Spectral-Galerkin Methods for High-Dimensional PDEs

PIs: Jie Shen(Dep. Mathematics, Purdue University)
Ahmed Sameh(Dep. Computer Science, Purdue University)
Student: Heejun Choi(Dep. Mathematics, Purdue Univerisity)

Project Description

Many scientific and engineering applications require solving high-dimensional partial differential equations. Examples include fundamental equations in physics such as Boltzmann equations, Fokker-Planck equations, non-linear Schrodinger equations, and mathematical models in finance. Current high-order methods for solving PDEs in multi-dimensions are usually based on tensor product formulation. Assuming M points are used in each direction, the total number of unknowns for the spatial variables is N=M^d where d is the dimension. Thus, even for problems with a moderate dimension, current computing power does not allow a reasonable full tensor product grid needed to resolve the physical solution. Hence, it is of critical important to develop new numerical methods which scale well with dimension. A popular, and often effective, approach is the Monte Carlo-based methods, whose performance is in principle independent of dimension. However, these methods are plagued with the problems of slow convergence and noisy output, so these are not suitable for problems with stingent accuracy requirements. The aim of this project is to develop efficient, high-order, multi-resolution methods for high-dimensional problems. Our method will be based on hyperbolic cross approximation and sparse grid interpolation.

nodes spectrum

Figure: Sparse grid and “hyperbolic cross” frequence space

Even with the use of sparse grids, high-dimensional problems such as those in Boltzmann equations and Fokker-Planck equations require substantial computing resource which can not be met on a workstation. Hence, it is necessary to perform parallel computations. However, it is highly non-trivial to parallelize the code generated from the multi-resolution and hyperbolic cross structures, as there is no suitable mesh structure which would allow a good load balence. Hence, an important task of the pro ject is to develop a parallel, iterative solution procedure based on the algebraic structure of the linear systems. It is hoped that this project will lead to numerical algorithms which allow us to perform direct numerical simulation for some high dimensional problems.

This research is supported by PRF Special Incentive Research Grant.