Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies

Journal Article · Thu Jan 01 04:00:00 EST 2015 · Journal of Computational Physics

DOI:https://doi.org/10.1016/J.JCP.2014.09.019· OSTI ID:22382160

Hampton, Jerrad; Doostan, Alireza

Sampling orthogonal polynomial bases via Monte Carlo is of interest for uncertainty quantification of models with random inputs, using Polynomial Chaos (PC) expansions. It is known that bounding a probabilistic parameter, referred to as coherence, yields a bound on the number of samples necessary to identify coefficients in a sparse PC expansion via solution to an ℓ{sub 1}-minimization problem. Utilizing results for orthogonal polynomials, we bound the coherence parameter for polynomials of Hermite and Legendre type under their respective natural sampling distribution. In both polynomial bases we identify an importance sampling distribution which yields a bound with weaker dependence on the order of the approximation. For more general orthonormal bases, we propose the coherence-optimal sampling: a Markov Chain Monte Carlo sampling, which directly uses the basis functions under consideration to achieve a statistical optimality among all sampling schemes with identical support. We demonstrate these different sampling strategies numerically in both high-order and high-dimensional, manufactured PC expansions. In addition, the quality of each sampling method is compared in the identification of solutions to two differential equations, one with a high-dimensional random input and the other with a high-order PC expansion. In both cases, the coherence-optimal sampling scheme leads to similar or considerably improved accuracy.

• OSTI ID: 22382160
• Journal Information: Journal of Computational Physics, Journal Name: Journal of Computational Physics Vol. 280; ISSN JCTPAH; ISSN 0021-9991
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
APPROXIMATIONS
CHAOS THEORY
COMPARATIVE EVALUATIONS
CONVERGENCE
EXPANSION
HERMITE POLYNOMIALS
LEGENDRE POLYNOMIALS
MARKOV PROCESS
MATHEMATICAL SOLUTIONS
MINIMIZATION
MONTE CARLO METHOD
PARTIAL DIFFERENTIAL EQUATIONS
PROBABILISTIC ESTIMATION
RANDOMNESS