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SUMMARY:High-Dimensional Collocation for Lognormal Diffusion Problems - Ol
 iver Ernst (Technische Universität Chemnitz)
DTSTART:20180209T113000Z
DTEND:20180209T123000Z
UID:TALK100393@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:Co-authors: Bj&ouml\;rn Sprungk (Universit&auml\;t Mannheim)\,
  Lorenzo Tamellini (IMATI-CNR Pavia)  Many UQ models consist of random dif
 ferential equations in which one or more data components are uncertain and
  modeled as random variables. When the latter take values in a separable f
 unction space\, their representation typically requires a large or countab
 ly infinite number of random coordinates. Numerical approximation methods 
 for such functions of an infinite number of parameters based on best N-ter
 m approximation have recently been proposed and shown to converge at an al
 gebraic rate. Collocation methods have a number of computational advantage
 s over best N-term approximation\, and we show how ideas successful there 
 can be used to show a similar convergence rate for sparse collocation of H
 ilbert-space-valued functions depending on countably many Gaussian random 
 variables.  Such functions appear as solutions of elliptic PDEs with a log
 normal diffusion coefficient. We outline a general L2-convergence theory b
 ased on previous work by Bachmayr et al. and Chen and establish an algebra
 ic convergence rate for sufficiently smooth functions assuming a mild grow
 th bound for the univariate hierarchical surpluses of the interpolation sc
 heme applied to Hermite polynomials. We verify specifically for Gauss-Herm
 ite nodes that this assumption holds and also show algebraic convergence w
 ith respect to the resulting number of sparse grid points for this case. N
 umerical experiments illustrate the dimension-independent convergence rate
 .
LOCATION:Seminar Room 1\, Newton Institute
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