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M Motamed, O Runborg (2009)
Highly Oscillatory Problems, London Mathematical Society Lecture Note Series
I Babuska, R Tempone, GE Zouraris (2005)
Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulationComput. Methods Appl. Mech. Eng., 194
D Xiu, GE Karniadakis (2002)
Modeling uncertainty in steady state diffusion problems via generalized polynomial chaosComput. Methods Appl. Mech. Eng., 191
TY Hou, X Wu (2011)
Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applicationsJ. Comput. Phys., 230
H Liu, J Ralston, O Runborg, NM Tanushev (2014)
Gaussian beam method for the Helmholtz equationSIAM J. Appl. Math., 74
M Motamed, F Nobile, R Tempone (2013)
A stochastic collocation method for the second order wave equation with a discontinuous random speedNumer. Math., 123
H Liu, J Ralston (2010)
Recovery of high frequency wave fields from phase space-based measurementsMultiscale Model. Sim., 8
L Hörmander (1983)
The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis
L Hörmander (1994)
The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators
NM Tanushev (2008)
Superpositions and higher order Gaussian beamsCommun. Math. Sci., 6
F Nobile, R Tempone, CG Webster (2008)
An anisotropic sparse grid stochastic collocation method for partial differential equations with random input dataSIAM J. Numer. Anal., 46
S Mishra, C Schwab (2012)
Sparse tensor multi-level Monte Carlo finite volume methods hyperbolic conservation laws with random initial dataMath. Comput., 81
O Runborg (2007)
Mathematical models and numerical methods for high frequency wavesCommun. Comput. Phys., 2
LN Trefethen (2008)
Is Gauss quadrature better than Clenshaw–Curtis?SIAM Rev., 50
P Tsuji, D Xiu, L Ying (2011)
Fast method for high-frequency acoustic scattering from random scatterersInt. J. Uncertain. Quantif., 1
MB Giles (2008)
Multilevel Monte Carlo path simulationOper. Res., 56
NM Tanushev, J Qian, JV Ralston (2007)
Mountain waves and Gaussian beamsMultiscale Model. Simul., 6
S Jin, P Markowich, C Sparber (2011)
Mathematical and computational models for semiclassical Schrödinger equationsActa Numer., 20
I Babuska, F Nobile, R Tempone (2010)
A stochastic collocation method for elliptic partial differential equations with random input dataSIAM Rev., 52
G Malenova, M Motamed, O Runborg, R Tempone (2016)
A sparse stochastic collocation technique for high-frequency wave propagation with uncertaintySIAM J. Uncertain. Quantif., 4
B Engquist, O Runborg (2003)
Computational high frequency wave propagationActa Numer., 12
H Liu, O Runborg, N Tanushev (2016)
Sobolev and max norm error estimates for Gaussian beam superpositionsCommun. Math. Sci., 14
M Motamed, O Runborg (2010)
Taylor expansion and discretization errors in Gaussian beam superpositionWave Motion, 47
M Motamed, F Nobile, R Tempone (2015)
Analysis and computation of the elastic wave equation with random coefficientsComput. Math. Appl., 70
J Beck, F Nobile, L Tamellini, R Tempone (2014)
Convergence of quasi-optimal stochastic Galerkin methods for a class of PDEs with random coefficientsComput. Math. Appl., 67
V Cervený, MM Popov, I Pšenčík (1982)
Computation of wave fields in inhomogeneous media—Gaussian beam approachGeophys. J. R. Astr. Soc., 70
D Xiu, JS Hesthaven (2005)
High-order collocation methods for differential equations with random inputsSIAM J. Sci. Comput., 27
H Liu, O Runborg, NM Tanushev (2013)
Error estimates for Gaussian beam superpositionsMath. Comp., 82
J Ralston (1982)
Gaussian beams and the propagation of singularitiesStud. Partial. Diff. Equ., 23
GS Fishman (1996)
Monte Carlo: Concepts, Algorithms, and Applications
We consider high-frequency waves satisfying the scalar wave equation with highly oscillatory initial data. The wave speed, and the phase and amplitude of the initial data are assumed to be uncertain, described by a finite number of random variables with known probability distributions. We define quantities of interest (QoIs), or observables, as local averages of the squared modulus of the wave solution. We aim to quantify the regularity of these QoIs in terms of the input random parameters, and the wave length, i.e., to estimate the size of their derivatives. The regularity is important for uncertainty quantification methods based on interpolation in the stochastic space. In particular, the size of the derivatives should be bounded independently of the wave length. In this paper, we are able to show that when these QoIs are approximated by Gaussian beam superpositions, they indeed have this property, despite the highly oscillatory character of the waves.
Research in the Mathematical Sciences – Springer Journals
Published: Jan 24, 2017
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