We present a technique for the rapid and reliable prediction of
linear-functional
outputs of elliptic coercive partial differential equations with affine
parameter dependence. The essential components are (i )
(provably) 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 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. In our earlier work we develop a rigorous a posteriori error bound framework for reduced-basis
approximations of elliptic coercive equations. The resulting error
estimates are, in some cases, quite sharp: the ratio of the estimated
error in the output to the true error in the output, or
effectivity , is close to (but always greater than) unity. However, in
other cases, the necessary “bound conditioners” – in essence,
operator preconditioners that (i ) satisfy an additional spectral
“bound” requirement, and (ii ) admit the reduced-basis
off-line/on-line computational stratagem – either can not be found, or
yield unacceptably large effectivities. In this paper we introduce a new
class of improved bound conditioners: the critical innovation is the
direct approximation of the parametric dependence of the
inverse of the operator (rather than the operator itself); we
thereby accommodate higher-order (e.g., piecewise linear) effectivity
constructions while simultaneously preserving on-line efficiency. Simple
convex analysis and elementary approximation theory suffice to prove the
necessary bounding and convergence properties.