Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized PDEs

Reduced basis methodology focuses on the rapid and reliable prediction of engineering outputs associated with parametrized partial differential equations.
In particular, we consider an "output of interest" — related to energies or forces, stresses or strains, flowrates or pressure drops, temperatures or fluxes — as a function of an "input parameter" — related to geometry, physical properties, boundary conditions, or loads.
The output of interest is a linear functional of the field variable. We thus arrive at an input-output statement, evaluation of which requires solution of a parametrized partial differential equation.
We consider partial differential equations for which the parametric dependence is strictly or approximately affine; "affine" dependence implies that the parametrized differential operator can be expressed as a sum of Q products of [parameter-dependent functions] x [parameter-independent operators].

As regards "rapid,'' the method minimizes the marginal cost associated with (approximate) input-output evaluation, and is thus most useful either (a) in the real-time or interactive context, or (b) in the limit of many queries. Engineering situations which satisfy these criteria include in-the-field robust parameter estimation (or inverse problems, or nondestructive evaluation), design and optimization, and control.
As regards "reliable,'' we provide certificates of fidelity with every
prediction: an estimate that rigorously bounds the error in the input-output evaluation or field variable relative to a highly accurate (and hence very expensive) "truth" finite element solution.

In many engineering situations, the certainty provided by these error bounds is crucial. For example, in the real-time context, critical decisions must be made in the field — quickly, without recourse to extensive Offline resources — that are at least feasible and safe if not optimal.

The essential components of our approach are threefold.

(i) Rapidly convergent global Reduced-Basis (RB) approximations —
(Galerkin) projection onto a space W_N spanned by solution of the governing partial differential equation at N (optimally) selected points S_N in the parameter set. Typically, N will be small, as we focus attention on the (smooth) low-dimensional parametrically-induced manifold of interest.
Our approach is premised upon a classical Finite Element (FE) method ''truth'' approximation space of (typically very large) dimension. It is the FE truth approximation upon which we build our RB approximation, and with respect to which we measure the RB error//.
(ii) Rigorous /a posteriori/ error estimation procedures — relaxations of the error-residual equation that provide inexpensive yet sharp bounds for the error in the RB field-variable approximation, and output(s) approximation.
(iii) Offline/Online computational procedures — decomposition stratagems which decouple the generation and projection stages of the RB
approximation: very extensive (parameter-independent) pre-processing performed Offline once that then prepares the way for subsequent very inexpensive calculations performed Online for each new input-output evaluation required.
The operation count for the Online stage depends only on N and the parametric complexity of the problem. The Online computational complexity and mathematical stability does not depend on the dimension of the underlying "truth" FE approximation space/; we may thus consider a highly accurate truth approximation.

In the seminar a demo of the rbMIT_System Sotware will be provided and the emphasis wil be given to the construction of lower bounds for the coercivity and inf-sup stability constants required in the a posteriori error analysis of reduced basis approximations to affinely parametrized partial differential equations. The method reduces the Online calculation to a small Linear Program: the objective is a parametric expansion of the underlying Rayleigh quotient; the constraints reflect stability information at optimally selected parameter points. The method is simple and general to implement, the Offline stage is based on standard eigenproblems that can be efficiently treated by the Lanczos method, and the Online Linear Program is typically of modest size.