High-Order Collocation Methods for Differential Equations with Random Inputs
Dongbin Xiu,Jan S. Hesthaven +1 more
TLDR
A high-order stochastic collocation approach is proposed, which takes advantage of an assumption of smoothness of the solution in random space to achieve fast convergence and requires only repetitive runs of an existing deterministic solver, similar to Monte Carlo methods.Abstract:
Recently there has been a growing interest in designing efficient methods for the solution of ordinary/partial differential equations with random inputs. To this end, stochastic Galerkin methods appear to be superior to other nonsampling methods and, in many cases, to several sampling methods. However, when the governing equations take complicated forms, numerical implementations of stochastic Galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations. On the other hand, the traditional sampling methods, e.g., Monte Carlo methods, are straightforward to implement, but they do not offer convergence as fast as stochastic Galerkin methods. In this paper, a high-order stochastic collocation approach is proposed. Similar to stochastic Galerkin methods, the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence. However, the numerical implementation of stochastic collocation is trivial, as it requires only repetitive runs of an existing deterministic solver, similar to Monte Carlo methods. The computational cost of the collocation methods depends on the choice of the collocation points, and we present several feasible constructions. One particular choice, based on sparse grids, depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs. Numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods.read more
Citations
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Book ChapterDOI
Stochastic Differential Equations
TL;DR: In this paper, the authors explore questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties, using diffusion processes as a model of a Markov process with continuous sample paths.
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.
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