Journal ArticleDOI
Dimension-adaptive tensor-product quadrature
Thomas Gerstner,Michael Griebel +1 more
TLDR
The dimension–adaptive quadrature method is developed and presented, based on the sparse grid method, which tries to find important dimensions and adaptively refines in this respect guided by suitable error estimators, and leads to an approach which is based on generalized sparse grid index sets.Abstract:
We consider the numerical integration of multivariate functions defined over the unit hypercube. Here, we especially address the high-dimensional case, where in general the curse of dimension is encountered. Due to the concentration of measure phenomenon, such functions can often be well approximated by sums of lower-dimensional terms. The problem, however, is to find a good expansion given little knowledge of the integrand itself.The dimension-adaptive quadrature method which is developed and presented in this paper aims to find such an expansion automatically. It is based on the sparse grid method which has been shown to give good results for low- and moderate-dimensional problems. The dimension-adaptive quadrature method tries to find important dimensions and adaptively refines in this respect guided by suitable error estimators. This leads to an approach which is based on generalized sparse grid index sets. We propose efficient data structures for the storage and traversal of the index sets and discuss an efficient implementation of the algorithm.The performance of the method is illustrated by several numerical examples from computational physics and finance where dimension reduction is obtained from the Brownian bridge discretization of the underlying stochastic process.read more
Citations
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Journal ArticleDOI
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
TL;DR: A rigorous convergence analysis is provided and exponential convergence of the “probability error” with respect to the number of Gauss points in each direction in the probability space is demonstrated, under some regularity assumptions on the random input data.
Journal ArticleDOI
A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data
TL;DR: This work demonstrates algebraic convergence with respect to the total number of collocation points and quantifies the effect of the dimension of the problem (number of input random variables) in the final estimates, indicating for which problems the sparse grid stochastic collocation method is more efficient than Monte Carlo.
Journal ArticleDOI
Adaptive sparse polynomial chaos expansion based on least angle regression
Géraud Blatman,Bruno Sudret +1 more
TL;DR: A non intrusive method that builds a sparse PC expansion, which may be obtained at a reduced computational cost compared to the classical ''full'' PC approximation.
ReportDOI
DAKOTA : a multilevel parallel object-oriented framework for design optimization, parameter estimation, uncertainty quantification, and sensitivity analysis. Version 5.0, user's manual.
Michael S Eldred,Keith R. Dalbey,William J. Bohnhoff,Brian M. Adams,Laura Painton Swiler,Patricia Diane Hough,John Eddy,Karen H. Haskell +7 more
TL;DR: This report serves as a reference manual for the commands specification for the DAKOTA software, providing input overviews, option descriptions, and example specifications.
Journal ArticleDOI
An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis
Géraud Blatman,Bruno Sudret +1 more
TL;DR: A non-intrusive method that builds a sparse PC expansion and an adaptive regression-based algorithm is proposed for automatically detecting the significant coefficients of the PC expansion in a suitable polynomial chaos basis.
References
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Dynamic Programming
TL;DR: The more the authors study the information processing aspects of the mind, the more perplexed and impressed they become, and it will be a very long time before they understand these processes sufficiently to reproduce them.
Journal ArticleDOI
Multivariate Adaptive Regression Splines
TL;DR: In this article, a new method is presented for flexible regression modeling of high dimensional data, which takes the form of an expansion in product spline basis functions, where the number of basis functions as well as the parameters associated with each one (product degree and knot locations) are automatically determined by the data.
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Spline models for observational data
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OtherDOI
Generalized Additive Models
Trevor Hastie,Robert Tibshirani +1 more
TL;DR: The generalized additive model (GA) as discussed by the authors is a generalization of the generalized linear model, which replaces the linear model with a sum of smooth functions in an iterative procedure called local scoring algorithm.