Moreover, under the compliance assumption (i.e. if A(·, ·; µ) is a symmetric bilinear form and
L(·; µ) ≡ F (·; µ)), the output converges as the “square” of the energy error, i.e. it verifies the
following property:
sN(µ) − sN(µ) = |||yN(µ) − yN(µ)|||2µ; (2.45)
in this case, the error estimator for the output are given by
∆sN(µ) := ∆2N(µ) ≡
kˆe(µ)k2X αN
LB(µ)
. (2.46)
Although this latter result depends critically on the compliance assumption, extensions via adjoint approximations to the noncompliant case are also possible (see e.g. [237, 256] and the application to the Stokes case in Sect. 3.4.2-3.4.3). Nevertheless, the error bounds (2.44)–(2.46) are of no utility without an accompanying Offline-Online computational approach. The computationally crucial component of all the error bounds introduced is the dual norm kˆe(µ)kX of the residual,
which can be computed through an Offline-Online procedure; details about these expressions are reported in Appendix A for the Stokes (Sect. A.2.1) and the Navier-Stokes (Sect. A.2.2) case.
2.4
Extension to nonaffine problems
We briefly discuss in this section the extension of the RB methodology to nonaffine problems, which are particularly relevant to our geometrical parametrizations and related applications. Also other (more general) classes of problems, such as noncompliant and noncoercive problems, can be tackled by the RB methodology. In the next chapter the case of Stokes and Navier-Stokes problems – representing one of the most remarkable cases of noncoercive problems – will be extensively treated, In the same chapter, an example of noncompliant problem is presented as well, when dealing with linear (noncompliant) or quadratic outputs of Stokes variables. For a general introduction to noncompliant and noncoercive problems the interested reader can refer to [237, 256].
As already remarked in Sect. 2.2.4, the assumption of affine parametric dependence (1.7) is fundamental in order to exploit the Offline-Online strategy and then minimize the marginal cost associated with each input-output evaluation. However, also nonaffine problems, i.e. problems in which condition (1.7) is not still valid, can be efficiently treated in the RB framework. In this case, we rely on the Empirical Interpolation Method (EIM) [21, 122, 90], which is based on adaptively chosen interpolation points and global shape functions, and allows to recover the assumption of affine parametric dependence in nonaffinely parametrized operators (e.g. linear, bilinear forms, etc.). In the case of a nonaffinely parametrized bilinear form A(v, w; µ), the latter is replaced by an affinely parametrized approximation of the form
A(v, w; µ) =
Q
X
q=1
Θq(µ)AqEIM(v, w) + εEIM(v, w; µ), (2.47)
where the error term εEIM needs to be controlled to an acceptable tolerance. We provide a short
presentation of the EIM procedure based on [21]. Let us denote by g(x, µ) ∈ C0(D; L∞(Ω)) a
nonaffine way; the extension to tensors through an element-wise procedure is straightforward. The goal is to find an approximate expansion of the form
gM(x, µ) = M
X
j=1
Θj(µ)ζj(x), (2.48)
where Θj(µ), j = 1, . . . , M , are parameter-dependent functions and ζ
j(x), j = 1, . . . , M , are
parameter-independent functions, denoted also shape functions. Being an interpolation procedure, the EIM procedure seeks a sequence of (nested) sets of interpolation points TM = {p1, . . . , pM}
(magic points), with pj ∈ Ω for each j = 1, . . . , M , and a set of shape functions ζj(x), in order to
compute the expansion (2.48) by solving the following Lagrange interpolation problem:
M X j=1 BM i,jΘ j(µ) = g(p i, µ), ∀ i = 1, . . . , M, whereBM ∈RM ×M is defined as (BM)
ij:= ζj(pi), i, j = 1, . . . , M . Let us denote by ΞEIMtrain ⊂ D
a large training set, Mmax the maximum number of terms, ε∗EIM a fixed tolerance, and select an
initial parameter value µ1. The EIM procedure [21] is as follows:
ζ1(x) := g(x, µ1); compute p1:= arg ess supx∈Ω|ζ1(x)|;
q1= ζ1(x)/ζ1(p1); G1:= span(ζ1), setB111= 1;
for M = 2 : Mmax
solve µM := arg maxµ∈ΞEIM
traininfv∈GM −1||g(·, µ) − v||L ∞(Ω)(linear programming) set ζM(x) := g(x, µM), GM := span(ζ1, . . . , ζM) solve PM −1j=1 σjM −1qj(pi) = ζM(pi), i = 1, . . . , M − 1; compute (residual) rM(x) := ζM(x) −P M −1 j=1 σ M −1 j ζj(x);
compute pM := arg ess supx∈Ω|rM(x)|;
set qM(x) = rM(x)/rM(pM), BMij = qj(pi), i, j = 1, . . . , M ; if maxµ∈ΞEIM traininfv∈GM||g(·, µ) − v||L ∞(Ω)< ε∗EIM Mmax= M − 1; end; end.
In the end, given an approximation gM(x, µ), M < Mmax, we denote the one point error estimator
the following quantity (very inexpensive to compute): ˆ
M(µ) = |g(pM +1; µ) − gM(pM +1; µ)|, (2.49)
corresponding to the difference between the function and the interpolant at the point pM +1,
which gives the largest residual rM(x). While not rigorous as a posteriori error bound, this
quantity proves to be an intuitive measure of the error committed by the EIM procedure [21]. Advances in error bounds developments have been presented in [90, 122, 200, 226].
In practice, if the problem9is not affinely parametrized (e.g. when the geometrical transformation
(2.15) is not affine, see Sect. 2.5.2, or the physical coefficients appearing in the tensors νo,k, χo,k,
ηo,k are nonaffine functions of x and µ), the parametrized tensors in (2.17) depend both on the
parameter µ and the spatial coordinate x. In this case, the operators can not be expressed as in
9For the sake of simplicity, we consider here a parametrized bilinear form A(·, ·; µ) corresponding to a pure
(1.7) and we thus need an additional pre-processing, before the FE assembling stage, in order to recover the affinity assumption. According to EIM – considering for instance the tensor νk –
each component νk
ij(x, µ) is approximated by an affine expression given by
νkij(x, µ) = Kk ijl X l=1 βlijk(µ)ηlijk(x) + εkij(x, µ); (2.50)
all the functions βlijk’s and ηijkl ’s are efficiently computable scalar functions and the error terms are guaranteed to be under some tolerance,
kεk
ij(·; µ)k∞≤ εEIMtol ∀µ ∈ D.
In this way, we can identify the µ-dependent functions βlijk(µ) in (2.50) as the functions ΘqA(µ) in (1.7), where q is a condensed index for (i, j, k, l), while the µ-independent functions will be treated as pre-factors in the integrals which give the µ-independent bilinear forms Aq(w, v). The nonaffine treatment is really important since many problems involving more complex geometrical parametrizations and/or more complex physical properties are hold by nonaffine parametric dependence. Not only, EIM has been exploited also to deal with nonlinear operators involving polynomial functions of the variables, for instance in [122].