Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation

  author = {Gerstner, T. and Griebel, M.},
  title = {Dimension--Adaptive Tensor--Product Quadrature},
  journal = {Computing},
  volume = {71},
  number = {1},
  pages = {65--87},
  pdf = { 1},
  abstract = { 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 developed
		  and presented in this paper aims to find such an expansion
		  automatically. It is based on the sparse grid method which
		  has been 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. },
  year = {2003},
  annote = {article,C2,ALM}