1 research outputs found
Rank-adaptive tensor methods for high-dimensional nonlinear PDEs
We present a new rank-adaptive tensor method to compute the numerical
solution of high-dimensional nonlinear PDEs. The new 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 algorithm 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 the covariance matrix of the 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.Comment: 24 pages, 10 figure