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REDUCED BASIS METHODS FOR QUADRATICALLY NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS WITH STOCHASTIC INFLUENCES

We present a new approach to solve nonlinear parametric partial differential equations (PPDEs) with stochastic coefficients, which is based on the Reduced Basis (RB) method. It is imposed that the problem formulation allows for an affine decomposition in the (deterministic) parameter. The uncertainties in the coefficients are modeled using Karhunen-Loève expansions. The nonlinearity is resolved using Newton’s method with Fréchet derivatives. For the computation of a linear output functional, its first and second moment, and its variance, we additionally solve some linear dual problems. Statistical analysis is then done using the Monte Carlo (MC) method. We investigate offline/online decompositions that enable efficient computations independent of the number of degrees of freedom of the full system. We provide efficient and rigorous a-posteriori error bounds for the state variable, the output functional, and the statistical quantities based upon the Brezzi-Rappaz-Raviart theory, dealing however with additional errors as a result of the uncertainty modeling. We present numerical experiments for a stationary quadratic convection-diffusion problem to illustrate different aspects of the proposed method.