Quantifying Uncertainty
Xiu, D., Kevrekidis, I.G., and Ghanem, R.G.. "An Equation-Free, Multiscale Approach to Uncertainty Quantification" Computing in Science and Engineering. 7
(3).
2005.
pp. 16--23.
Recently, interest has grown in developing efficient computational methods (both sampling and nonsampling) for studying ordinary or partial differential equations with random inputs. Stochastic Galerkin (SG) methods based on generalized polynomial chaos (gPC) representations have several appealing features. However, when the model equations are complicated, the numerical implementation of such algorithms can become highly nontrivial, and care is needed to design robust and efficient solvers for the resulting systems of equations. The authors' equation- and Galerkin-free computational approach to uncertainty quantification (UQ) for dynamical systems lets them conduct UQ computations without explicitly deriving the SG equations for the gPC coefficients. They use short bursts of appropriately initialized ensembles of simulations with the basic model to estimate the quantities required in SG algorithms.
Xiu, D. and Kevrekidis, I.G.. "Equation-free, multiscale computation for unsteady random diffusion" Multiscale Modeling & Simulation. 4
(3).
2005.
pp. 915--935.
We present an ``equation-free'' multiscale approach to the simulation of unsteady diffusion in a random medium. The diffusivity of the medium is modeled as a random field with short correlation length, and the governing equations are cast in the form of stochastic differential equations. A detailed fine-scale computation of such a problem requires discretization and solution of a large system of equations and can be prohibitively time consuming. To circumvent this difficulty, we propose an equation-free approach, where the fine-scale computation is conducted only for a (small) fraction of the overall time. The evolution of a set of appropriately defined coarse-grained variables (observables) is evaluated during the fine-scale computation, and ``projective integration'' is used to accelerate the integration. The choice of these coarse variables is an important part of the approach: they are the coefficients of pointwise polynomial expansions of the random solutions. Such a choice of coarse variables allows us to reconstruct representative ensembles of fine-scale solutions with ``correct'' correlation structures, which is a key to algorithm efficiency. Numerical examples demonstrating accuracy and efficiency of the approach are presented.