# A Posteriori Error Bounds for Reduced-Basis Approximation of Parametrized Noncoercive and Nonlinear Elliptic Partial Differential Equations

Abstract: 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. © 2003 by the American Institute of Aeronautics and Astronautics, Inc.

## Summary (5 min read)

### 1 Introduction

- The optimization, control, and characterization of an engineering component or system requires the prediction of certain “quantities of interest,” or performance metrics, which the authors shall denote outputs — for example deflections, heat transfer rates, or drags.
- The authors goal is the development of computational methods that permit rapid and reliable evaluation of this partial-differential-equation-induced input-output relationship in the limit of many queries — that is, in the design, optimization, control, and characterization contexts.
- The reduced-basis method recognizes that the field variable is not, in fact, some arbitrary member of the infinite-dimensional solution space associated with the partial differential equation; 1 of 18 American Institute of Aeronautics and Astronautics Paper 2003-3847 rather, it resides, or “evolves,” on a much lowerdimensional manifold induced by the parametric dependence.
- The work6,8, 9, 14,15,20 differs from these earlier efforts in several important ways: first, the authors develop global approximation spaces; second, they introduce rigorous a posteriori error estimators; and third, they exploit off-line/on-line computational decompositions (see2 for an earlier application of this strategy.).

### 2.1 Preliminaries

- (2) More general inner products and norms can (and should) be considered, as discussed in Section 2.4.2.
- The authors now introduce their parametrized bilinear form.

### 2.2.1 Weak Statement

- In the language of the introduction, s(µ) is their output, and u(µ) is their field variable.
- The authors assume that N is chosen sufficiently large that sN (µ) and uN (µ) may be effectively equated with s(µ) and u(µ), respectively.

### 2.2.2 Reduced-Basis Approximation

- The focus of the current paper is a posteriori error estimation.
- The authors shall thus take their reduced-basis approximation as given.
- The discrete inf-sup parameter associated with the latter may not be “good,” with corresponding detriment to the accuracy of uN (µ) and hence sN (µ).
- More sophisticated minimumresidual8,18 and in particular Petrov-Galerkin7,18 approaches restore stability, albeit at some additional complexity and cost.

### 2.3.1 Error Bound

- The authors note that their proof (or bound) does not exploit any special properties of e(µ) (or uN (µ)).
- It remains to develop their lower bound construction, β̂(µ), and to demonstrate that both β̂(µ) and ‖Yr( · ;µ)‖Y may be computed efficiently (that is, in complexity independent of N ).

### 2.3.2 Inf-Sup Lower Bound Construction

- Many of the most obvious eigenvalue approximation concepts are not relevant here, since the authors require a lower, not upper, bound.
- The authors can now state Proposition 2 The construction β̂(µ) of (42) satisfies the inequality (31).
- In essense, the equation for the error e(µ), (34), permits relaxations — and hence rigorous yet inexpensive bounds — that can not be directly applied to the original equation for u(µ), (24).

### 2.3.3 Offline/Online Computational Procedure

- The central computational aspect of their reduced-basis approach is an offline/online computational decomposition which separates the requisite calculations into two distinct stages.
- The authors develop here similar estimates for ∆N (µ) (and hence ∆sN (µ)).
- The authors also briefly address the associated offline complexity.
- (In many cases, domain decomposition may be exploited to further reduce the Q dependence — from 5 of 18 American Institute of Aeronautics and Astronautics Paper 2003-3847 quadratic to linear.

### 2.4.2 Bound Conditioner

- This can be remedied by better choice of their bound conditioner.
- Χ1(µ) is high-wavenumber, and thus the authors may add a significant L2 contribution to their bound conditioner without adversely affecting the inf-sup parameter; this additional L2 term does, however, significantly improve their continuity constants — on which their lower bound construction is critically dependent, also known as The reason is simple.
- These arguments apply to Helmholtz problems generally; however, for larger ranges of frequency, the authors will need different bound conditioners for different subdomains of Dµ — if they wish to retain the Dµ-independence of J . 6 of 18 American Institute of Aeronautics and Astronautics Paper 2003-3847.

### 2.4.3 Deflation

- The second debilitating aspect of (55) is the − ln(εs) dependence.
- The latter should improve their bounds and effectivity; but, more importantly, it will remove the εs dependence from (55) — their regions will be generally larger, and will not shrink to zero as the authors approach resonances (or, at most, except very near resonances).
- It remains to address two issues concerning δDM (µ).

### 2.5.1 Model Helmholtz Problem: P = 2

- The authors assume that the boundary of the memrane is “pinned” except on the “stress-free” crack.
- The offline expense will be increased somewhat, not so much due to the λ2(µj), χ2(µj) (say for M = 1) — in particular, since J will now be much smaller — but rather due to the JQ2(M2 + MN)N operations required for the inner products associated with the deflation correction (60).
- Clearly, an elastic plate (and more realistic outputs) would be a much more relevant model; their methodology directly applies to this case as well.
- For a given Loc , the bilinear form is, apart from several scaling factors, identical to (53) of Section 2.4.1.
- In particular, unlike in Section 2.4, the authors can no longer characterize β(ω2, Loc) in terms of a few (more generally, denumerable) “resonance” eigenvalues — their lower bound constructions are now required.

### 2.5.2 The Inf-Sup Lower Bound

- First, the authors observe that the “correct” bound conditioner (I → II) considerably increases the size of the regions; furthermore, this effect will be even more dramatic for higher frequency ranges.
- Second, the authors observe that some deflation (II→ III) further improves the situation; and sufficient deflation (III → V) greatly improves the situation, in particular as they approach resonance.
- Note that although IV performs better than III, only with V do the authors have sufficient deflation in the sense that all dangerous modes are neutralized — it is clear from Figure 3 that three modes are “active” near the end of their segment (25, 0.4)(50, 0.6).
- In actual fact, β(µ) varies significantly only in the one direction perpendicular to the P − 1 dimensional “resonance” manifolds.

### 2.5.3 Error Bounds and Effectivity

- The authors consider here a point µTEST which lies within a region Rµj ,τ for all cases I, II, III, IV, and V.
- As expected from the arguments of Section 2.4.2, the bound conditioner has little effect on the effectivity.
- And, as expected from the arguments of Section 2.3.3, deflation has a modest 9 of 18 American Institute of Aeronautics and Astronautics Paper 2003-3847 (respectively, significant) positive effect on the error (respectively, effectivity).
- Second, round-off errors will become increasingly important, and ultimately dominant, in the very immediate vicinity of resonances; in particular, as the authors approach extremely close to a resonance, they may observe effectivities below unity.
- Exact orthogonalization recovers the theoretical result — ηN (µ) ≥ 1; in more realistic models, damping will provide the necessary “cut-off.”.

### 3.2.1 Weak Statement

- In the language of the introduction, s(µ) is their output, and u(µ) is their field variable.
- As for their Helmholtz problem, in actual practice the authors replace s(µ) and u(µ) with corresponding “truth” Galerkin approximations sN (µ) and uN (µ), respectively (see Section 2.2.1).

### 3.2.2 Reduced-Basis Approximation

- The focus of the current paper is a posteriori error estimation.
- The authors shall thus take their reduced-basis approximation as given.

### 3.3.2 Coercivity Lower Bound Construction

- The authors approach to the inf-sup lower bound, described in Section 2.3.2, can also be adapted to general coercive problems.
- For their purposes here, however, the authors consider a simple variant that exploits the monotonicity of α(µ).
- Further details on these and related bound conditioners for coercive problems may be found elsewhere.

### 3.3.3 Offline/Online Computational Procedure

- The authors nonlinear problem admits an offline/online decomposition quite similar to that for linear problems.
- The authors focus here will be on efficient (or as efficient as possible) treatment of these new terms.
- Obviously, näıve treatment of (96) directly yields N6 operations.
- In general, T(N,κ) ∼ N6/κ!: in relative terms, higher order (e.g., uκ) nonlinearities thus enjoy greater economies; however, in absolute terms, T(N,κ) will grow very rapidly with N for larger κ.

### 3.4.1 Model Problems

- The authors model problem has already been specified in Sections 3.1 and 3.2.
- Note the nonlinearity will be most significant for µ1 and µ2 small.

### 3.4.2 Adaptive Reduced-Basis Approximation

- Given the higher powers of N that now appear in their complexity estimates, it is crucial (both as regards online and offline effort) to control N more tightly.
- The authors typically choose εpriord ε post d (µ) since their prior test sample is not exhaustive; and therefore, typically, Npost(µ) ≤ Nprior.
- The authors present in Table 3 the normalized error ‖e(µ∗)‖Y /‖u(µ∗)‖Y , as a function of N , for the (log) random and adaptive sampling processes (note that, in the results for the random sampling process, the sample SN is different for each N).
- The calculations were performed on a Pentium r©4 2.4GHz processor.
- Of course, in actual practice, the savings indicated in Table 3 can only be realized if their error estimators are true bounds (ηN (µ) ≥ 1), and good bounds (ηN (µ) ≈ 1).

### 4.2.1 Weak Statement

- For sufficiently large µ, (99), (110) — and the incompressible NavierStokes equations — have a unique solution; for smaller µ, the authors can encounter non-uniqueness — multiple solution branches may exist.
- As for their Helmholtz problem, in actual practice the authors replace s(µ) and u(µ) with corresponding “truth” Galerkin approximations sN (µ) and uN (µ), respectively (see Section 2.2.1).

### 4.2.2 Reduced-Basis Approximation

- Note uI(µn) refers to solutions of (99), (110), which are assumed to reside on a “first” branch; although the authors do not dwell here on possible bifurcation structure, other “parametric manifolds” (say, uII(µ)) may, in general, exist.
- The discrete inf-sup parameter associated with the latter may not be “good,” with corresponding detriment to the accuracy of uN (µ) and hence sN (µ).
- More sophisticated minimumresidual8,18 and in particular Petrov-Galerkin7,18 approaches restore stability, albeit at some additional complexity and cost.
- The authors comment that, for the case in which geometry is fixed and only viscosity varies, their reduced-basis approximation (and associated error estimation) procedure for the Burgers equation directly translates to the full incompressible Navier-Stokes equations — in particular, a divergence- (and hence pressure-) free formulation of the incompressible Navier-Stokes equations.

### 4.3.1 Preliminaries

- In what follows, the authors will explicitly highlight the N -dependence of β(µ), γ(µ), and σ(w;µ) only in those places where this dependence is either not obvious or potentially problematic.
- Note that T0 and the Tn are parameter-independent.

### 4.3.2 Error Bound

- In the quadratic case, the proof above is simpler and slightly sharper.
- In short, the error bound “sees” only the residual, which in turn “sees” only the branch-independent projection of uI(µ) (or uII(µ)), f(v).
- There is a dark side: one can not rigorously preclude the possibility that ‖uI(µ)−uN (µ)‖Y ≥ ΥN (µ).the authors.
- Clearly, in actual practice, the relative (and absolute) magnitude of ΥN (µ) will directly affect their comfort level in choosing (120).

### 4.3.2 Inf-Sup Lower Bound Construction

- Note if the authors include both (say, in the case of two branches) branches, uI(µn), uII(µn), in WN , then they will typically obtain good reduced-basis approximations to both branches — uIN (µ), uIIN (µ).
- The proof is almost identical to the proof of Proposition 2 for the Helmholtz inf-sup lower bound construction.

### 4.3.3 Offline/Online Computational Procedure

- All the elements of the offline/online procedure for the construction of Burgers a posteriori error bounds have already been introduced in the context of the Helmholtz and cubically nonlinear Poisson problems.
- Rather, the authors can make plausible continuity assumptions to construct these intervals, and then verify this condition, a posteriori , online.
- Second, the computationally most intensive online calculation (for large N) is precisely this ‖uN (µ)−uN (µj)‖L4(Ω) evaluation; however, by invoking the symmetry summation techniques developed in Section 3.3.3, the authors can reduce the relevant operation count to 124N 4 — typically not dominant for the small N realized by their adaptive sampling process.
- Third, for Burgers equation in R1, their reduced-basis approach is not competitive (even as regards marginal cost) with standard techniques, that is, direct computation of sN (µ).
- Their complexity estimates also apply to incompressible Navier-Stokes in R2,3, in which case the authors effect very considerable savings relative to finite element calculation of sN (µ).

### 4.4 Numerical Results

- All results presented are for the adaptive sampling procedure.
- It is possible that deflation techniques — similiar to those introduced in the context of the Helmholtz problem in Section 2.4.3 — could considerably increase the effective inf-sup parameter, and hence considerably decrease J .
- The authors observe that the reduced-basis approximation converges very rapidly; that at least in this particular case, the “good” choice, (120), obtains — ‖e(µ)‖Y ≤ ∆N (µ), ∀ N ∈ N; that the effectivities are, as desired, quite close to unity; and that ΥN (µ) is (constant and) very large.
- (Recall that these results are for the adaptive sampling procedure; in the case of a random sample, condition (119) is not satisfied for all N .).

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