## 15. ÖMG-Kongress

Jahrestagung der Deutschen Mathematikervereinigung

#### 16. bis 22. September 2001 in Wien

**Minisymposium Zahlentheoretische Algorithmen und ihre Anwendungen**

Dienstag, 18. September 2001, 17.20, Audimax der Universität Wien

**Combinatorial problems arising in high dimensional integration and approximation**

**Art Owen**,

**Stanford University**

Integrating and approximating a function over the
unit hyper-cube in dimension d, are a fundamental numerical
tasks with many applications. Integration methods are
widely used in valuation of financial derivatives.
Approximation methods are used in the design of complex
products like semiconductors and aircraft.

For small d, simple methods that evaluate the target function
on a grid work very well. As the dimension d increases, such
tensor product methods become computationally infeasible.

Sampling methods, such as Monte Carlo, are much
less sensitive to the dimension effect, and are the
practical methods of choice for large d. Better sampling
methods, such as quasi-Monte Carlo sampling, use combinatorics
to define the input points.

This talk surveys recent work in sampling methods for
integration and approximation, and presents some challenges
for combinatorial methods. One challenge is to develop
point sets with some equidistribution properties of (t,m,s)-nets
but for which the equidistribution is much better for
relatively important dimensions. Another is to study performance
bounds for equidistribution, analogous to the bounds on orthogonal
array strength, for very high dimensions. A third challenge
is to construct point sets in the unit cube that are better
for quasi-regression approximation than are random points.
A fourth challenge is to construct point sets that are better
for integrating very smooth functions.

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