[grads] [Sem-coll] Correction to room for Wed. Jan 17 4:40pm talk: Room E1 103

[Robert Ellis]

The correct room for tomorrow's faculty candidate colloquium is E1 103 
(not 106).  Please bear with us as we negotiate with the registrar for 
room assignments for the candidate talks.

Wednesday, Jan 17
4:40pm E1 103 Clayton Webster (Florida State University)
An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data

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AM Colloquium: Wednesday, Jan 17 4:40pm E1 106

Speaker: Clayton Webster (Florida State University)

Title: An Anisotropic Sparse Grid Stochastic Collocation Method for 
Partial Differential Equations with Random Input Data

Abstract: This work proposes and analyzes an anisotropic sparse grid 
stochastic collocation method for differential equations with random 
coefficients and forcing terms (input data of the model). Here, we 
especially address the situation where the input data are assumed to 
depend on a moderately large number of random variables, where in general 
the curse of dimensionality is encountered.

This method can be viewed as an extension of the Sparse Grid Stochastic 
Collocation method proposed in [Nobile-Tempone-Webster, Technical report 
#85, MOX, Dipartimento di Matematica, 2006] which consists of a Galerkin 
approximation in space and a collocation, in probability space, at the 
zeros of sparse tensor product spaces utilizing either Clenshaw-Curtis or 
Gaussian interpolants. As a consequence of the collocation approach our 
techniques naturally lead to the solution of uncoupled deterministic 
problems as in the Monte Carlo method.

Our previous sparse collocation procedure is very effective for problems 
whose input data depend on a moderate number of random variables, which 
"weigh equally" in the solution. For such an isotropic situation the 
displayed convergence is faster than standard collocation techniques built 
upon full tensor product spaces. On the other hand, the convergence rate 
deteriorates when we attempt to solve highly anisotropic problems, such as 
those appearing when the input random variables come e.g. from 
Karhunen-Loeve-type truncations of "smooth" random fields. In such cases, 
a full anisotropic tensor product approximation may still be more 
effective for a small or modest number of random variables. However, if 
the number of random variables is large, the construction of the full 
tensor product spaces becomes infeasible, since the dimension of the 
approximating space grows exponentially fast in the number of random 
variables.

Instead, this work proposes the use of anisotropic sparse tensor product 
spaces constructed from the Smolyak algorithm utilizing suitable 
abscissas. This approach is particularly attractive in the case of 
truncated expansions of random fields, since the anisotropy can be tuned 
to the decay properties of the expansion. We will present a priori and a 
posteriori procedures for choosing the anisotropy of the sparse grids 
which are extremely effective for the problems under study.

This work also provides a rigorous convergence analysis of the fully 
discrete problem and demonstrates: (sub)-exponential convergence of the 
probability error in the asymptotic regime and algebraic convergence of 
the probability error in the pre-asymptotic regime, with respect to the 
total number of collocation points. Numerical examples exemplify the 
theoretical results and are used to compare this approach with several 
others, including standard Monte Carlo. In particular, for moderately 
large dimensional problems, the sparse grid approach with a properly 
chosen anisotropy seems to be very efficient and superior to all examined 
methods.
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