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POD-Galerkin model order reduction for parametrized nonlinear time-dependent optimal flow control: an application to shallow water equations

Maria Strazzullo, Francesco Ballarin and Gianluigi Rozza

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.

MSC 2010: 65M60; 49M41; 76N99

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.

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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:

0TΩvtϰdΩdt+μ1a1((v,h),(ϰ,ξ))+μ20Ta1nl((v,h),(v,h),(ϰ,ξ))dt+0Ta2(v,h),(ϰ,ξ)dt=0TΩuϰdΩdt0TΩhtξdΩdt+0Ta2nl((v,h),(v,h),(ϰ,ξ))dt=0.

The forms a1(,),a2(,),a1nl(,,),a2nl(,,) are defined as follows:

a1:y×yRa1((v,h),(ϰ,ξ))=Ωv:ϰdΩa2:y×yRa2((v,h),(ϰ,ξ))=Ωgϰ:hdΩa1nl:y×y×yRa1nl((v,h),(w,φ),(ϰ,ξ))=Ωϰ:(v)wdΩa2nl:y×y×yRa2nl((v,h),(w,φ),(ϰ,ξ))=Ωξdiv(φv)dΩ.

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

α0TΩuτdΩdt=0TΩχτdΩdtτU

while the weak adjoint equation, reads

0TΩvzdΩdt0TΩχtzdΩdt+μ10Ta1((χ,λ),(z,q))dt+μ20Ta1nl((v,h),(χ,λ),(z,q))dt+0Ta2nl((v,h),(χ,λ),(z,q))dt=0TΩvdzdΩdt0TΩhqdΩdt0TΩhtqdΩdt+0Ta3nll((v,h),(χ,h),(z,q))dt+0Ta3((χ,λ),(z,q))dt=0TΩhdqdΩdt

for all (z, q) ∈ 𝒴, where the involved forms are defined as

a1a1:y×yRa1((χ,λ),(z,q))=Ωχ:zdΩa1nl:y×y×yRa1nl((v,h),(χ,λ),(z,q))=Ω(v)χzdΩ+0TΩ(v)TχzdΩdta2nl:y×y×yRa2nl((v,h),(χ,λ),(ϰ,ξ))=ΩhλzdΩa3nl:y×y×yRa3nl((v,h),(χ,λ),(z,q))=ΩvλqdΩa3:y×yRa3((χ,λ),(z,q))=gΩdiv(χ)qdΩ.

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 K,D,HvkN,HˉvkN,GvkN,GvkN, and FhhN, where

(K)ij=a1φj,φj,φi,φi(D)ij=a2φj,φj,φi,φiHvkNij=a1nlvkN,hkN,φj,φj,φi,φiHˉvkNij=ΩvkNTφiφjdΩGvkNij=a2nlvkN,hkN,φj,φj,φi,φiGvkNij=a3nlvkN,hkN,φi,φi,φj,φjFhkNij=a2nlvkN,hkN,φi,φi,φj,φj.

Moreover, for the sake of notation, let us define the operators SvkN=μ1ΔtK+μ2ΔtHvkN and SvkN= μ1ΔtKμ2ΔtHvkN+μ2ΔtHˉvkN. Then, the explicit form for the residual vector is

(B.1) RwN;μ=ΔtMvvˉ+K1vNX+K2hNλˉΔtMhhˉ+K3χˉ+K4vNλˉαΔtMuuˉΔtMuχˉK1vNv+K2hˉΔtMuuˉK4vNhˉG(wˉ;μ)wˉΔtMvvˉdΔtMhhˉd0Mvvˉ0Mhhˉ0fˉ

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 K2=diag{ΔtD,,ΔtD},K2hN= diagΔtFh1N,,ΔtFhNtN,K3=diagΔtDT,,ΔtDT and the matrices

K 1 v N = M v + S v 1 N 0 0 M v M v + S v 2 N 0 0 0 M v M v + S v 3 N 0 0 0 0 M v M v + S v N t N K 4 v N = M h + Δ t G v 1 N 0 0 M h M h + Δ t G v 2 N 0 0 0 M h M h + Δ t G v 3 N 0 0 0 0 M h M h + Δ t G v N t N K 1 v N = M v + S v 1 N M v 0 0 M v + S v 2 N M v 0 0 0 M v + S v N t 1 N M v 0 0 0 M v + S v N t N

and

K4vN=Mh+Gv1NMh00Mh+Gv2NMh000Mh+GvNt1NMh000Mh+GvNtN.

Thus, the residual ℛ(wN; μ) can be also written in the following compact form

(B.1) RwN;μ=ΔtM[vˉ,hˉ]+KvN,hN[χˉ,λˉ]αΔtMuuˉΔtMuχˉKvN[vˉ,hˉ]ΔtMu0[uˉ,0]ΔtMvˉd,hˉd0Mvˉ0,hˉ0

where

M=Mv,00,Mh,KvN=K1vNK20K4vN,KvN,hN=K1vNK2hNK3K4vN,Mu0=Mu,00,0.

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: HˉvkN, with HˉvkNij=a1nlφj,φj,vkN,hkN,φi,φi, and FhkN with FhkNij=a2nlφi,φj,vkN,hkN,φi,φi. Then, defining

SvkND:=DSvkNvk=μ1ΔtK+μ2ΔtHvkN+μ2ΔtHˉvkN

we can differentiate the state equation as follows:

DKvNvˉhˉΔtMuuˉ.

The process will affect only nonlinear terms and will lead to a linearized system of the form

(B.3) K D v N , h N v h _ Δ t M u u _

with KD=K1DvNK2K3DhNK4vN, where K3D=diagFh1N,,FhNtN and

K1D=Mv+Sv1ND00MvMv+Sv2ND000MvMv+Sv3ND0000MvMv+SvNtND

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: HχkN, HˉχkN,FˉλkN, and GˉλkN, where

HχkNij=ΩφiχkNφjdΩ,HˉχkNij=ΩφiNTχkNφjdΩFˉλkNij=a2nlφj,φj,χkN,λkN,φj,φj

and

GˉλkNij=a2nlφj,φj,χkN,λkN,φj,φj.

Thanks to these quantities, we can perform the differentiation of the adjoint equation

D Δ t M v ¯ h ¯ + K v N , h N χ ¯ λ ¯

which will result in the following linearized system

(B.4) Δ t M + K D χ N , λ N M D χ N , λ N v h _ + K v N , h N χ λ _

whereMD=M1DχNM2DλNM3DλNM4D with the block diagonal matrices defined by

M1DχN=diagMv+μ2Hχ1N+μ2Hˉχ1N,,Mv+μ2HχNtN+μ2HˉχNtN

withNNNNN2DλN=diagFˉλ1N,,FˉλNtN,N3DλN=diagGˉλ1N,,GˉλNtN, and M4D=diagMh,,Mh.

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 v ¯ h ¯ u ¯  and with  p ¯ = χ ¯ λ ¯ the adjoint variable, and defining

A=ΔtMD00αΔtMu,B=KDΔtMu

it is simple to remark that the Frechét derivative can be read in the following saddle point framework:

(B.5) D(wˉ;μ)wˉ=ABTB0χˉpˉ.

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.

Received: 2020-12-08
Revised: 2021-08-05
Accepted: 2021-10-10
Published Online: 2022-03-04
Published in Print: 2022-03-28

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