#
**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.

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.