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Title: Multi-fidelity stochastic collocation method for computation of statistical moments

Abstract

We present an efficient numerical algorithm to approximate the statistical moments of stochastic problems, in the presence of models with different fidelities. The method extends the multi-fidelity approximation method developed in . By combining the efficiency of low-fidelity models and the accuracy of high-fidelity models, our method exhibits fast convergence with a limited number of high-fidelity simulations. We establish an error bound of the method and present several numerical examples to demonstrate the efficiency and applicability of the multi-fidelity algorithm.

Authors:
 [1];  [2];  [3]
  1. Department of Mathematics, University of Iowa, Iowa City, IA 52242 (United States)
  2. Department of Mathematics, University of Utah, Salt Lake City, UT 84112 (United States)
  3. Department of Mathematics, The Ohio State University, Columbus, OH 43210 (United States)
Publication Date:
OSTI Identifier:
22622309
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Computational Physics; Journal Volume: 341; Other Information: Copyright (c) 2017 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICAL METHODS AND COMPUTING; ACCURACY; ALGORITHMS; APPROXIMATIONS; COMPUTERIZED SIMULATION; CONVERGENCE; EFFICIENCY; ERRORS; STATISTICAL MODELS; STOCHASTIC PROCESSES

Citation Formats

Zhu, Xueyu, E-mail: xueyu-zhu@uiowa.edu, Linebarger, Erin M., E-mail: aerinline@sci.utah.edu, and Xiu, Dongbin, E-mail: xiu.16@osu.edu. Multi-fidelity stochastic collocation method for computation of statistical moments. United States: N. p., 2017. Web. doi:10.1016/J.JCP.2017.04.022.
Zhu, Xueyu, E-mail: xueyu-zhu@uiowa.edu, Linebarger, Erin M., E-mail: aerinline@sci.utah.edu, & Xiu, Dongbin, E-mail: xiu.16@osu.edu. Multi-fidelity stochastic collocation method for computation of statistical moments. United States. doi:10.1016/J.JCP.2017.04.022.
Zhu, Xueyu, E-mail: xueyu-zhu@uiowa.edu, Linebarger, Erin M., E-mail: aerinline@sci.utah.edu, and Xiu, Dongbin, E-mail: xiu.16@osu.edu. 2017. "Multi-fidelity stochastic collocation method for computation of statistical moments". United States. doi:10.1016/J.JCP.2017.04.022.
@article{osti_22622309,
title = {Multi-fidelity stochastic collocation method for computation of statistical moments},
author = {Zhu, Xueyu, E-mail: xueyu-zhu@uiowa.edu and Linebarger, Erin M., E-mail: aerinline@sci.utah.edu and Xiu, Dongbin, E-mail: xiu.16@osu.edu},
abstractNote = {We present an efficient numerical algorithm to approximate the statistical moments of stochastic problems, in the presence of models with different fidelities. The method extends the multi-fidelity approximation method developed in . By combining the efficiency of low-fidelity models and the accuracy of high-fidelity models, our method exhibits fast convergence with a limited number of high-fidelity simulations. We establish an error bound of the method and present several numerical examples to demonstrate the efficiency and applicability of the multi-fidelity algorithm.},
doi = {10.1016/J.JCP.2017.04.022},
journal = {Journal of Computational Physics},
number = ,
volume = 341,
place = {United States},
year = 2017,
month = 7
}
  • This paper presents a data-driven framework for performing uncertainty quantification (UQ) by choosing a stochastic model that accurately describes the sources of uncertainty in a system. This model is propagated through an appropriate response surface function that approximates the behavior of this system using stochastic collocation. Given a sample of data describing the uncertainty in the inputs, our goal is to estimate a probability density function (PDF) using the kernel moment matching (KMM) method so that this PDF can be used to accurately reproduce statistics like mean and variance of the response surface function. Instead of constraining the PDF tomore » be optimal for a particular response function, we show that we can use the properties of stochastic collocation to make the estimated PDF optimal for a wide variety of response functions. We contrast this method with other traditional procedures that rely on the Maximum Likelihood approach, like kernel density estimation (KDE) and its adaptive modification (AKDE). We argue that this modified KMM method tries to preserve what is known from the given data and is the better approach when the available data is limited in quantity. We test the performance of these methods for both univariate and multivariate density estimation by sampling random datasets from known PDFs and then measuring the accuracy of the estimated PDFs, using the known PDF as a reference. Comparing the output mean and variance estimated with the empirical moments using the raw data sample as well as the actual moments using the known PDF, we show that the KMM method performs better than KDE and AKDE in predicting these moments with greater accuracy. This improvement in accuracy is also demonstrated for the case of UQ in electrostatic and electrothermomechanical microactuators. We show how our framework results in the accurate computation of statistics in micromechanical systems.« less
  • Abstract not provided.
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