Reliable Real-Time Solution of Parametrized partial Differential Equations: Reduced-Basis Output Bound Methods

摘要

We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic (and parabolic) partial differential equation with affine parameter dependence. The essential components are (i) (provable) rapidly convergent global reduced-basis approximations-Galerkin projection onto a space W_N spanned by solutions of the governing partical differential equation at N selected point in parameter space; (ii ) a posteriori error estimation-relaxations of the error-residual equation that provide inexensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures methods which decouple the generation and projection stages of the approximation process. The operation cound for the on-live stage in which, given a new parameter value, we calculate the outpur of interest and associated error bound, depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control