RT Journal Article
ID 06f4ea93770add7d
A1 Stoyanov, Miroslav
A1 Webster, Clayton G.
T1 A GRADIENT-BASED SAMPLING APPROACH FOR DIMENSION REDUCTION OF PARTIAL DIFFERENTIAL EQUATIONS WITH STOCHASTIC COEFFICIENTS
JF International Journal for Uncertainty Quantification
JO IJUQ
YR 2015
FD 2015-03-24
VO 5
IS 1
SP 49
OP 72
K1 representation of uncertainty
K1 stochastic model reduction method
K1 stochastic sensitivity analysis
K1 high-dimensional approximation
K1 stochastic partial differential equations
K1 Karhunen-Loeve expansion
K1 Monte Carlo
AB 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.
PB Begell House
LK http://dl.begellhouse.com/journals/52034eb04b657aea,5303738564693bb8,06f4ea93770add7d.html