in many areas of engineering, physical sciences and mathematics. Computing their solution numerically is a challenging problem that requires approximating high-dimensional functions and developing appropriate numerical schemes to compute such functions accurately.

In this advancement, 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 rank-adaptive algorithm based on a thresholding criterion that limits the component of the PDE velocity vector normal to the FTT tensor manifold. This 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 stability issues which arise at locations on the tensor manifold with high curvature. Numerical applications are presented and discussed for linear and nonlinear advection problems in two dimensions, and for a four-dimensional Fokker-Planck equation.

We also describe our ongoing research efforts to build upon our existing method and apply our methods to high-dimensional problems.

Daniele Venturi