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A Sparse Stochastic Collocation Technique for High-Frequency Wave Propagation with Uncertainty
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.ORCID iD: 0000-0002-6321-8619
2016 (English)In: SIAM/ASA Journal on Uncertainty Quantification, ISSN 1560-7526, E-ISSN 2166-2525, Vol. 4, no 1, p. 1084-1110Article in journal (Refereed) Published
Abstract [en]

We 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, u(epsilon), 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 |u(epsilon)|(2) are smooth, with derivatives in the stochastic space uniformly bounded in epsilon, where epsilon 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.

Place, publisher, year, edition, pages
Society for Industrial and Applied Mathematics Publications , 2016. Vol. 4, no 1, p. 1084-1110
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-214529DOI: 10.1137/15M1029230ISI: 000407996700042Scopus ID: 2-s2.0-85033578756OAI: oai:DiVA.org:kth-214529DiVA, id: diva2:1145663
Note

QC 20170929

Available from: 2017-09-29 Created: 2017-09-29 Last updated: 2018-11-14Bibliographically approved
In thesis
1. Uncertainty quantification for high frequency waves
Open this publication in new window or tab >>Uncertainty quantification for high frequency waves
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

We consider high frequency waves satisfying the scalar wave equation with highly oscillatory initial data represented by a short wavelength ε. The speed of propagation of the medium as well as the phase and amplitude of the initial data is assumed to be uncertain, described by a finite number of independent random variables with known probability distributions. We introduce quantities of interest (QoIs) as spatial and/or temporal averages of the squared modulus of the wave solution, or its derivatives. The main focus of this work is on fast computation of the statistics of those QoIs in the form of moments like the average and variance. They are difficult to obtain numerically by standard methods, as the cost grows rapidly with ε−1 and the dimension of the stochastic space. We therefore propose a fast approximation method consisting of three techniques: the Gaussian beam method to approximate the wave solution, the numerical steepest descent method to compute the QoIs and the sparse stochastic collocation to evaluate the statistics.

The Gaussian beam method is introduced to avoid the considerable cost of approximating the wave solution by direct methods. A Gaussian beam is an asymptotic solution to the wave equation localized around rays, bicharacteristics of the wave equation. This setup allows us to replace the PDE by a set of ODEs that can be solved independently of ε.

The computation of QoIs includes evaluations of highly oscillatory integrals. The idea of the numerical steepest descent method is to change the integration path in the complex plane such that the integrand is non-oscillatory along it. Standard quadrature methods can be then utilized. We construct such paths for our case and show error estimates for the integral approximation by the Gauss-Laguerre and Gauss-Hermite quadrature.

Finally, the evaluation of statistical moments of the QoI may suffer from the curse of dimensionality.  The sparse grid collocation method introduces a framework where certain large group of points can be neglected while only slightly reducing the convergence rate. The regularity of the QoIs in terms of the input random parameters and the wavelength is important for the method to be efficient.  In particular, the size of the derivatives should be bounded independently of ε. We show that the QoIs indeed have this property, despite the highly oscillatory character of the waves.

Abstract [sv]

Vi studerar högfrekventa lösningar till den skalära vågekvationen med mycket oscillerande begynnelsedata, given av en kort våglängd ε. Utbredningshastigheten i mediumet samt fasen och amplituden av begynnelsedatan antas vara osäkra, och kunna beskrivas av ett ändligt antal oberoende slumpvariabler med kända sannolikhetsfördelningar. Vi definierar funktionaler som lokala medelvärden av beloppskvadraten av lösningen, eller av dess derivator.

Huvudfokus i detta arbete ligger på att snabbt beräkna statistiska mått som medelvärde och varians av dessa funktionaler. Kostnaden att beräkna funktionalerna med direkta metoder växer snabbt med ε−1. Därför föreslår vi en beräkningsmetod bestående av tre tekniker: Gaussian beam-metoden, numeriska gradientmetoden och glesa beräkningsnät.

Gaussian beam-metoden används för att undvika komplexiteten att approximera lösningen med direkta metoder. En gaussian beam är en asymptotisk lösning till vågekvationen lokaliserad runt strålar, dvs. bikarakteristikor till vågekvationen. Denna ansats låter oss ersätta den ursprungliga partiella differentialekvationen med en uppsättning ordinära differentialekvationer, vilka kan lösas oberoende av ε.

Beräkningen av funktionaler kräver beräkningen av mycket oscillerande integraler. Idén bakom numeriska gradientmetoden är att ändra integrationskurvan för dessa integraler till en kurva i komplexa planet längs med vilken integranden inte oscillerar. Dessa senare integraler kan beräknas med standardmetoder.

Beräkningskostnaden av funktionalernas statistiska mått växer mycket snabbt med problemets dimension. För att undkomma detta problem använder vi glesa beräkningsnät. Med hjälp av dessa nät kan vi ignorera stora delar av gridpunkterna, samtidigt som konvergenshastigheten inte påverkas allt för negativt.

Regulariteten hos dessa funktionaler i termer av de givna slumpmässiga parametrarna och våglängden är viktig för att metoden ska vara effektiv. Speciellt så måste derivatorna begränsas uppifrån oberoende av ε. Vi visar att funktionalerna har denna egenskap, trots att lösningen själv är mycket oscillerande

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2018. p. 53
Series
TRITA-SCI-FOU ; 2018:52
National Category
Computational Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-238878 (URN)
Public defence
2018-12-07, K1, Teknikringen 56, Stockholm, 10:00 (English)
Opponent
Supervisors
Note

QC 20181114

Available from: 2018-11-14 Created: 2018-11-13 Last updated: 2018-11-14Bibliographically approved

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Malenova, GabrielaRunborg, Olof

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