Polynomial-Time Algorithms for Multivariate Linear Problems with Finite-Order Weights: Average Case Setting |
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Authors: | G W Wasilkowski H Wo?niakowski |
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Institution: | (1) Department of Computer Science, University of Kentucky, Lexington, KY 40506, USA;(2) Department of Computer Science, Columbia University, New York, NY 10027, USA;(3) Institute of Applied Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland |
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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
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
for approximation.
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Keywords: | Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 65D15 H1A46 H1A63 H1A65 |
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