Dimitrios V. Rovas

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

We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced basis approximations, Galerkin projection onto a space W(sub N) spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation, relaxations of the error-residual equation that provide inexpensive 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 count for the on-line stage, in which, given a new parameter value, we calculate the output 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.

A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations

We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic partial differential equations with affine parameter dependence. The essential components are (i) rapidly convergent global reduced-basis approximations - (Galerkin) projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation relaxations of the error-residual equation that provide inexpensive yet sharp 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 count for the on-line stage - in which, given a new parameter value, we calculate the output of interest and associated error bound - depends only on N (typically very small) and the parametric complexity of the problem. In this paper we develop new a posteriori error estimation procedures for noncoercive linear, and certain nonlinear, problems that yield rigorous and sharp error statements for all N. We consider three particular examples: the Helmholtz (reduced-wave) equation; a cubically nonlinear Poisson equation; and Burgers equation - a model for incompressible Navier-Stokes. The Helmholtz (and Burgers) example introduce our new lower bound constructions for the requisite inf-sup (singular value) stability factor; the cubic nonlin-earity exercises symmetry factorization procedures necessary for treatment of high-order Galerkin summations in the (say) residual dual-norm calculation; and the Burgers equation illustrates our accommodation of potentially multiple solution branches in our a posteriori error statement. Numerical results are presented that demonstrate the rigor, sharpness, and efficiency of our proposed error bounds, and the application of these bounds to adaptive (optimal) approximation.

Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems

Abstract We propose a new reduced-basis output bound method for the symmetric eigenvalue problem. The numerical procedure consists of two stages: the pre-processing stage, in which the reduced basis and associated functions are computed—“off-line”—at a prescribed set of points in parameter space; and the real-time stage, in which the approximate output of interest and corresponding rigorous error bounds are computed—“on-line”—for any new parameter value of interest. The real time calculation is very inexpensive as it requires only the solution or evaluation of very small systems. We introduce the procedure; prove the asymptotic bounding properties and optimal convergence rate of the error estimator; discuss computational considerations; and, finally, present corroborating numerical results.