negative. With the constant α > 0, we can increase the weight of this factor, making it crucial for the selection of the parameter value.
We stop the Adaptive RBF algorithm when the addition of a new parameter value does not essentially change the radial basis interpolant of the stability factor. That is when for a prescribed tolerance, the norm of the difference of two consecutive iterates is under this tolerance.
Related to the computational complexity of this algorithm, note that the evaluation of C(µ) is independent of Nh, and it only depend on the number of elements of DI and Ddisc. On the other hand, the online computation of βI(µ) only needs to solve a linear system with a symmetric matrix of dimension nI+ P + 1. This allows us to compute a fast approximation of the stability factor.
1.6 Empirical Interpolation Method
In this section we introduce the Empirical Interpolation Method (EIM), which was first presented in [9], and then extended in [45]. This algorithm allows the treatment of non-linear and non-affine terms (with respect to the parameter) in the reduced basis method. This method relies in the computation of a Lagrangian interpolant function over a set of points, called magic points (cf. [68]), properly selected.
When we arise with non-affine and non-linear problems, being able to approx-imate those terms is crucial. With the EIM we are able to tensorize non-affine and non-linear terms, decoupling the parameter dependency with respect to the spatial variables.
For the suitability of the EIM, we define a family of parameter-dependent functions W = {g(·; µ), µ ∈ D} ⊂ L∞(Ω). With the EIM, we construct the reduced-space WM = span{q1(x), . . . , qM(x)} ⊂ W, by selecting properly snap-shots of functions of the space W. The selection of those snapsnap-shots is given by a greedy procedure, in the same way as the algorithm explained in section 1.3. The EIM also provides a set of interpolation points TM = {x1, . . . , xM} and a set of parameter values SM = {µ1EIM, . . . , µMEIM}, defined during the EIM procedure.
Thank to the EIM, we can approximate any function g ∈ W as a linear combination of elements of the EIM-reduced space WM, i.e.,
g(x; µ) ≈ IM[g] =
M
X
k=1
σk(µ)qk(x), (1.34)
where we are denoting by IM[g] the empirical interpolant of the function g ∈ W, with respect to the space WM.
Let Mmaxthe maximum number of basis we consider for the EIM. We describe the EIM algorithm below:
1. First, we randomly select a parameter value µ1 ∈ DEIM, where here DEIM is the discrete set where we select the possible values of µ in the EIM. We set
x1 = arg sup
x∈Ω
|g1(x)|, q1(x) = g1(x)
g1(x1), B111 = 1, (1.35) and define S1 = {µ1}, T1 = {x1} and W1 = span{q1}.
2. Then, for 2 ≤ M ≤ Mmax, solve for each µ ∈ DEIM the linear system
M −1
X
j=1
BM −1ij σj(µ) = g(xi; wh(µ)), 1 ≤ i ≤ M − 1. (1.36)
We define the empirical interpolation of g(µ) on WM −1 as
IM −1[g(µ)] =
M −1
X
j=1
σj(µ)qj(x).
3. Select the next parameter value as µM = arg max
µ∈DEIM
kg(µ) − IM −1[g(µ)]k∞, (1.37)
4. Finally, we define rM(x) = g(x; µM) −
M −1
X
j=1
σjM −1(µM)qj(x). Finally, we set
xM = arg sup
x∈Ω
|rM(x)|, qM(x) = rM(x)
rM(xM), BMij = qj(xi), 1 ≤ i, j ≤ M, and set SM = {µ1, . . . , µM}, TM = {x1, . . . , xM} and WM = span{q1, . . . , qM}.
5. Stop if max
µ∈DEIM
kg(µ) − IM[g(µ)]k∞ ≤ εEIM.
The EIM algorithm is supported for the following result (see e.g. [9] or [90]
for the proof).
Lemma 1.1. The construction of the interpolation points is well-defined, and the functions {q1, . . . , qM} form a basis of WM, which has dimension M . Moreover, the matrix BM is lower triangular with (BM)ii = 1, for i = 1, . . . , M .
1.6. Empirical Interpolation Method 27
In [9] there is also the numerical analysis of the error for the EIM, with an a posteriori error estimator for regular non-linear functions. The a posteriori error bound estimator for the EIM defined in [9] is given by
∆EIMM = |g(xM +1; µ) − IM[g(xM +1; µ)]|, (1.38) with xM +1the magical point in the (M + 1)-th iteration of the Greedy algorithm in the EIM. The deduction of this a posteriori error bound estimator can be found in [9] (Section 3.2). In practice, this a posteriori error bound is useful only when the non-linear function only depends on the parameter and the domain.
For high non-linear problems, where we have to approximate via EIM a function that depends on the solution, this a posteriori error bound is no longer valid.
Alternatively to the EIM, the Discrete Empirical Interpolation Method (DEIM) (cf. [29]) is also used for the approximation of non-linear and non-affine terms.
The philosophy of DEIM is the same that the EIM, but in the DEIM the con-struction of the space WM ∈ W is done with a POD sample instead a Greedy algorithm like in the EIM.
Chapter 2
A Smagorinsky Reduced Basis turbulence model
2.1 Introduction
In this chapter we present a reduced basis Smagorinsky turbulence model for steady flows, that is a Large Eddy Simulation (LES) model. We first start defining the FE problem, and then, by selecting snapshots properly with the greedy algo-rithm, we obtain the low-dimensional spaces for which we define the Smagorinsky RB model. The eddy viscosity is modeled by a non-linear term, that depends on the mesh size and the modulus of the velocity gradient. We approximate this non-linear term using the Empirical Interpolation Method (section 1.6), in order to obtain a linearised decomposition of the reduced basis Smagorinsky model.
We address the construction of a Reduced Basis (RB) Smagorinsky turbulence model (cf. [103]), which is the basic Large Eddy Simulation (LES) turbulence model, in which the effect of the subgrid scales on the resolved scales is modeled by eddy diffusion terms (cf. [28, 99]). It is an intrinsically discrete model, since the eddy viscosity term depends on the mesh size. Although it is well known that the Smagorinsky model is over-diffusive, the construction of this RB Smagorinsky turbulence model represents a first step complex enough, essential for the RB models presented in the following Chapters.
The RB Smagorinsky model is based upon an a posteriori error estimation, essential in the snapshots selection in the greedy algorithm. The theoretical de-velopment of the a posteriori error estimation is based on [36] and [72], according to the Brezzi-Rappaz-Raviart stability theory, and adapted for the non-linear eddy diffusion term, extending the work done for the Navier-Stokes equations to the Smagorinsky turbulence model.
Thanks to the EIM, the RB Smagorinsky turbulence model can be decoupled 29
in a online/offline procedure. First, in the offline stage, we construct hierarchical bases in each iteration of the Greedy algorithm, by selecting the snapshots which have the maximum a posteriori error estimation value. To assure the Brezzi inf-sup condition on our Reduced Basis space, we have to define a supremizer operator on the pressure solution, and enrich the reduced velocity space. Then, in the Online stage, we are able to compute a speedup solution of our problem, with a good accuracy.
We have performed several tests of the reduced model to solve 2D step and cavity flows, with Reynolds number ranging in intervals in which a steady solution is known to exist. We obtain speed-up rates of several thousands, with normalized errors with respect to the finite element solution below 10−4.
This chapter is structured as follows: in section 2.2 we present the Finite El-ement Smagorinsky turbulence model (subsection 2.2.1) and the Reduced Basis Smagorinsky turbulence model (subsection 2.2.2). After that, in section 2.3, we present the numerical analysis that we need in order to assure the well-possedness, based on the BRR theory, of the discrete problem presented in subsection 2.2.1.
The construction and analysis of the a posteriori error bound estimator is pre-sented in section 2.4. Then, in section 2.5, we explain more in detail how we treat the non-linear eddy viscosity corresponding with the Smagorinsky term. Finally, we present some numerical results in section 2.6, where we show the reduction of the computational time in two different tests.