Reduced basis method for parametrized optimal control problems … · 2017. 5. 10. · Gianluigi...

Introduction Saddle-point formulation RB method Results

Reduced basis method for

parametrized optimal control problems

governed by PDEs

Gianluigi Rozzajoint work with Federico Negri, Alfio Quarteroni, Andrea Manzoni

MATHICSE - CMCS Modelling and Scientific Computing

Ecole Polytechnique Federale de Lausanne

J.-L. Lions and E. Magenes Memorial Days Workshop

University Pierre et Marie Curie (UPMC Paris VI) Laboratoire Jacques-Louis Lions

Paris, France, 14-15 December 2011.

Acknowledgements: A.T. Patera (MIT)

Sponsors: European Research Council - Mathcard Project, Progetto Roberto Rocca Politecnico di Milano-MIT, Swiss National Science Foundation


Outline

1. Introduction

Motivation and ingredients

2. Parametrized linear-quadratic Optimal Control Problems

Abstract formulation

Saddle-point formulation

3. Reduced Basis method for Optimal Control Problems

Reduced Basis (RB) methodology

Approximation, stability, a posteriori error bounds

4. Preliminary numerical results

Optimal control of parametrized advection-diffusion problems

Optimal control of parametrized Stokes problems

An (idealized) application for blood flows


Reduction strategies for Parametrized Optimal Control Problems

The characterization of a system in

terms of physical quantities and/or

geometrical configuration usually depends

on a set of parameters

the system response will be

parameter dependent as well,

and so will be the optimal control

Parametrized optimal control problems: the prediction of optimal control inputs and

the optimization of given output of interests is required for each different

value of the parameters.

The computational effort may be unacceptably high and, often, unaffordable when

performing the optimization process for many different parameter values (many-query

context)

for a given new configuration, we want to compute the solution in a rapid way

(real-time context)

Goal: to achieve the accuracy and reliability of a high fidelity approximationbut at greatly reduced cost of a low order model


Previous works and improvements on parametrized optimal control

early works by Ito and Ravindran [1998 and 2001]: optimal control of Navier-Stokes

equations

the RB method has been applied to parametrized linear-quadratic advection

diffusion optimal control problems in different contexts:

[Quarteroni, R., Quaini, 07], [Tonn, Urban, Volkwein, 11], [Grepl, Karcher, 11]

considering low-dimensional control variable.

we aim at developing a reduced framework that enables to handle with general

control functions, i.e. infinite dimensional distributed and/or boundary control

functions.

an efficient and rigorous a posteriori error estimation, necessary both for

constructing the reduced order model and measuring its accuracy, is still partially

missing for a large class of optimal control problems. Previuous preliminary works by

[Dede, 2010] [Tonn, Urban, Volkwein, 11] [Grepl, Karcher, 11]


Optimal control problems [Lions, 1971]

In general, an optimal control problem (OCP) consists of:

a control function u, which can be

seen as an input for the system,

a controlled system, i.e. an

input-output process: E(y , u) = 0,

beign y the state variable

an objective functional to be

minimized: J (y , u)

STATE

PROBLEM

STATE

PROBLEM

µ

Output J (y, u)

Output

yd (µ) (data)

J (y, u; µ)

y(u)

Optimization:

update control u

u











STATE

PROBLEM

STATE

PROBLEM

µ

Output J (y, u)

Output

yd (µ) (data)

J (y, u; µ)

y(u)

Optimization:

update control u

u

find the optimal control u∗ and the state y(u∗) such that the cost functional

J (y , u) is minimized subject to E(y , u) = 0(OCP)











STATE

PROBLEM

STATE

PROBLEM

µ

Output J (y, u)

Output

yd (µ) (data)

J (y, u; µ)

y(u)

Optimization:

update control u

u

find the optimal control u∗ and the state y(u∗) such that the cost functional

J (y , u) is minimized subject to E(y , u) = 0(OCP)

We restrict attention to:

quadratic cost functionals, e.g. J(y , u) = 12‖y − yd‖2 + α

2‖u‖2

stationary, linear, elliptic state equations: in particular (scalar coercive)

advection-diffusion equations and then (vectorial noncoercive) Stokes equations


Parametrized optimal control problems

A parametrized optimal control problem (OCPµ) consists of:

a control function u(µ), which can

be seen as an input for the system,


input-output process:

E(y(µ), u(µ);µ) = 0,


minimized: J (y(µ), u(µ);µ)

STATE

PROBLEM

STATE

PROBLEM

µ

Output J (y, u)

Output

yd (µ) (data)

J (y, u; µ)y(u(µ))

Optimization:

update control u

u(µ)

given µ ∈ D, find the optimal control u∗(µ) and the state y∗(µ) such that the

cost functional J (y(µ), u(µ);µ) is minimized subject to E(y(µ), u(µ);µ) = 0(OCPµ)

where µ ∈ D ⊂ Rp denotes a p-vector whose components can represent:

coefficients in boundary conditions

geometrical configurations

physical parametrization

data (observation)



for optimal control problems


The abstract optimization problem

Functional setting: Y state space, U control space, Q(≡ Y ) adjoint space,

Z observation (data) space s.t. Y ⊂ Z,

In the following: y , z ∈ Y u, v ∈ U, p, q ∈ Q.


The abstract optimization problem: quadratic cost functional




we consider the following quadratic cost functional to be minimized

J(y , u;µ) =1

2m(y − yd (µ), y − yd (µ);µ) +

α

2n(u, u;µ)

where yd (µ) ∈ Z is a given function, α > 0 regularization parameter.

We make the following assumptions: for any µ ∈ D

m(·, ·;µ) : Z × Z → R is a symmetric, continuous and positive bilinear form

n(·, ·;µ) : U × U → R is a symmetric, continuous and coercive bilinear form


The abstract optimization problem: linear constraint




the linear constraint is expressed by the following state equation (elliptic PDE in

weak form)

a(y , q;µ) = c(u, q;µ) + 〈G(µ), q〉 ∀q ∈ Q

where G(µ) ∈ Q ′ for all µ ∈ D.

We make the following assumptions: for any µ ∈ D

a(·, ·;µ) : Y × Q → R is a continuous and (weakly) coercive bilinear form

c(·, ·;µ) : U × Q → R is a continuous bilinear form

Holding these assumptions, given u ∈ U, the state equation is uniquely solvable

for any µ ∈ D.


The abstract optimization problem

Parametrized optimal control problem

minimize J(y , u;µ) =1

2m(y − yd (µ), y − yd (µ);µ) +

α

2n(u, u;µ)

s.t. a(y , q;µ) = c(u, q;µ) + 〈G(µ), q〉 ∀q ∈ Q.

In order to develop the Reduced Basis (RB) method

we firstly recast the problem in the framework of saddle-point problem

[Gunzburger, Bochev, 2004]

we then apply the well-known Brezzi theory [Brezzi, Fortin, 1991]

→ existence, uniqueness, stability and optimality conditions

This way we can exploit the analogies with the already developed theory of RB

method for Stokes-type problems [R., Veroy, 2007; R., Huynh, Manzoni 2010]



Let X ≡ Y × U the state and control space

Q the adjoint (Lagrange multiplier) space as before

and let x = (y , u) ∈ X , w = (z , v) ∈ X , with ‖x‖2X = ‖y‖2

Y + ‖u‖2U

Let A(x ,w ;µ) = m(y , z ;µ) + αn(u, v ;µ)

〈F (µ),w〉 = m(yd (µ), z ;µ)

t(µ) =1

2m(yd (µ), yd (µ);µ)

the cost functional can be expressed as

J(x ;µ) = 12A(x , x ;µ) − 〈F (µ), x〉 + t(µ) = J (x ;µ) + t(µ)

Let B(w , q;µ) = a(z , q;µ)− c(v , q;µ),

the constraint equation can be expressed as

B(x , q;µ) = 〈G(µ), q〉 ∀q ∈ Q


Saddle-point formulation: applying Brezzi theory

The optimal control problem can be recast in the form: given µ ∈ D,

min J (x ;µ) =1

2A(x , x ;µ)− 〈F (µ), x〉, subject to

B(x , q;µ) = 〈G(µ), q〉 ∀q ∈ Q.

Let X0 = w ∈ X : B(w , q;µ) = 0 ∀q ∈ Q ⊂ X , the assumptions we have made

imply that (e.g. [Gunzburger, Bochev, 2004]):

the bilinear form A(·, ·;µ) is continuous over X × X and coercive over X0

the bilinear form B(·, ·;µ) is continuous over X × Q and satisfies the following

inf-sup condition

∃β0 > 0 : β(µ) = infq∈Q

supw∈X

B(w , q;µ)

‖w‖X‖q‖Q≥ β0, ∀µ ∈ D

the bilinear form A(·, ·;µ) is symmetric and non-negative over X


Saddle-point formulation: applying Brezzi theory

the optimal control problem

minx∈X

J (x ;µ) subject to B(x , q;µ) = 〈G(µ), q〉 ∀q ∈ Q.

has a unique solution x = (y , u) ∈ X for any µ ∈ D

that solution can be determined by solving the optimality systemA(x(µ),w ;µ) + B(w , p(µ);µ) = 〈F (µ),w〉 ∀w ∈ X ,

B(x(µ), q;µ) = 〈G(µ), q〉 ∀q ∈ Q,

Compact form

given µ ∈ D, find U(µ) ∈ X s.t:

B(U(µ),W;µ) = F(W;µ) ∀W ∈ X .

X = X × Q, U = (x , p), W = (w , q)

B(U,W;µ) = A(x ,w ;µ) +B(w , p;µ) +B(x , q;µ)

F(W;µ) = 〈F (µ),w〉+ 〈G(µ), q〉

at this point we may apply the Galerkin-FE approximation


Reduced Basis methodfor parametrized optimal control problems


RB method: Construction

Pb(µ; U(µ))

µ-OCP, optimality system

U(µ) ∈ X : B(U(µ),W;µ) = F(W) ∀W ∈ X

PbN (µ; UN (µ))

Truth approximation (FEM)

UN (µ) ∈ XN : B(UN (µ),W;µ) = F(W) ∀W ∈ XN

Truth Hypothesis: UN (µ) “indistinguishable” from U(µ).

RB Motivation: µ→ UN (µ), JN (µ) too expensive and slow in

many-query and real-time contexts.

Sampling (Greedy)

Space Construction

(Hierarchical Lagrange basis)

OFFLINE

SN = µi , i = 1, . . . ,N

XN = spanUN (µi ), i = 1, . . . ,N

dim(XN ) = N N = dim(XN )

PbN (µ; UN (µ))

Galerkin projection

ONLINE

Reduced Basis (RB) approximation

UN (µ) ∈ XN : B(UN (µ),W;µ) = F(W) ∀W ∈ XN

[R., Huynh, Patera 2008, Talk by A. Manzoni]


RB method: smooth parametric dependency

Guess: MN is low dimensional - smooth

dependence on µ (Lipschitz continuity)

(Adaptive) sampling procedure for parameter

exploration

reduced basis made of optimal solutions of the

FE “truth” problem

Evaluation procedure: (optimal) Galerkin

projection

[Noor, Peters, Porshing, Lee, ...]

MN

M

XN

M = U(µ) ∈ X : µ ∈ D

MN = UN (µ) ∈ XN : µ ∈ D

XN = spanUN (µi ), i = 1, . . . ,N

How to be rigorous, rapid and reliable?

depends on the sampling procedure for parameter exploration (greedy algorithm)

exploits an Offline/Online stratagem based on the affinity assumption:

B(U,W;µ) =

Qb∑q=1

Θqa (µ)Bq(U,W), F(W;µ) =

Qf∑q=1

Θqf (µ)Fq(W)

relies on a posteriori error analysis


RB method: smooth parametric dependency

Guess: MN is low dimensional - smooth

dependence on µ (Lipschitz continuity)

(Adaptive) sampling procedure for parameter

exploration

reduced basis made of optimal solutions of the

FE “truth” problem

Evaluation procedure: (optimal) Galerkin

projection

[Noor, Peters, Porshing, Lee, ...]

MN

XN

M = U(µ) ∈ X : µ ∈ D

MN = UN (µ) ∈ XN : µ ∈ D

UN (µ1)

UN (µN )

XN = spanUN (µi ), i = 1, . . . ,N

How to be rigorous, rapid and reliable?

depends on the sampling procedure for parameter exploration (greedy algorithm)

exploits an Offline/Online stratagem based on the affinity assumption:

B(U,W;µ) =

Qb∑q=1

Θqa (µ)Bq(U,W), F(W;µ) =

Qf∑q=1

Θqf (µ)Fq(W)

relies on a posteriori error analysis


Reduced Basis Method: approximation stability

Reduced Basis (RB) approximation: given µ ∈ D, find (xN (µ), pN (µ)) ∈ XN × QN :A(xN (µ),w ;µ) + B(w , pN (µ);µ) = 〈F (µ),w〉 ∀w ∈ XN

B(xN (µ), q;µ) = 〈G(µ), q〉 ∀q ∈ QN

(∗)

How to define the reduced basis spaces?

we have to provide a spaces pair XN ,QN that

guarantee the fulfillment of an equivalent parametrized Brezzi inf-sup condition

βN (µ) = infq∈QN

supw∈XN

B(w , q;µ)

‖w‖X ‖q‖Q≥ β0, ∀µ ∈ D.

For the state and adjoint variables: aggregated spaces

YN ≡ QN = span

yN (µn), pN (µn)N

n=1

For the control variable:

UN = span

uN (µn)N

n=1

Let XN = YN × UN , we can prove that

βN (µ) ≥ αN (µ) > 0 being αN (µ) the coercivity constant associated to the FE

approximation of the PDE operator

Brezzi theorem =⇒ for any µ ∈ D, the RB approximation (∗) has a unique solution

depending continuously on the data


Reduced Basis Method: approximation stability

Reduced Basis (RB) approximation: given µ ∈ D, find (xN (µ), pN (µ)) ∈ XN × QN :A(xN (µ),w ;µ) + B(w , pN (µ);µ) = 〈F (µ),w〉 ∀w ∈ XN

B(xN (µ), q;µ) = 〈G(µ), q〉 ∀q ∈ QN

(∗)

How to define the reduced basis spaces? we have to provide a spaces pair XN ,QN that

guarantee the fulfillment of an equivalent parametrized Brezzi inf-sup condition

βN (µ) = infq∈QN

supw∈XN

B(w , q;µ)

‖w‖X ‖q‖Q≥ β0, ∀µ ∈ D.

For the state and adjoint variables: aggregated spaces

YN ≡ QN = span

yN (µn), pN (µn)N

n=1

For the control variable:

UN = span

uN (µn)N

n=1

Let XN = YN × UN , we can prove that

βN (µ) ≥ αN (µ) > 0 being αN (µ) the coercivity constant associated to the FE

approximation of the PDE operator

Brezzi theorem =⇒ for any µ ∈ D, the RB approximation (∗) has a unique solution

depending continuously on the data


RB method: algebraic formulation

We can express the RB state, control and adjoint solutions as

xN (µ) =3N∑j=1

xNj (µ)σj , denoting XN = spanσj

3N

j=1

pN (µ) =2N∑j=1

pNj (µ)τj , denoting QN = spanτj

2N

j=1

recalling the assumption of affine parametric dependence (also for the RHS)

A(x ,w ;µ) =

Qa∑q=1

Θqa (µ)Aq(x ,w), B(w , p;µ) =

Qb∑q=1

Θqb(µ)Bq(w , p)

for a new parameter µ, the RB solution can be written as a combination of basis functions

with weights given by the following reduced basis linear system:

3N∑j=1

Qa∑q=1

Θqa (µ)Aq

ij xNj (µ) +2N∑l=1

Qb∑q=1

Θqb(µ)Bq

li pNl (µ) =

Qf∑q=1

Θqf (µ)F q

i , 1 ≤ i ≤ 3N

3N∑j=1

Qb∑q=1

Θqb(µ)Bq

lj xNj (µ) =

Qg∑q=1

Θqg (µ)G q

l , 1 ≤ l ≤ 2N

size: 5N × 5N


RB method: Offline/Online decomposition

3N∑j=1

Qa∑q=1

Θqa (µ)Aq


Qb∑q=1

Θqb(µ)Bq

li pNl (µ) =

Qf∑q=1

Θqf (µ)F q

i , 1 ≤ i ≤ 3N,

3N∑j=1

Qb∑q=1

Θqb(µ)Bq

lj xNj (µ) =

Qg∑q=1

Θqg (µ)G q

l , 1 ≤ l ≤ 2N,

Offline pre-processing: compute and store the basis function σii and τjj

store the matrices Aq and Bq given by

(AN )qij = Aq(σj , σi ), (BN )q

li = Bq(σi , τl ), 1 ≤ i , j ≤ 3N, 1 ≤ l ≤ 2N,

as well as the RHS F qi , G q

l .

Operation count: depends on N, Qa, Qb, Qf , Qg and N

Online evaluations: evaluate coefficients Θq∗(µ), assemble RB matrices and vectors

AN (µ) =∑Qa

q=1 Θqa (µ)Aq

N , BN (µ) =∑Qb

q=1 Θqb(µ)Bq

N

FN (µ) =∑Qf

q=1 Θqf (µ)F q

N , GN (µ) =∑Qg

q=1 Θqf (µ)G q

N ;

and solve

(AN (µ) BT

N (µ)

BN (µ) 0

)︸︷︷︸

KN (µ)

(xN (µ)

pN (µ)

)︸︷︷︸

UN (µ)

=

(FN (µ)

GN (µ)

)

Operation count: O((5N)3 + (Qa + Qb)N2 + (Qf + Qg )N)

independent of N , N N


RB method: Offline/Online decomposition

3N∑j=1

Qa∑q=1

Θqa (µ)Aq


Qb∑q=1

Θqb(µ)Bq

li pNl (µ) =

Qf∑q=1

Θqf (µ)F q

i , 1 ≤ i ≤ 3N,

3N∑j=1

Qb∑q=1

Θqb(µ)Bq

lj xNj (µ) =

Qg∑q=1

Θqg (µ)G q

l , 1 ≤ l ≤ 2N,

Offline pre-processing: compute and store the basis function σii and τjj

store the matrices Aq and Bq given by

(AN )qij = Aq(σj , σi ), (BN )q

li = Bq(σi , τl ), 1 ≤ i , j ≤ 3N, 1 ≤ l ≤ 2N,

as well as the RHS F qi , G q

l . Operation count: depends on N, Qa, Qb, Qf , Qg and N

Online evaluations: evaluate coefficients Θq∗(µ), assemble RB matrices and vectors

AN (µ) =∑Qa

q=1 Θqa (µ)Aq

N , BN (µ) =∑Qb

q=1 Θqb(µ)Bq

N

FN (µ) =∑Qf

q=1 Θqf (µ)F q

N , GN (µ) =∑Qg

q=1 Θqf (µ)G q

N ;

and solve

(AN (µ) BT

N (µ)

BN (µ) 0

)︸︷︷︸

KN (µ)

(xN (µ)

pN (µ)

)︸︷︷︸

UN (µ)

=

(FN (µ)

GN (µ)

)

Operation count: O((5N)3 + (Qa + Qb)N2 + (Qf + Qg )N)

independent of N , N N


RB Method: a posteriori error estimation

Goal: provide rigorous, sharp and inexpensive estimators for the error on the solution

variables and for the error on the cost functional

General noncoercive (Babuska) framework (as Stokes problems, e.g. [Rozza et al., 2010]):

Recall the compact (weakly coercive) formulation: given µ ∈ D,

find U(µ) ∈ X s.t: B(U(µ),W;µ) = F(W;µ) ∀W ∈ X .

Note that ∃ β0 > 0 : β(µ) = infW∈X

supU∈X

B(U,W;µ)

‖U‖X ‖W‖X≥ β0, ∀µ ∈ D.

the FE and RB approximations satisfy an equivalent inf-sup condition

define the residual r(W;µ) = F(W;µ)− B(UN ,W;µ) and apply Babuska theorem

A posteriori error estimation on the solution variables

(‖xN (µ)− xN (µ)‖2

X + ‖pN (µ)− pN (µ)‖2Q

)1/2 ≤‖r(·;µ)‖X ′βLB(µ)

:= ∆N (µ)

0 < βLB(µ) ≤ βN (µ) is a constructible lower bound given by the

successive constraint method (SCM) [R., Huynh, Manzoni 2010]

thanks to the affinity assumption, we can provide the standard

Offline/Online stratagem for the efficient computation of ‖r(·;µ)‖X ′


RB Method: a posteriori error estimation

Goal: provide rigorous, sharp and inexpensive estimators for the error on the solution

variables and for the error on the cost functional

Define the error on the cost functional as

JN (µ)− JN (µ) = J(yN (µ), uN (µ);µ)− J(yN (µ), uN (µ);µ)

with standard arguments (e.g. [Becker et al., 2000; Dede, 2010]) we can easily show that

JN (µ)− JN (µ) =1

2r(UN (µ)− UN (µ);µ)

A posteriori error estimation on the cost functional

|JN (µ)− JN (µ)| ≤1

2‖r(·;µ)‖X ′‖UN (µ)− UN (µ)‖X ≤

1

2

‖r(·;µ)‖2X ′

βLB(µ):= ∆J

N (µ).

The error estimator ∆JN (µ) does not need any additional ingredients than those already

discussed:

- the efficient computation of ‖r(·;µ)‖X ′

- the calculation of the lower bound βLB(µ)


RB Method: the “complete game”

FE kernel

affine decomposition

assembly KqN , Fq

N

successive constraint

method (SCM)

βLB(µ)

basis selection

by greedy algorithm

assembly KqN, Fq

N

a posteriori

error estimation

∆N (µ)

assembly

KN (µ), FN (µ)

solution of the RB-OCPµ

KN (µ)UN (µ) = FN (µ)Θq∗(µ)µ ∈ D

certification

solution UN (µ),∆N (µ)

functional JN (µ),∆JN (µ)

offline

online

Offline stage involves precomputation of FE structures required for the RB space

construction and the certified error estimates.

Online stage has complexity only depending on N and allows resolution of the Optimal

Control Problem for any µ ∈ D with a certified error bound.


Numerical results - IOptimal control of parametrized advection-diffusion equations


Boundary control for a Graetz convection-diffusion problem

Ω1o Ω2

o(µ)

(1 + µ2, 0)

(1 + µ2, 1)(1, 1)(0, 1)

(0, 0) (1, 0)

ΓoN

ΓoC

ΓoCΓo

D

ΓoD

ΓoD

Ωo

Ωo the desired function is parameter dependent:

yd (µ) = µ3χΩo

parameter domain: D = [6, 20]× [1, 3]× [0.5, 3]

We consider the following optimal control problem:

minimize J(yo (µ), uo (µ);µ) =1

2‖yo (µ)− yd (µ)‖2

L2(Ωo )+α

2‖uo (µ)‖2

L2(ΓoC

)

subject to

−1

µ1∆yo (µ) + xo2(1− xo2)

∂yo (µ)

∂xo1= 0 in Ωo (µ)

yo (µ) = 1 on ΓoD

1

µ1∇yo (µ) · n = uo (µ) on Γo

C (µ)

1

µ1∇yo (µ) · n = 0 on Γo

N (µ),

- µ1 Peclet number, µ2 heigth of the channel

- the flow has an imposed temperature at the inlet and a parabolic convection field

- observation in the thermal layer Ωo



Ω1o Ω2

o(µ)

(1 + µ2, 0)

(1 + µ2, 1)(1, 1)(0, 1)

(0, 0) (1, 0)

ΓoN

ΓoC

ΓoCΓo

D

ΓoD

ΓoD

Ωo

Ωo

min J(yo (µ), uo (µ); µ) =1

2‖yo (µ)− yd (µ)‖2 +

α

2‖uo (µ)‖2

s.t.

−1

µ1∆yo (µ) + xo2(1− xo2)

∂yo (µ)

∂xo1= 0 in Ωo (µ)

1

µ1∇yo (µ) · n = uo (µ) on Γo

C (µ)

+ BCs

Functional spaces: Yo = H1ΓD

(Ωo ) Uo = L2(ΓoC ) Qo ≡ Yo

observation space: Zo = L2(Ωo )

I the problem is mapped to a reference domain Ω = Ωo (µref) with µref = (·, 1, ·)

I we obtain an affine decomposition with Qa = 1, Qb = 5, Qf = 1, Qg = 4

I we fixed α = 0.07 and we used P1 finite elements for the FE approximation of the

state, adjoint and control variables



1 1.4 1.8 2.2 2.6 330.1

0.2

0.3

0.4

0.5

0.6

comparison of Brezzi inf-sup constant βN (µ) and

coercivity constant of the state equation αN (µ)

as a function of µ2

Number of FE dof N 8915

Number of parameters P 3

Number of RB functions N 39

Dimension of RB linear system 39 · 5Affine operator components Q 6

Linear system dimension reduction 50:1

FE evaluation tFE (s) 14.5

RB evaluation tonlineRB (s) 0.1

RB evaluation tofflineRB (s) 3970

state yN adjoint yN

1 1.4 1.8 2.2 2.6 3

0

0.5

1

1.5

optimal control uN on ΓC

Representative solution for µ = (12, 2, 2.5).



1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

βLB

(µ)

βRB

(µ)

βFE

(µ)

Lower bound βLB(µ) for the Babuska inf-sup constant

β(µ) as a function of the parameter µ1




Dimension of RB linear system 39 · 5Affine operator components Q 6





5 10 15 20 25 30 3510

−6

10−4

10−2

100

102

104

106

average error

∆N average

max error

∆N max

5 10 15 20 25 30 3510

−10

10−5

100

105

average error

∆N

J average

Error estimation (•) and true error (•) for the solution (left) and the cost functional (right)


Numerical results - IIOptimal control of parametrized Stokes problems


How to extend the method

minimize J(v, p, u;µ) =1

2m(v − vd (µ), v − vd (µ);µ) +

α

2n(u, u;µ) subject to

a(v, ξ;µ) + b(ξ, p;µ) = 〈F (µ), ξ〉+ c(u, ξ;µ) ∀ξ ∈ V ,

b(v, τ ;µ) = 〈G(µ), τ〉 ∀τ ∈ M,

Functional setting: V = [H1(Ω)]2 M = L2(Ω) velocity and pressure spaces

Y = V ×M state space, Q ≡ Y adjoint space, U control space

two nested saddle-point

• outer: optimal control • inner: Stokes constraint

reduced basis functions computed by solving N times the FE approximation (with

stable spaces pair for velocity and pressure variables)

stability of the RB approximation of the Stokes constraint fulfilled by introducing

suitable supremizer operators [R., Veroy, 2007; R., Huynh, Manzoni 2010]

stability of the RB approximation of the whole optimal control problem fulfilled by

defining suitable aggregated spaces for the state and adjoint variables


Test 1: confirming theoretical results (Couette flow)

Ωo(µ)

(1, 0)

(1, µ1)(0, µ1)

(0, 0)

ΓoN

ΓoD

ΓoD

ΓoD

µ1: height of the pipe

forcing term state equation: fo (µ) = (0,−µ2)

parameter domain: D = [0.5, 2]× [0.5, 1.5]

We consider the following optimal control problem

minimize J(vo (µ), po (µ), uo (µ)) =1

2‖vo1(µ)− xo2‖2

L2(Ωo )+α

2‖uo (µ)‖2

L2(Ωo )

s.t.

− ν∆vo +∇po = fo (µ) + uo in Ωo (µ)

div vo = 0 in Ωo (µ)

vo1 = xo2, vo2 = 0 on ΓoD (µ)

− po no1 + ν∂vo1

∂no1= 0, vo2 = 0 on Γo

N (µ),

Example of uncontrolled flow

velocity on the left

pressure on the right



Ωo(µ)

(1, 0)

(1, µ1)(0, µ1)

(0, 0)

ΓoN

ΓoD

ΓoD

ΓoD

min J(vo (µ), po (µ), uo (µ)) =1

2‖vo1(µ)− xo2‖2 +

α

2‖uo (µ)‖2

s.t.

− ν∆vo +∇po = fo (µ) + uo in Ωo (µ)

div vo = 0 in Ωo (µ)

vo1 = xo2, vo2 = 0 on ΓoD (µ)

− po no1 + ν∂vo1

∂no1= 0, vo2 = 0 on Γo

N (µ),

Functional spaces: velocity Vo = [H1ΓD

(Ωo )]2 and pressure Mo = L2(Ωo )

state Yo = Vo ×Mo adjoint Qo ≡ Yo control Uo = [L2(Ωo )]2

I the problem is mapped to a reference domain Ω = Ωo (µref) with µref = (1, ·)

I we obtain an affine decomposition with Qa = 1, Qb = 4, Qf = 1, Qg = 5

I we fixed α = 0.008, ν = 0.1 and used P2-P1 Taylor-Hood finite elements for the FE

approximation of the velocity and pressure variables (for the control variable we used P2

finite elements)



0.5 1 1.5 2

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

comparison of Brezzi inf-sup constant βN (µ) and

Babuska in-sup constant αN (µ) of the Stokes

operator.




Dimension of RB linear system 225

Affine operator components Q 5





Representative solutions for µ = (1.7, 1.5):

(state) pressure on the left, (state) velocity in the middle, control on the right.



0,5 1 1,5 20

0.005

0.01

0.015

0.02

0.025

0.03

0.035

βLB

(µ)

βFE

(µ)

Lower bound βLB(µ) for the Babuska inf-sup constant

β(µ) as a function of the parameter µ1




Dimension of RB linear system 225






2 4 6 8 10 12 14

10−6

10−4

10−2

100

102

average error

∆N

average

2 4 6 8 10 12 14

10−10

10−5

100

average error

∆N

J average

Error

estimation (•) and true error (•) for the solution (left) and the cost functional (right)


An (idealized) application in haemodynamics: a data assimilation problem

we consider an inverse boundary problem in hemodynamics, inspired by the recent works

[D’Elia et al., 2011; Perego et al., 2011]

simplified model of an arterial bifurcation

we suppose to have a measured velocity profile on the red section, but not the Neumann

flux on ΓC that will be our control variable

starting from the velocity measures we want to find the control variable in order to

retrieve the velocity and pressure fields in the whole domain.

120 Chapter 5. Reduced Basis Method for Parametrized Stokes Optimal Control problems

−4 −3 −2 −1 0 1 2 3

−2

−1

0

1

2

3

Γin

ΓD

ΓD

ΓC

ΓD

ΓC

ΓD

ΓD

ΓD

given new geometrical configuration (µgeom)and parametrized measurements µmeas

on the red section

−4 −3 −2 −1 0 1 2 3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

−0.2

0

0.2

0.4

0.6

−4 −3 −2 −1 0 1 2 3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

0

0.2

0.4

0.6

0.8

Online

vd(µmeas)g(µin)

find the unknown Neumann boundarycondition on ΓC and retrieve

the whole velocity and pressure fields

pressure

velocity

Figure 5.6: An (idealized) example of inverse boundary problem in haemodynamics. Given ageometrical configuration and some velocity measurements on some sections of the domain (bothobtainable via medical image and data assimilation devices, e.g. MRI), we want to retrieve the wholepressure and velocity fields in order to detect possible pathologies, e.g. occlusions or flow disturbancein arterial bifurcations.

The state velocity and pressure variables vo, po satisfy the following Stokes problem in theoriginal domain Ωo(µ):

−ν∆vo +∇po = 0 in Ωo(µ),

divvo = 0 in Ωo(µ),

v = 0 on ΓoD(µ),

vo = g(µin) on Γoin(µ),

−pono + ν∂vo∂no

= uo on ΓoC(µ),

(5.5.1)

where uo is the control variable; the inlet velocity profile g(µin) is given by the following

−4 −3 −2 −1 0 1 2 3

−2

−1

0

1

2

3

Γin

ΓD

ΓD

ΓC

ΓD

ΓC

ΓD

ΓD

ΓD

Γoss

(a) Boundary conditions: no-slip conditions onΓD(µ), Poiseuille velocity profile g(µin) on Γin,unknown Neumann flux on the outflow sections.

−4 −3 −2 −1 0 1 2 3

−3

−2

−1

0

1

2

µ3

µ4

µ6

µ5

µ1 µ

2

(b) Parametrization of the original do-main.

Figure 5.7: Original domain Ωo(µ) of the arterial bifurcation.



−4 −3 −2 −1 0 1 2 3

−2

−1

0

1

2

3

Γin

ΓD

ΓD

ΓC

ΓD

ΓC

ΓD

ΓD

ΓD

Γoss

−4 −3 −2 −1 0 1 2 3

−3

−2

−1

0

1

2

µ3

µ4

µ6

µ5

µ1 µ

2

The state velocity and pressure variables vo , po satisfy the

following Stokes problem in the original domain Ωo (µ):

−ν∆vo +∇po = 0 in Ωo (µ),

div vo = 0 in Ωo (µ),

vo = 0 on ΓoD (µ),

vo = g(µin) on Γoin(µ),

−po no + ν∂vo

∂no= uo on Γo

C (µ),

Then we consider the following (parametrized) cost functional to be minimized

J(vo (µ), po (µ), uo (µ);µ) =1

2

∫Γo

obs

|vo (µ)− vd (µ)|2 dΓo

+α1

2

∫Γo

C

|∇uo (µ)to |2dΓo +α2

2

∫Γo

C

|uo (µ)|2dΓo



5 10 15 20 25 30 35 40

10−4

10−3

10−2

10−1

100

101

Average computed error between the truth FE solution

and the RB approximation.




Dimension of RB linear system 43 · 13


FE evaluation tFE (s) 22


µ = (1, π/5, π/6, 1, 1.7, 2.2, 0.8, 1). µ = (1.2, π/6, π/6, 0.8, 2.5, 2.1, 0.3, 1).


Conclusions

we provided a certified RB method for the solution of parametrized optimal

control problems with high dimensional control variable

reduction to low-dimensionality of the whole control problem and not just of the

state equation

key ingredient: saddle-point formulation of the optimal control problem

full Offline/Online decomposition strategy

...

References

Bibliography I

Barrault, M., Maday, Y., Cuong, N. Nguyen, & Patera, A.T. 2004. An ‘empirical interpolation’ method: application to

efficient reduced-basis discretization of partial differential equations. C. R. Acad. Sci. Paris. Ser. I Math., 339(9), 667

– 672.

Becker, R., Kapp, H., & Rannacher, R. 2000. Adaptive Finite Element Methods for Optimal Control of Partial Differential

Equations: Basic Concept. SIAM J. Control Optim., 39, 113–132.

Brezzi, F., & Fortin, M. 1991. Mixed and Hybrid Finite Elements Methods. New York: Springer-Verlag.

Dede, L. 2010. Reduced Basis Method and A Posteriori Error Estimation for Parametrized Linear-Quadratic Optimal

Control Problems. SIAM J. Sci. Comput., 32, 997–1019.

Dede, L. 2011. Reduced Basis Method and Error Estimation for Parametrized Optimal Control Problems with Control

Constraints. Journal of Scientific Computing, 1–19.

D’Elia, M., Mirabella, L., Passerini, T., Perego, M., Piccinelli, M., Vergara, C., & Veneziani, A. 2011. Applications of

Variational Data Assimilation in Computational Hemodynamics. In: Ambrosi, D., Quarteroni, A., & Rozza, G. (eds),

Modelling of Physiological Flows. MS&A Series, vol. 5. Springer.

Grepl, M., Maday, Y., Nguyen, N., & Patera, A.T. 2007. Efficient reduced-basis treatment of nonaffine and nonlinear partial

differential equations. Esaim Math. Model. Numer. Anal., 41(3), 575–605.

Grepl, M.A., & Karcher, M. 2011. Reduced basis a posteriori error bounds for parametrized linear-quadratic elliptic optimal

control problems. C. R. Math. Acad. Sci. Paris, 349(15-16), 873 – 877.

Gunzburger, M.D., & Bochev, P.B. 2004. Finite element methods for optimization and control problems for the Stokes

equations. Comp. Math. Appl., 48, 1035–1057.

Lions, J.L. 1971. Optimal Control of Systems governed by Partial Differential Equations. Berlin Heidelberg: Springer-Verlag.

References

Bibliography II

Perego, M., Veneziani, A., & Vergara, C. 2011. A variational approach for estimating the compliance of the cardiovascular

tissue: an inverse fluid-structure interaction problem. SIAM J. Sci. Comput., 33(3), 1181–1211.

Quarteroni, A., Rozza, G., & Quaini, A. 2007. Reduced basis methods for optimal control of advection-diffusion problems.

Pages 193–216 of: Advances in Numerical Mathematics.

Rozza, G., & Veroy, K. 2007. On the stability of the reduced basis method for Stokes equations in parametrized domains.

Comput. Methods Appl. Mech. Engrg., 196(7), 1244 – 1260.

Rozza, G., Huynh, D.B.P., & Patera, A.T. 2008. Reduced Basis Approximation and a Posteriori Error Estimation for

Affinely Parametrized Elliptic Coercive Partial Differential Equations. Arch. Comput. Methods Eng., 15, 229–275.

Rozza, G., Huynh, D.B.P., & Manzoni, A. 2010. Reduced basis approximation and a posteriori error estimation for Stokes

flows in parametrized geometries: roles of the inf-sup stability constants. Tech. rept. 22.2010. MATHICSE.

Submitted.

Tonn, T., Urban, K., & Volkwein, S. 2011. Comparison of the reduced-basis and POD a-posteriori estimators for an elliptic

linear-quadratic optimal control problem. Math. Comput. Model. Dyn. Syst., 17(1), 355–369.

For more information see

http://sma.epfl.ch/∼rozza

http://cmcs.epfl.ch/

http://augustine.mit.edu

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