## Abstract

We study multivariate linear problems in the average case setting with respect to a zero-mean Gaussian measure whose covariance kernel has a finite-order weights structure. This means that the measure is concentrated on a Banach space of d-variate functions that are sums of functions of at most q ^{*} variables and the influence of each such term depends on a given weight. Here q ^{*} is fixed whereas d varies and can be arbitrarily large. For arbitrary finite-order weights, based on Smolyak's algorithm, we construct polynomial-time algorithms that use standard information. That is, algorithms that solve the d-variate problem to within ε using of order -pd q function values modulo a power of ln∈ε ^{-1}. Here p is the exponent which measures the difficulty of the univariate (d=1) problem, and the power of ln∈ε ^{-1} is independent of d. We also present a necessary and sufficient condition on finite-order weights for which we obtain strongly polynomial-time algorithms, i.e., when the number of function values is independent of d and polynomial in ε ^{-1}. The exponent of ε ^{-1} may be, however, larger than p. We illustrate the results by two multivariate problems: integration and function approximation. For the univariate case we assume the r-folded Wiener measure. Then p=1/(r+1) for integration and p=1/(r+ 1 2) for approximation.

Original language | English |
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Pages (from-to) | 105-132 |

Number of pages | 28 |

Journal | Foundations of Computational Mathematics |

Volume | 9 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2009 |

## Keywords

- Average case setting
- Finite-order weights
- Multivariate linear problems
- Polynomial-time algorithms
- Small effective dimension
- Tractability

## ASJC Scopus subject areas

- Analysis
- Computational Mathematics
- Computational Theory and Mathematics
- Applied Mathematics