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International Journal for Uncertainty Quantification
IF: 0.967 5-Year IF: 1.301 SJR: 0.531 SNIP: 0.8 CiteScore™: 1.52

ISSN Print: 2152-5080
ISSN Online: 2152-5099

Open Access

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2014010945
pages 49-72

A GRADIENT-BASED SAMPLING APPROACH FOR DIMENSION REDUCTION OF PARTIAL DIFFERENTIAL EQUATIONS WITH STOCHASTIC COEFFICIENTS

Miroslav Stoyanov
Applied Mathematics Group, Computer Science and Mathematics Division, Oak Ridge National Laboratory, 1 Bethel Valley Road, P.O. Box 2008, Oak Ridge TN 37831-6164
Clayton G. Webster
Department of Computational and Applied Mathematics, Oak Ridge National Laboratory, One Bethel Valley Road, P.O. Box 2008, MS-6164, Oak Ridge, Tennessee 37831-6164, USA

ABSTRACT

We develop a projection-based dimension reduction approach for partial differential equations with high-dimensional stochastic coefficients. This technique uses samples of the gradient of the quantity of interest (QoI) to partition the uncertainty domain into "active" and "passive" subspaces. The passive subspace is characterized by near-constant behavior of the quantity of interest, while the active subspace contains the most important dynamics of the stochastic system. We also present a procedure to project the model onto the low-dimensional active subspace that enables the resulting approximation to be solved using conventional techniques. Unlike the classical Karhunen-Loeve expansion, the advantage of this approach is that it is applicable to fully nonlinear problems and does not require any assumptions on the correlation between the random inputs. This work also provides a rigorous convergence analysis of the quantity of interest and demonstrates: at least linear convergence with respect to the number of samples. It also shows that the convergence rate is independent of the number of input random variables. Thus, applied to a reducible problem, our approach can approximate the statistics of the QoI to within desired error tolerance at a cost that is orders of magnitude lower than standard Monte Carlo. Finally, several numerical examples demonstrate the feasibility of our approach and are used to illustrate the theoretical results. In particular, we validate our convergence estimates through the application of this approach to a reactor criticality problem with a large number of random cross-section parameters.