Abstract
In the present paper we propose reduced order methods as a reliable strategy to efficiently solve parametrized optimal control problems governed by shallow waters equations in a solution tracking setting. The physical parametrized model we deal with is nonlinear and time dependent: this leads to very time consuming simulations which can be unbearable, e.g., in a marine environmental monitoring plan application. Our aim is to show how reduced order modelling could help in studying different configurations and phenomena in a fast way. After building the optimality system, we rely on a POD-Galerkin reduction in order to solve the optimal control problem in a low dimensional reduced space. The presented theoretical framework is actually suited to general nonlinear time dependent optimal control problems. The proposed methodology is finally tested with a numerical experiment: the reduced optimal control problem governed by shallow waters equations reproduces the desired velocity and height profiles faster than the standard model, still remaining accurate.
Acknowledgment
We acknowledge the support by European Union Funding for Research and Innovation (Horizon 2020 Program) in the framework of European Research Council Executive Agency: Consolidator Grant H2020 ERC CoG 2015 AROMA-CFD project 681447 ‘Advanced Reduced OrderMethods with Applications in Computational Fluid Dynamics’. We also acknowledge the INDAM-GNCS project ‘Advanced intrusive and non-intrusive model order reduction techniques and applications’ and the PRIN 2017 ‘Numerical Analysis for Full and Reduced Order Methods for the efficient and accurate solution of complex systems governed by Partial Differential Equations’ (NA-FROM-PDEs). The computations in this work have been performed with RBniCS [52] library, developed at SISSA mathLab, which is an implementation in FEniCS [42] of several reduced order modelling techniques; we acknowledge developers and contributors to both libraries.
References
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A Details on the continuous model
We here propose a more detailed discussion on the problem formulation introduced in Section 2.2. First of all, the state equation (2.1) can be interpret in weak formulation: Given a parameter μ ∈ 𝒫, find x ∈ X which minimizes (2.2) constrained to 𝒮(x, (ϰ, ξ); μ) = 0 for all (ϰ, ξ) ∈ 𝒴, where 𝒮(x, (ϰ, ξ); μ) = 0 reads as follows:
The forms
The nonlinear nature of the bilinear forms have been specified through the apex ‘nl’. Considering the adjoint variable (χ, λ) := (χ(μ), λ(μ)) ∈ 𝒴, it is clear that the weak form of the optimality equation is given by
while the weak adjoint equation, reads
for all (z, q) ∈ 𝒴, where the involved forms are defined as
B Details on the space-time formulation
We here propose a detailed discussion of the algebraic structure briefly presented in Section 2.3. First of all,we specify the nature of the residual vector ℛ(wN; μ). Then, we aim at underlining the saddle point structure of J(̄w; μ). This concept is fundamental for our formulation to comply with classical references for optimization problems such as [11, 29, 30, 62, 63]. Moreover, the saddle point structure arising from linearization justifies the reduction techniques proposed in Section 3.3, already exploited for linear steady OCP(μ)s in [47, 48, 56], and for time dependent problems in [65, 66].
To this purpose, we define Mv, Mu, and Mh as mass matrices with respect to the variables v, u, and h, respectively, and
Moreover, for the sake of notation, let us define the operators
where ℳv,ℳh, and ℳu are block diagonal matrices with diagonal entries {Mv , . . . , Mv}, {Mh , . . . , Mh}, and {Mu, . . . , Mu}, respectively.
In addition, we define the block diagonal matrices given by
and
Thus, the residual ℛ(wN; μ) can be also written in the following compact form
where
We remark that the first, second, and last row of ℛ(wN, μ) represent adjoint, optimality, and state equations, respectively.We want now focus our attention on J(̄w; μ). For the sake of clarification, we underline that with the notation (⋅)D,we denote a quantitywhich derives from the differentiation of operators. The differentiation will be applied to general space-time variables that are denoted with v, h, u, χ, λ. Let us start our analysis with the state equation. New operators are needed:
we can differentiate the state equation as follows:
The process will affect only nonlinear terms and will lead to a linearized system of the form
with
The differentiation of the optimality equation (2.7) leads to the same equation, due to its linearity. Let us differentiate the adjoint equationk. In Norder to write the linearized systemwe define four more operators:
and
Thanks to these quantities, we can perform the differentiation of the adjoint equation
which will result in the following linearized system
where
withNNNN
We underline that, in (B.2) the 𝒦∗ =𝒦T, due to the nonlinearity of the involved forms in the system, then no saddle point structure arises. However, we can recast the linearized problem in a saddle formulation since 𝒦∗ ≡ 𝒦DT. Indeed, calling with ̄x the state-control space-time vector variable
it is simple to remark that the Frechét derivative can be read in the following saddle point framework:
We remark that J(w̄; μ) is actually a generalized saddle point matrix (see [11] as references), where A ≠ AT. Still, we will always talk about saddle point structure from now on, since the generalization does not affect the reduced strategy used [14]: indeed, the solvability condition remains the fulfillment of the inf-sup condition [5, 13, 15] over the state equation for the symmetric part of A. Moreover, the saddle point structure does not depend on the discretization scheme used: it can be generalized for other space and time approximations.
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