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dc.contributor.author
Cohen, Albert
dc.contributor.author
DeVore, Ronald
dc.contributor.author
Schwab, Christoph
dc.date.accessioned
2017-11-20T15:08:36Z
dc.date.available
2017-06-14T01:34:10Z
dc.date.available
2017-06-09T00:04:05Z
dc.date.available
2017-11-20T14:09:39Z
dc.date.available
2017-11-20T15:08:36Z
dc.date.issued
2009-01
dc.identifier.uri
http://hdl.handle.net/20.500.11850/210602
dc.identifier.doi
10.3929/ethz-a-010406289
dc.description.abstract
Deterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain D_Rd are introduced and their convergence rates are estimated. The approximations are based on expansions of the random diffusion coefficients in L2(D)-orthogonal bases, and on viewing the coefficients of these expansions as random parameters y=y(!)=(yi(!)). This yields an equivalent parametric deterministic PDE whose solution u(x,y) is a function of both the space variable x 2 D and and the in general countably many parameters y. We establish new regularity theorems decribing the smoothness properties of the solution u as a map from y 2 U = (-1, 1) to V = H1 0 (D). These results lead to analytic estimates on the V norms of the coefficients (which are functions of x) in a so-called "generalized polynomial chaos”(gpc) expansion of u. Convergence estimates of approximations of u by best N-term truncated V -valued polynomials in the variable y 2 U are established. These estimates are of the form N-r, where the rate of convergence r depends only on the decay of the random input expansion. It is shown that r exceeds the benchmark rate 1/2 afforded by Monte-Carlo simulations with N "samples" (i.e. deterministic solves) under mild smoothness conditions on the random diffusion coefficients. A class of fully discrete approximations is obtained by Galerkin approximation from a hierarchic family (V) 1 1=0_V of finite element spaced in D of the coefficients in the N-term truncated gpc expansions of u(x,y). In contrast to previous works, the level l of spatial resolution is adapted to the gpc coefficients. New regularity theorems decribing the smoothness properties of the solution u as a map from y 2 U = (-1,1)1 to a smoothness space W_V are established leading to analytic estimates on the W norms of the gpc coefficients and on their space discretization error. The space W coincides with H2(D)/H1 0 (D) in the case where D is a smooth or convex domain. Our analysis shows that in realistic settings a convergence rate N-s d.o.f in terms of the total number of degrees of freedom Nd.o.f can be obtained. Here the rate s is determined by both the best N-term approximation rate r and the approximation order of the space discretization in D.
en_US
dc.language.iso
en
en_US
dc.publisher
ETH Zurich, Seminar for Applied Mathematics
en_US
dc.rights.uri
http://rightsstatements.org/page/InC-NC/1.0/
dc.subject
Stochastic and parametric elliptic equations
en_US
dc.subject
Wiener polynomial chaos
en_US
dc.subject
approximation rates
en_US
dc.subject
nonlinear approximation
en_US
dc.subject
sparsity
en_US
dc.title
Convergence rates of best N-term Galerkin approximations for a class of elliptic sPDEs
en_US
dc.type
Report
dc.rights.license
In Copyright - Non-Commercial Use Permitted
ethz.journal.title
Research Report
ethz.journal.volume
2009-02
en_US
ethz.size
30 p.
en_US
ethz.code.ddc
5 - Science::510 - Mathematics
en_US
ethz.publication.place
Zurich
en_US
ethz.publication.status
published
en_US
ethz.leitzahl
ETH Zürich::00002 - ETH Zürich::00012 - Lehre und Forschung::00007 - Departemente::02000 - Dep. Mathematik / Dep. of Mathematics::02501 - Seminar für Angewandte Mathematik / Seminar for Applied Mathematics
en_US
ethz.leitzahl.certified
ETH Zürich::00002 - ETH Zürich::00012 - Lehre und Forschung::00007 - Departemente::02000 - Dep. Mathematik / Dep. of Mathematics::02501 - Seminar für Angewandte Mathematik / Seminar for Applied Mathematics
en_US
ethz.identifier.url
https://www.math.ethz.ch/sam/research/reports.html?id=57
ethz.date.deposited
2017-06-09T00:04:22Z
ethz.source
ECOL
ethz.source
ECIT
ethz.identifier.importid
imp59366b73a773335087
ethz.identifier.importid
imp59364cc1a61e259368
ethz.ecolpid
eth:47569
ethz.ecitpid
pub:33519
ethz.eth
yes
en_US
ethz.availability
Open access
en_US
ethz.rosetta.installDate
2017-11-20T14:02:45Z
ethz.rosetta.lastUpdated
2018-11-06T02:30:46Z
ethz.rosetta.versionExported
true
dc.identifier.olduri
http://hdl.handle.net/20.500.11850/20914
dc.identifier.olduri
http://hdl.handle.net/20.500.11850/155036
ethz.COinS
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