Selection techniques based on restriction procedures

We have implemented four techniques within the restriction paradigm for reducing the geometric design parameter space and selecting a suitable set of control points in the FFD parametrization case. A restriction paradigm is better suited in this case, where a starting set of candidate parameters is given by the lattice of control points. Instead, an adaptive paradigm should be more feasible with RBF parametrizations, where the control points can be freely positioned.

Let us assume to be able to construct a trial parametrization of the reference domain Ω with a priori large number of parameters, p0  10, and a large parameter space D0 _⊂Rp0

, which can effectively explore all the parametric variability of the PDE system we are interested in. In our case a trial parametrization is obtained by FFD, by considering sufficiently many control points. The goal is to select a reduced parameter set D ⊂ D0 of all the possible parameters µ ∈ D ⊂Rp, with p  p0, such that the parametric variability of a given quantity of interest s(µ) – e.g., the cost functional s(µ) = S(y(µ)) to be minimized – is still well explored in the lower-dimensional parameter space. Since the selection procedure has to be run before the construction of a reduced-order approximation, the exploration of the larger parameter space D0 relies on the full-order approximation – i.e. on the input/output evaluation µ → sN(µ) = S(yN(µ)). The number of these expensive input/output evaluations has to be as small as possible.

Without loss of generality, we can assume that the output functional S : X →R (and its discrete version) are under the form

S(y) = 1

2Q(y, y), S(y

N_{) =} 1

2(Qy, y),

where Q : X × X →R is a symmetric and coercive bilinear form and Q ∈ RN ×N a symmetric positive definite matrix, respectively, and y ∈RN the vector of degrees of freedom representing

yN ∈ XN _{over a basis of the discrete space X}N_{. In particular, let us denote (y, w)}

Q= (Qy, w)

the inner product defined over XN byQ and Q = HHT its Cholesky factorization. In the following we sketch the four selection recipes:

1. User’s experience-based (UEB) selection

In this case, the reduced set of retained parameters is selected according to empirical rules, based e.g. on user’s experience or problem-dependent. In fact, the closer the con- trol point to the shape, the more influent its effect on shape deformation when it is displaced.

2. One-at-a-time (OAT) selection by experimental orthogonal design

Here we exploit the simplest technique for variable screening, by adding to the retained set of parameters the ones with the largest observed sensitivities, with respect to varying one parameter at a time while keeping the others fixed. Let us define the parametric variations

µi,min and µi,maxfor i = 1, . . . , p0, by varying one parameter at a time while keeping the

other parameters fixed at a reference value ¯µ:

µi,min= ( ¯µ1, . . . , µmini , . . . , ¯µp0), µi,max= ( ¯µ₁, . . . , µmax_i , . . . , ¯µ_p0), ∀i = 1, . . . , p0.

Then, let us evaluate the parametric range w.r.t to each parameter by solving the FE approximations for yN_(µi,min_{) and y}N_(µi,max_{) for each i = 1, . . . , p}0_{, and retain the p}

parameters with the largest observed parametric sensitivities, such that     ∂s ∂µi ( ¯µ)     ≈ |S(y

N_(µi,max_{)) − S(y}N_(µi,min_))|

|µi,max_{− µ}i,min_| > ε

OAT

tol , (4.49)

where εOAT_tol > 0 is a prescribed tolerance (otherwise, we fix a priori the number p of param-

eters to be selected). We underline that the calculation of the exact parametric sensitivities would need to approximate the solution ∂y(µ)/∂µ of the sensitivity equations, and thus the parametric derivatives ∂Θ•/∂µ for a very large number p0 of parameter components

– rather unfeasible for such a preparatory stage. In any case, such an approach may fail when some design parameters are strongly correlated.

3. Morris screening procedure (MR-OAT)

Morris’ randomized one-at-a-time design is a screening procedure based on individual randomized OAT designs, accounting also for interactions among the factors [218, 57]. The basic idea is related to a sample of independently observed elementary effects, which are exploited to measure the output sensitivity for a particular input. Let us suppose that each component µi, i = 1, . . . , p0 is scaled to have a region of interest equal to [0, 1] and can

take k discrete values in the set {0, 1/(k − 1), . . . , 1}. The experimental domain is then a

p0-dimensional k-level grid in D0. The elementary effect of the i-th factor at a given point in the design space is defined as

δi(µ) =

s(µ1, . . . , µi−1, µi+ ∆, µi+1, . . . , µp0)

∆

where ∆ is a fixed multiple of 1/(k − 1). The ultimate goal of the Morris procedure is to estimate the mean and the standard deviation of the distribution of elementary effects associated with each input11_{, which can be obtained by producing a random sample of R}

elementary effects for each i = 1 . . . , p0. Here we do not provide the construction of the procedure (see [218] for details). We just mention that, by the MR-OAT procedure, R elementary effects are produced for each input parameter µi at a total cost of R(p0+ 1)

input/output evaluations (which is a linear function of the numbers of factors involved), corresponding to a sample B = {µl,r_{} ⊂ D, l = 1, . . . , p}0_{+ 1, r = 1, . . . , R.}

11_{A large absolute value for the mean of δ}

i(µ) implies that the corresponding parameter component µihas an important overall effect on the output, whereas a high standard deviation indicates that the effect of µiis not constant, which may be implied by a parameters interacting with other parameters or whose effect is nonlinear.

In this way, at the r-th step, (p0+ 1) parameters values {µl,r_}

l=1,...,p0₊₁ originate a design

matrix accounting for possible interactions among the parameters. Once the R(p0+ 1) input/output evaluations have been performed, we rank the parameters µ1, . . . , µp0 according

to its absolute mean effect

¯ δ(i) = 1 R R X r=1 |δi(µi,r)| ≈ 1 |D0_| Z D     ∂s(µ) ∂µ     dµ (4.50)

and retain as active parameter only the p  p0 most effective ones.

4. Approximate POD (APOD)

In this case we consider a greedy-like procedure over the large parameter space D0, based on a set of computed snapshots. Let us define the parametric variations µi,min _{and µ}i,max

as in (OAT) case, by varying one parameter at a time while keeping the other parameters fixed at a reference value ¯µ. Then:

• evaluate the parametric range w.r.t to each parameter by computing yN_(µi,min_{) and}

yN(µi,min), for each i = 1, . . . , p0, and set the initial parametric directions as the

vectors corresponding to yN(µi,min) and yN(µi,min):

ζmin_i = y(µi,min), ζmax_i = y(µi,max), i = 1, . . . , p0;

• sample a suitable training set Ξtrain ⊂ D0 and form the matrix of snapshots Y =

[ζ₁| . . . |ζ_S]T _{by computing}

ζs:= y(µs) − y( ¯µ), {µs}Ss=1⊂ Ξtrain.

Once these quantities have been stored, we can set the following iterative procedure:

set P0= {1, . . . , p0} P = ∅; repeat

for i ∈ P0 compute theQ − correlation factors ∆min_i := kYHζ min i k2 kζmin_i k2 , ∆max_i := kYHζ max i k2 kζmax i k2

select ˆp = arg maxi{∆mini , ∆maxi } (largest total correlation)

set P = P ∪ ˆp, P0= P0\ ˆp;

deflate all the other parametric directions: for q 6= ˆp

ζmin / maxq 7−→ ζ

min / max

q −

(ζmin / max_pˆ , ζmin / max_q )K

(ζmin / max_pˆ , ζmin / max_pˆ )K

ζmin / max_pˆ ; until maxi{∆mini , ∆

max

i } < ε AP OD

tol .

being εAP OD

tol > 0 a prescribed tolerance. We denoted this procedure as approximate

POD. In fact, the (K-weighted) POD modes are given by the right singular vectors ξ_i of YH = UΣVT_{, whereas their significance is measured by the singular values σ}

i. We can

consider the quantity ζmin / max_i corresponding to the largestQ-correlation factor ∆min / max_i at each step as an approximation to the largest singular vector of YH in the subspace

spanned by the current set of ζmin / max_i .

We point out that if all parameters are independent and the snapshots are taken exactly as the parametric directions yN(µi,min), yN(µi,max), then APOD and OAT strategies coincide.

We compare these techniques applied to a simple test case in the forthcoming section. We remark that, concerning instead RBFs parametrizations, the best choice of control points (or centers) to approximate a given function has been considered as a problem of pure approximation theory. A possible greedy algorithm has been proposed in [292]; see e.g. [269] for a possible application of efficient RBF parametrizations in the framework of a shape parameterization context. We are currently investigating selection procedures for RBF centers [182].