Title: What some people do to their "high-dimensional" integrals
Authors: Nuyens, Dirk # ×
Issue Date: Apr-2009
Conference: Departemental seminar location:School of Mathematical and Physical Sciences, University of Newcastle, Australia date:23 April 2009
Abstract: This talk gives an introduction to quasi-Monte Carlo methods
for high-dimensional integrals.
Such methods apply low-discrepancy points in an equal-weights quadrature rule,
in contrast to the Monte Carlo method which uses random points.
Low-discrepancy point sets are deterministic point sets dependent on some parameters and with a specific structure.

We first start by motivating the usage of Monte Carlo and then quasi-Monte Carlo
after which we then explore some of the recent developments.
These topics include: worst-case errors in reproducing kernel Hilbert spaces,
construction of lattice rules/sequences, the effective dimension, and higher order
of convergence.

In the minds of many, quasi-Monte Carlo methods seem to share the bad stanza of the Monte Carlo method: a brute force method of last resort with slow order of convergence, i.e., $O(N^{-1/2})$.
This is not so.
While the standard rate of convergence for quasi-Monte Carlo is rather slow, being $O(N^{-1})$, the theory shows that these methods achieve the optimal rate of convergence in many interesting function spaces.
E.g., in function spaces with higher smoothness one can have $O(N^{-\alpha})$, $\alpha > 1$.
This will be illustrated by numerical examples.

This talk is meant for a general audience.
Publication status: published
KU Leuven publication type: AMa
Appears in Collections:NUMA, Numerical Analysis and Applied Mathematics Section
× corresponding author
# (joint) last author

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