8 - Tractability of Multivariate Problems

Summary

Abstract

In this paper I present a history of tractability of continuous problems, which has its beginning in the successful numerical tests for highdimensional integration of finance problems. Tractability results will be illustrated for two multivariate problems, integration and linear tensor products problems, in the worst case setting. My talk at FoCM'08 in Hong Kong and this paper are based on the book Tractability of Multivariate Problems, written jointly with Erich Novak. The first volume of our book has been recently published by the European Mathematical Society.

Introduction

Many people have recently become interested in studying the tractability of continuous problems. This area of research addresses the computational complexity of multivariate problems defined on spaces of functions of d variables, with d that can be in the hundreds or thousands; in fact, d can even be arbitrarily large. Such problems occur in numerous applications including physics, chemistry, finance, economics, and the computational sciences.

As with all problems arising in information-based complexity, we want to solve multivariate problems to within ∈, using algorithms that use finitely many functions values or values of some linear functionals. Let n(∈, d) be the minimal number of function values or linear functionals that is needed to compute the solution of the d-variate problem to within ∈.

For many multivariate problems defined over standard spaces of functions n(∈, d) is exponentially large in d.