Rapid Reliable Solution
of the Parametrized Partial Differential Equations
of Continuum Mechanics and Transport


Methodology: Reduced Basis Methods and A Posteriori Error Estimators

Here we describe our approach to reduced basis approximation and rigorous a posteriori error estimation.


Domain of Relevance

Our methodology focuses on the rapid and reliable prediction of engineering outputs associated with parametrized partial differential equations.
In particular, we consider a (say, single) "output of interest" $s \in \mathbb{R}$ — related to energies or forces, stresses or strains, flowrates or pressure drops, temperatures or fluxes — as a function of an "input parameter" $P$-vector $\mu$ — related to geometry, physical properties, boundary conditions, or loads. The input parameter domain — the set of possible inputs — is denoted ${\cal D}$, which is a subset of $\mathbb{R}^P$ .

The output of interest $s (\mu)$ is a (say, linear) functional $\ell$ of a field variable $u (\mu)$, $s(\mu) = \ell(u(\mu))$. Here $u (\mu)$ — say displacement, velocity, or temperature — satisfies a partial differential equation parametrized with respect to $\mu$. We thus arrive at an input-output statement $\mu \to s(\mu)$ 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,'' our 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.
Many educational situations also satisfy these criteria — from in-class demonstrations that require extensive parameter exploration and immediate gratification to homework assignments and projects that must be completed (by many parties) rapidly on modest platforms.

As regards "reliable,'' we provide certificates of fidelity with every prediction: an estimate that rigorously bounds the error in our (rapid, approximate) 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.
Educational situations also demand certainty: a demonstration or project founded upon a-physical numerical artifacts is obviously anathema to the development of sound engineering principles and practices.
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Ingredients

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 $\cal D$. Typically, will be small, as we focus attention on the (smooth) low-dimensional parametrically-induced manifold of interest. The RB approximations to the field variable and output are denoted $u_N (\mu)$ and $s_N (\mu)$, respectively.
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 — our$u(\mu)$ introduced above — upon which we build our RB approximation, and with respect to which we measure the RB error (see (ii) below).

(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, $u_N (\mu)$, and output(s) approximation, $s_N (\mu)$. Our error indicators are rigorous upper bounds for the error (relative to the FE truth approximation) for all$\mu \in {\cal D}$ and for all $N$; furthermore, in many cases, we can prove that the effectivity of our error estimators — the ratio of the error bound to the true error — is O(1)–O(10).
Our inexpensive error estimators also serve to construct the optimal RB samples and spaces which ensure an efficient and well-conditioned RB 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 — in which, given a new parameter value $\mu$, the RB Online Evaluator calculates the RB output and associated error bound (relative to the expensive FE truth approximation) — depends only on $N$ and the parametric complexity of the problem. The Online computational complexity and mathematical stability does not depend on ${\cal N}$, the dimension of the underlying "truth" FE approximation space; we may thus consider a highly accurate truth approximation.
(Note that for visualization of the RB field variable approximation, the complexity does scale with ${\cal N}$ — roughly as $N {\cal N}$ — since we must recreate and render the field over the entire physical domain.)
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