A Sparse Stochastic Collocation Technique for High-Frequency Wave Propagation with Uncertainty
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AbstractWe consider the wave equation with highly oscillatory initial data, where there is uncertainty in the wave speed, initial phase, and/or initial amplitude. To estimate quantities of interest related to the solution and their statistics, we combine a high-frequency method based on Gaussian beams with sparse stochastic collocation. Although the wave solution, uϵ, is highly oscillatory in both physical and stochastic spaces, we provide theoretical arguments for simplified problems and numerical evidence that quantities of interest based on local averages of |uϵ|2 are smooth, with derivatives in the stochastic space uniformly bounded in ϵ, where ϵ denotes the short wavelength. This observable related regularity makes the sparse stochastic collocation approach more efficient than Monte Carlo methods. We present numerical tests that demonstrate this advantage.
CitationMalenova G, Motamed M, Runborg O, Tempone R (2016) A Sparse Stochastic Collocation Technique for High-Frequency Wave Propagation with Uncertainty. SIAM/ASA Journal on Uncertainty Quantification 4: 1084–1110. Available: http://dx.doi.org/10.1137/15M1029230.
SponsorsThe work of the first author was partially supported by the Swedish Research Council under grant 2012-3808. The work of the fourth author was supported by the King Abdullah University of Science and Technology (KAUST) SRI Center for Uncertainty Quantification in Computational Science and Engineering.