2017/12/19 by Alexander D. Gilbert, Gilbert, Alexander D., Frances Y. Kuo +5
Decision Sciences · Mathematics · #FOS: Mathematics #Mathematical Approximation and Integration #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design #Scientific Measurement and Uncertainty Evaluation
paper · pdf · doi:10.48550/arxiv.1712.06782
openalex publication_date 2017/12/19 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
In this paper we focus on efficient implementations of the Multivariate\nDecomposition Method (MDM) for approximating integrals of \∞-variate\nfunctions. Such \∞-variate integrals occur for example as expectations in\nuncertainty quantification. Starting with the anchored decomposition f =\n\∑_ mathfraku\⊂\ℕ f_ mathfraku, where the sum is over all\nfinite subsets of \ℕ and each f_ mathfraku depends only on the\nvariables xj with j\∈ mathfraku, our MDM algorithm approximates the\nintegral of f by first truncating the sum to some `active set' and then\napproximating the integral of the remaining functions f_ mathfraku\nterm-by-term using Smolyak or (randomized) quasi-Monte Carlo (QMC) quadratures.\nThe anchored decomposition allows us to compute f_ mathfraku explicitly by\nfunction evaluations of f. Given the specification of the active set and\ntheoretically derived parameters of the quadrature rules, we exploit structures\nin both the formula for computing f_ mathfraku and the quadrature rules to\ndevelop computationally efficient strategies to implement the MDM in various\nscenarios. In particular, we avoid repeated function evaluations at the same\npoint. We provide numerical results for a test function to demonstrate the\neffectiveness of the algorithm.\n