Reduced-basis output bound methods for parametrized partial differential equations

Abstract

An efficient and reliable method for the prediction of outputs of interest of partial differential equations with affine parameter dependence is presented. To achieve efficiency we employ the reduced-basis method: a weighted residual Galerkin-type method, where the solution is projected onto low-dimensional spaces with certain problem-specific approximation properties. Reliability is obtained by a posteriori error estimation methods - relaxations of the standard error-residual equation that provide inexpensive but sharp and rigorous bounds for the error in outputs of interest. Special affine parameter dependence of the differential operator is exploited to develop a two-stage off-line/on-line blackbox computational procedure. In the on-line stage, for every new parameter value, we calculate the output of interest and an associated error bound. The computational complexity of the on-line stage of the procedure scales only with the dimension of the reduced-basis space and the parametric complexity of the partial differential operator; 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. The theory and corroborating numerical results are presented for: symmetric coercive problems (e.g. problems in conduction heat transfer), parabolic problems (e.g. unsteady heat transfer), noncoercive problems (e.g. the reduced-wave, or Helmholtz, equation), the Stokes problem (e.g flow of highly viscous fluids), and certain nonlinear equations (e.g. eigenvalue problems).