A dimension-adaptive approach to multivariate numerical integration
Thomas Gerstner (Uni Bonn)
We consider the numerical integration of multivariate functions defined
over the unit hypercube. Here, we especially address the high-dimensional
case, where in general the curse of dimension is encountered. Due
to the concentration of measure phenomenon, such functions can often be well
approximated by sums of lower-dimensional terms. The problem, however,
is to find a good expansion given little knowledge of the integrand itself.
The dimension-adaptive quadrature method which is presented here
aims to find such an expansion automatically. It is based on the
sparse grid method which has shown to give good results for low- and
moderate-dimensional problems. The dimension-adaptive quadrature
method tries to find important dimensions and adaptively refines in this
respect guided by suitable error estimators. This leads to an approach
which is based on generalized sparse grid index sets. We propose efficient
data structures for the storage and traversal of the index sets and discuss
an efficient implementation of the algorithm.
The performance of the method is illustrated by several numerical examples
from computational physics and finance where dimension reduction is obtained
from the Brownian bridge discretization of the underlying stochastic process.