ASGarD: Adaptive Sparse Grid Discretization

Abstract

Many areas of science exhibit physical processes that are described by high dimensional partial differential equations (PDEs), e.g., the 4D, 5D and 6D models describing magnetized fusion plasmas, models describing quantum chemistry, or derivatives pricing. Such problems are affected by the so-called “curse of dimensionality” where the number of degrees of freedom (or unknowns) required to be solved for scales as N D where N is the number of grid points in any given dimension D. A simple, albeit naive, 6D example is demonstrated in the left panel of Figure 1. With N = 1000 grid points in each dimension, the memory required just to store the solution vector, not to mention forming the matrix required to advance such a system in time, would exceed an exabyte - and also the available memory on the largest of supercomputers available today. The right panel of Figure 1 demonstrates potential savings for a range of problem dimensionalities and grid resolution. While there are methods to simulate such high-dimensional systems, they are mostly based on Monte-Carlo methods, which rely on a statistical sampling such that the resulting solutions include noise. Since the noise in such methods can only be reduced at a rate proportional to $\sqrt{N_p}$ where N p is the number of Monte-Carlo samples, there is a need for continuum, or grid/mesh-based methods for high-dimensional problems, which both do not suffer from noise and bypass the curse of dimensionality. We present a simulation framework that provides such a method using adaptive sparse grids.