A reduced framework for optimization and inverse problems

The main focus of this thesis is the development, analysis and application of a reduced integrated setting for efficient solution of optimization problems based on parametrized PDEs. This framework exploits the Reduced Basis (RB) method [253, 237, 280] for a rapid and reliable approximation of parametrized PDEs, built upon a full-order discretization technique such as the

FE method. Of course, this is not the only available option, since many other techniques – such as, for instance, Proper Orthogonal Decomposition (POD) [15, 33, 144] – can be exploited to achieve computational reduction. In practice, for inverse problems such as optimal control and shape optimization, very often the standard adjoint-based approach turns to be too computationally expensive; the presence of uncertainty e.g. in measurements or in model parameters make the solution of inverse problems even less feasible.

Substantial computational saving becomes possible thanks to a reduced order model (ROM) which relies on two reduction steps: (i) parametrization of the control inputs and (ii) substitution of the full-order solution of the state problem with a reduced-order solution, which in our case is obtained through the reduced basis method. In particular, the parametrization of control variables in OC problems, as well as the parametrization of (simple) uncertain quantities in a context where shape variation are not involved, is usually straightforward, and is made a priori when the model is set. In this case, by expressing for instance the set of admissible controls as a family of functions depending on the parameter vector µ ∈ D, i.e.

Uad= {u = u(µ) ∈ U : µ ∈ Dad⊆ D},

the optimal control problem (1.10) can be rewritten in a parametric framework as follows:

ˆ

µ = arg min

µ∈Dad

J (µ) s.t. A(y(µ), w) = F (w; µ), ∀w ∈ X, (1.14)

where y = y(µ) is the state of the system – now expressed directly as a function of the control parameter vector µ, A(·, ·) the (in case, parameter dependent) operator corresponding to a parametrized PDE (see equation (1.2)) and J (µ) = ˜J (y(µ), µ) the parametrized version of the cost functional to be minimized. In the same way, by expressing the set of uncertain features affecting the state system as

Qad= {q = q(µ) ∈ Q : µ ∈ Dad⊆ D},

the inverse identification problem (1.13) can be rewritten as follows:

ˆ

µ = arg min

µ∈Qad

J (µ) = kz(y(µ)) − z∗k2

Z s.t. A(y(µ), w; µ) = F (w; µ), ∀w ∈ X. (1.15)

Thus, once the control function or the uncertain features have been expressed with respect to some parameters, the problems at hand are rewritten as parametric optimization problems, where each input/output evaluation requires the solution of a parametrized PDE fitting the framework presented in Sect. 1.2. These problems are defined on a fixed geometry, which does not affect the problem and does not require any treatment. On the other hand, parametrization of shapes requires particular care (and additional work) in case of shape optimization and shape-related inverse problems – our ultimate goal. Here we address the main blocks and guidelines of this framework; specific ingredients will be detailed within the next chapters.

Shape parametrization techniques for geometrical complexity reduction

A first key to simplification consists of making use of a fixed-domain approach in shape-related problems. To do this, we introduce a reference configuration on which every problem is brought back and solved at each iteration of the optimization process, whereas geometry variations are accounted for the equation coefficients. In order to represent a set of admissible shapes through a set of geometrical parameters, we rely on suitable (low-dimensional) shape parametrizations: in this way, domains Ωo(µ) corresponding to different shape configurations are obtained from the

fixed, reference configuration Ω through a parametric map T ( · ; µ) : Ω → Ωo(µ). Several options

Hence, by mapping the problem (1.11) back to the reference domain, and considering the set of admissible shapes Oadas

Oad= {Ωo(µ) = T (Ω; µ) ∈ O : µ ∈ Dad⊆ D},

we obtain the following formulation3_:

ˆ

µ = arg min

µ∈Dad

J (µ) s.t. A(y(µ), w; µ) = F (w; µ), ∀w ∈ X, (1.16)

where Dad⊆ D is the subset of admissible parameters. We underline that at this stage the PDE

problem – as well as the functional space – has been defined on the reference domain Ω, where the new linear/bilinear forms can be expressed from the original ones through a suitable change of variables. Clearly, if the parametric map T ( · ; µ) is sufficiently flexible and low-dimensional, shape deformations can be easily handled by acting on a small number of geometrical parameters – thus yielding to an effective geometric reduction.

The introduction of such a map is also instrumental to the use of reduced basis methods, since basis solutions corresponding to different shape configurations Ωo(µ) can only be compared and

combined on a reference, parameter-independent domain. Once the original problem has been traced back to a reference configuration (and the admissible shapes have been parametrized), it results in a parametrized problem where the effect of geometry variations is traced back onto its parametrized transformation tensors – thus fitting the parametrized framework of Sect. 1.2.

Figure 1.3: Parametric optimization problems (optimal control or shape optimization).

Reduced basis method for computational reduction

To solve the parametrized PDEs appearing in the formulations (1.14)–(1.16) in a very efficient way, we rely on a suitable reduced-order method, which in our case is the reduced basis method. Thanks to a suitable computational Offline-Online decomposition (see Sect. 2.2.4), these methods provide rapid results at a greatly reduced cost. At the outer level, the solution of parametrized PDEs and the rapid input/output evaluation is exploited within a suitable iterative procedure for the (now, parametric) optimization, provided e.g. by some nonlinear programming techniques such as sequential quadratic programming and various quasi-Newton methods [23, 143].

3_{As before, we consider for the sake of simplicity in the exposition a linear state problem. The analysis}

Roughly, reduced-order methods for parametrized PDE problems are low-dimensional approx- imations to the solution of a parametrized PDE, built by choosing a basis ζi, i = 1, . . . , N of

full-order discretized solutions computed for some selected parameter values, i.e. ζi = y(µi),

where N is – hopefully – much smaller than the dimension (say N ) of the full-order discretization. Next, the reduced approximation to the state solution for a new parameter value µ is defined as

yN(µ) = N

X

j=1

yN j(µ)ζj

so that yN(µ) ∈ XN = span{ζ1 . . . , ζN}, where the coefficients {yN j} are determined through

a Galerkin projection of the state equation onto the reduced space XN. The cost of such a

computation is thus very small – compared to a full-order approximation – if N is small. Hence, if the state solution is approximated in the reduced space XN, then the cost of each optimization

step will be very small compared to that involving the full-order state approximation. Hence, throughout the optimization, only the small RB-based discrete solutions are used; this entails relevant savings in CPU costs when compared to the use of a full-order discrete system within the optimization process. We emphasize that a reduced basis method requires the solution of some full-order and therefore very expensive discrete equations. The key-idea is that these demanding calculations can be done Offline, before the optimization with respect to the control parameters is attempted. Hence, a RB-based strategy will be convenient whenever the optimization procedure requires as many steps as the number of basis functions which have to be computed Offline. However, very often the same reduced basis for the approximation of the state system can be exploited for the optimization with respect to many different cost functionals – this greatly enhances the computational saving offered by RB-based strategies. The principal features of RB methods will be introduced in Chapter 2; the RB approximation of fluid dynamics problems modelled by Stokes and Navier-Stokes equations will be presented in Chapter 3.

Computational reduction strategies such as RB methods or POD have already been employed to speedup the solution of optimal control, simpler shape optimization and other inverse problems dealing with parametrized PDEs. First examples of optimal control problems solved by exploiting computational reduction techniques have been addressed by Ito and Ravindran, in the context either of (a preliminary version of) the reduced basis method [152, 153, 154] or of the proper orthogonal decomposition method [264, 265, 267]. A more systematic approach has been firstly provided by Maday and Patera [198, 253]. More recent contributions dealing with RB methods have been presented in both the elliptic case by Quarteroni, Rozza and Quaini [257], by Tonn and Urban [309], Grepl and Kärcher [121], and the parabolic case by Dedè [73, 74]. Other recent works dealing with optimal control problems through POD techniques have been addressed for instance by Kunisch and Volkwein [176], Tonn, Urban and Volkwein [310] dealing with POD techniques.

Shape optimization problems solved by means of RB techniques have been addressed for the first time by Rozza [273] in the case of simple shape parametrizations and afterwards by Lassila and Rozza [185] in a more complex case; other inverse design problems solved through POD techniques have been presented by Willcox and Ghattas [51, 52], whereas a more recent treatment of uncertainties in aerodynamic design dealing with POD-based reduced order models to inverse has been addressed by Schulz and Schillings [295, 296]. Other reduced models, such as balanced truncation [14], have been applied in the last two years to tackle shape optimization problems by Heinkenschloss, Hoppe, Sorensen, Antil [13, 12].

Furthermore, reduced models have been recently applied to real-time parameter estimation by Nguyen and many others [225, 123], as well as to statistical inverse problems related with uncertainty quantification by Willcox, Ghattas et al. [106, 191], Patera, Huynh, Knezevic, [147], Nguyen and Rozza [227], Grepl and Veroy [124]. An exhaustive review of reduced order models in this context can be found in [101].