An Adaptive ANOVA-Based Data-Driven Stochastic Method for Elliptic PDE with Random Coefficients

2012 ◽  
Author(s):  
Xin Hu ◽  
Guang Lin ◽  
Thomas Y. Hou ◽  
Pengchong Yan
Keyword(s):  
Random Coefficients ◽  
Stochastic Method ◽  
Data Driven ◽  
Elliptic Pde

A Data-Driven Stochastic Method for Elliptic PDEs with Random Coefficients

2013 ◽  
Vol 1 (1) ◽  
pp. 452-493 ◽  
Author(s):  
Mulin Cheng ◽  
Thomas Y. Hou ◽  
Mike Yan ◽  
Zhiwen Zhang
Keyword(s):  
Random Coefficients ◽  
Stochastic Method ◽  
Data Driven ◽  
Elliptic Pdes

scholarly journals A Multiscale Data-Driven Stochastic Method for Elliptic PDEs with Random Coefficients

2015 ◽  
Vol 13 (1) ◽  
pp. 173-204 ◽  
Author(s):  
Zhiwen Zhang ◽  
Maolin Ci ◽  
Thomas Y. Hou
Keyword(s):  
Random Coefficients ◽  
Stochastic Method ◽  
Data Driven ◽  
Elliptic Pdes

An Adaptive ANOVA-Based Data-Driven Stochastic Method for Elliptic PDEs with Random Coefficient

2014 ◽  
Vol 16 (2) ◽  
pp. 571-598 ◽  
Author(s):  
Zhiwen Zhang ◽  
Xin Hu ◽  
Thomas Y. Hou ◽  
Guang Lin ◽  
Mike Yan
Keyword(s):  
Stochastic Method ◽  
Data Driven ◽  
High Dimensional ◽  
Compact Representation ◽  
Elliptic Pdes ◽  
Decomposition Technique ◽  
Stochastic Problem ◽  
Forcing Function ◽  
Speed Up ◽  
Stochastic Basis

AbstractIn this paper, we present an adaptive, analysis of variance (ANOVA)-based data-driven stochastic method (ANOVA-DSM) to study the stochastic partial differential equations (SPDEs) in the multi-query setting. Our new method integrates the advantages of both the adaptive ANOVA decomposition technique and the data-driven stochastic method. To handle high-dimensional stochastic problems, we investigate the use of adaptive ANOVA decomposition in the stochastic space as an effective dimension-reduction technique. To improve the slow convergence of the generalized polynomial chaos (gPC) method or stochastic collocation (SC) method, we adopt the data-driven stochastic method (DSM) for speed up. An essential ingredient of the DSM is to construct a set of stochastic basis under which the stochastic solutions enjoy a compact representation for a broad range of forcing functions and/or boundary conditions.Our ANOVA-DSM consists of offline and online stages. In the offline stage, the original high-dimensional stochastic problem is decomposed into a series of low-dimensional stochastic subproblems, according to the ANOVA decomposition technique. Then, for each subproblem, a data-driven stochastic basis is computed using the Karhunen-Loève expansion (KLE) and a two-level preconditioning optimization approach. Multiple trial functions are used to enrich the stochastic basis and improve the accuracy. In the online stage, we solve each stochastic subproblem for any given forcing function by projecting the stochastic solution into the data-driven stochastic basis constructed offline. In our ANOVA-DSM framework, solving the original highdimensional stochastic problem is reduced to solving a series of ANOVA-decomposed stochastic subproblems using the DSM. An adaptive ANOVA strategy is also provided to further reduce the number of the stochastic subproblems and speed up our method. To demonstrate the accuracy and efficiency of our method, numerical examples are presented for one- and two-dimensional elliptic PDEs with random coefficients.

scholarly journals BLP-2LASSO for aggregate discrete choice models with rich covariates

2019 ◽  
Vol 22 (3) ◽  
pp. 262-281 ◽  
Author(s):  
Benjamin J Gillen ◽  
Sergio Montero ◽  
Hyungsik Roger Moon ◽  
Matthew Shum
Keyword(s):  
Ad Hoc ◽  
Demographic Data ◽  
Discrete Choice Models ◽  
Random Coefficients ◽  
Data Driven ◽  
Preference Heterogeneity ◽  
High Dimensional ◽  
Differentiated Products ◽  
Estimation Algorithms ◽  
Data Driven Approach

Summary We introduce the BLP-2LASSO model, which augments the classic BLP (Berry, Levinsohn, and Pakes, 1995) random-coefficients logit model to allow for data-driven selection among a high-dimensional set of control variables using the 'double-LASSO' procedure proposed by Belloni, Chernozhukov, and Hansen (2013). Economists often study consumers’ aggregate behaviour across markets choosing from a menu of differentiated products. In this analysis, local demographic characteristics can serve as controls for market-specific preference heterogeneity. Given rich demographic data, implementing these models requires specifying which variables to include in the analysis, an ad hoc process typically guided primarily by a researcher’s intuition. We propose a data-driven approach to estimate these models, applying penalized estimation algorithms from the recent literature in high-dimensional econometrics. Our application explores the effect of campaign spending on vote shares in data from Mexican elections.

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