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Uncertainty quantification: a data-driven stochastic multi-scale method

PDE and Applied Math Seminar

Speaker: Michael Yan, Caltech
Location: 1147 MSB
Start time: Wed, Oct 6 2010, 4:10PM

Uncertainty quantification (UQ) is important in many areas, such as wave scattering, multiscale materials modeling, and flow-structure interactions. Incorporating UQ into the conceptual and preliminary design phases is particularly important for many practical applications. In mathematics, uncertainty can be represented in probabilistic terms by transforming a PDE-based model to stochastic PDE, in particular, uncertainty arises in random initial/boundary conditions, random coefficients of differential equations, or even random geometries. In this talk, we will first survey some popular methods developed in the last twenty years. Then I will introduce a data-driven multi-scale method for a class of SPDE, in which we construct the empirical basis offline based on the Karhunen-Loève decomposition of the Monte Carlo solution, and solve the a general SPDE using this basis. For certain class of stochastic PDEs, we demonstrate that only a small number of random bases is required to accurately represent the stochastic solution, thus providing significant saving in the online computational effort compared with other existing methods. Numerical results will be provided to demonstrate the effectiveness of the method.