Article2012

Stochastic Collocation Methods on Unstructured Grids in High Dimensions via Interpolation

73 citations0/0 files storedSIAM Journal on Scientific Computing

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Probabilistic and Robust Engineering Design Mathematical functions and polynomials Mathematical Approximation and Integration

Abstract

In this paper we propose a method for conducting stochastic collocation on arbitrary sets of nodes. To accomplish this, we present the framework of least orthogonal interpolation, which allows one to construct interpolation polynomials based on arbitrarily located grids in arbitrary dimensions. These interpolation polynomials are constructed as a subspace of the family of orthogonal polynomials corresponding to the probability distribution function on stochastic space. This feature enables one to conduct stochastic collocation simulations in practical problems where one cannot adopt some popular node selections such as sparse grids or cubature nodes. We present in detail both the mathematical properties of the least orthogonal interpolation and its practical implementation algorithm. Numerical benchmark problems are also presented to demonstrate the efficacy of the method.

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References

1991Stochastic Finite Elements: A Spectral Approach 2002The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations 2005High-Order Collocation Methods for Differential Equations with Random Inputs 2007A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data 2008A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data 2009An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations 2011A non-adapted sparse approximation of PDEs with stochastic inputs 2011On the convergence of generalized polynomial chaos expansions

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2014Computational Aspects of Stochastic Collocation with Multifidelity Models 2015Stochastic Collocation on Unstructured Multivariate Meshes 2013Minimal multi-element stochastic collocation for uncertainty quantification of discontinuous functions 2017Multi-fidelity stochastic collocation method for computation of statistical moments 2015Constructing Surrogate Models of Complex Systems with Enhanced Sparsity: Quantifying the Influence of Conformational Uncertainty in Biomolecular Solvation

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