A LOW LEVEL INTRODUCTION TO HIGH-DIMENSIONAL SPARSE GRIDS

John Burkardt
School of Computational Science
Florida State University

This talk is intended to be a very accessible introduction 
to some of the ideas involved in Smolyak's sparse grid 
construction for estimating integrals in high dimension.

The integral collects many small items into a single whole.
It is the natural way to describe, analyze and simulate
geometry, physics, probability, and finance.  Although it
was 'discovered' as a tool for physical space, it is 
essential in higher dimensional abstract spaces,
particularly in simulation or the analysis of stochastic
processes.  

Approximation techniques must be used for such problems.
In high dimensions, every choice we make comes with very
high costs.  We will discuss the accuracy, efficiency,
and flexibility of product rules, Monte Carlo, and Quasi 
Monte Carlo methods.

We will present a sparse grid method that is somewhat
like the product rule, with a similar guarantee of accuracy,
but at a drastically reduced cost.  We will discuss an
application involving a stochastic partial differential
equation, in which the higher dimensions represent degrees
of freedom in the stochastic disturbance.