Optimal Solvers for Elliptic Optimal Control Problems...Optimal Control of Partial Diﬀerential...

Motivating Examples Existence and Uniqueness Optimize first, then discretize Numerical results

Optimal Solvers for Elliptic Optimal ControlProblems

F. Ibrahim, S. Turek and M. Köster

Institut for Applied Mathematics and Numerics (LSIII)Technische Universität Dortmund

[email protected]

ICAAMMP 2014Istanbul, Turkey, 20.08.2014

TU Dortmund University

Optimal Solvers for Elliptic Optimal Control Problems


Outline

Motivating Examples

Existence and Uniqueness

Optimize first, then discretize

Numerical results




Outline

Motivating Examples



Numerical results




Outline

Motivating Examples



Numerical results




Outline

Motivating Examples



Numerical results




References

Optimal Control of System Governed by Partial DifferentialEquations: Lions (1971)Optimal Steuerung partial DifferentialgleichungenTheorie:Tröltzsch (2005)

Exact and Approximate Controllability for DistributedParameter Systems: A Numerical Approach:Lions, Glowinskiand He (2008)Optimization with PDE Constraints:Ulbrich and others (2009)Optimal Control of Partial Differential Equations:Theory,Methods and Applications Tröltzsch (2010)




References

Optimal Control of System Governed by Partial DifferentialEquations: Lions (1971)Optimal Steuerung partial DifferentialgleichungenTheorie:Tröltzsch (2005)Exact and Approximate Controllability for DistributedParameter Systems: A Numerical Approach:Lions, Glowinskiand He (2008)

Optimization with PDE Constraints:Ulbrich and others (2009)Optimal Control of Partial Differential Equations:Theory,Methods and Applications Tröltzsch (2010)




References

Optimal Control of System Governed by Partial DifferentialEquations: Lions (1971)Optimal Steuerung partial DifferentialgleichungenTheorie:Tröltzsch (2005)Exact and Approximate Controllability for DistributedParameter Systems: A Numerical Approach:Lions, Glowinskiand He (2008)Optimization with PDE Constraints:Ulbrich and others (2009)

Optimal Control of Partial Differential Equations:Theory,Methods and Applications Tröltzsch (2010)




References

Optimal Control of System Governed by Partial DifferentialEquations: Lions (1971)Optimal Steuerung partial DifferentialgleichungenTheorie:Tröltzsch (2005)Exact and Approximate Controllability for DistributedParameter Systems: A Numerical Approach:Lions, Glowinskiand He (2008)Optimization with PDE Constraints:Ulbrich and others (2009)Optimal Control of Partial Differential Equations:Theory,Methods and Applications Tröltzsch (2010)




References

Optimal Control of System Governed by Partial DifferentialEquations: Lions (1971)Optimal Steuerung partial DifferentialgleichungenTheorie:Tröltzsch (2005)Exact and Approximate Controllability for DistributedParameter Systems: A Numerical Approach:Lions, Glowinskiand He (2008)Optimization with PDE Constraints:Ulbrich and others (2009)Optimal Control of Partial Differential Equations:Theory,Methods and Applications Tröltzsch (2010)




Part 1: Motivating Examples




Optimal stationary boundary temperature: Heating of a body Ωbya controlled boundary temperature u to reach the desiredtemperature yd

infy,u

J(y, u) =1

2||y − yd||2L2(Ω) +

α

2||u||2L2(Γ) (P)

subject to state equation

−∆y = 0 in Ω

y = u on Γ,

and control constraints

a ≤ u ≤ b on Γ




Optimal stationary heat source: The body Ω is heated e.g. bymicrowaves. The goal is to find u that minmizes the distance of yand yd

infy,u

J(y, u) =1

2||y − yd||2L2(Ω) +

α

2||u||2L2(Ω) (P1)

subject to state equation

−∆y = u in Ω

y = 0 on Γ,




Notation and dataJ is the objective functional and yd is the desired stateα > 0 is regularization parameterOptimal control problems with linear state equation andquadratic objective functional called linear-quadratic(P) is a linear-quadratic elliptic boundary control Problem(P1) is a linear-quadratic elliptic distributed control Problem




Part 2:Existence and Uniqueness of optimal controlproblem




Problem P1: Defining the control- to- state map G : u → yfrom L2(Ω) to H1

0 (Ω) , a linear and bounded operator whichassigns to a control u ∈ L2(Ω) the unique solution y = y(u)of the state equation,writing the state y = Gu

introduce the reduced objective functional

infy,u

Jred(u) :=1

2||Gu− yd||2L2(Ω) +

α

2||u||2L2(Ω)




Problem P1: Defining the control- to- state map G : u → yfrom L2(Ω) to H1

0 (Ω) , a linear and bounded operator whichassigns to a control u ∈ L2(Ω) the unique solution y = y(u)of the state equation,writing the state y = Gu

introduce the reduced objective functional

infy,u

Jred(u) :=1

2||Gu− yd||2L2(Ω) +

α

2||u||2L2(Ω)




TheoremThe reduced optimal control problem has a unique optimal solution(y, u) ∈ H1

0 (Ω)× L2(Ω) .

ProofDirect method of the calculus of variations(minimizing sequence) .




Part 3: Optimize first, then discretize




Theorem

Let (y, u) ∈ H10 (Ω)× L2(Ω) be the optimal solution of Problem

P1. Then, there exists an adjoint state p ∈ H10 (Ω) as a weak

solution of−∆p = y − yd in Ω

p = 0 on Ω

and the control equation

p+ αu = 0 in Ω.




Finite Element Approximation of Distributed Control ProblemLet Th := Th(Ω) be a shape-regular, quadrilateral triangulation andlet

Vh := vh ∈ C(Ω) | vh|T ∈ Q1(T ) , T ∈ Th(Ω), | vh|Γ = 0

be the FE space of continuous piecewise linear finite elements.Then a possible FE approximation of (P1) is as follows:

inf J(yh, uh) :=1

2

∫Ω

|yh − yd|2 dx +α

2

∫Ω

|uh|2 dx

subject toa(yh, vh) = (uh, vh), ∀vh ∈ Vh




Optimality Conditions for the FE Distributed Control Problemthere exists an adjoint state ph ∈ Vh such that the triple(yh, uh, ph) satisfies the following optimality conditions,

a(yh, vh) = (uh, vh), ∀vh ∈ Vh

a(ph, vh) = (yh − yd, vh), ∀vh ∈ Vh

(uh, vh) = −α−1(ph, vh) ∀vh ∈ Vh




The discrete KKT system

Ah 0 −Mh

−Mh Ah 00 Mh αMh

yhphuh

=

0−ydh0




A Reduced KKT systemIf we substitute uin the state equation by means of the controlequation according to u = −α−1 ph, the discrete KKT system canbe stated as (

Ah α−1Mh

Mh Ah

)(yhph

)=

(0

−ydh

).

There exist also solution methods, which rely on a reduction of theKKT system. In this method, the optimality system is reduced to asingle integral equation for the control u




A Reduced KKT systemIf we substitute uin the state equation by means of the controlequation according to u = −α−1 ph, the discrete KKT system canbe stated as (

Ah α−1Mh

Mh Ah

)(yhph

)=

(0

−ydh

).

There exist also solution methods, which rely on a reduction of theKKT system. In this method, the optimality system is reduced to asingle integral equation for the control u




The integral equation methodWe introduce the operator A : y 7→ (−∆y) and the adjointoperator A∗ : p 7→ (−∆p). The KKT system reads in the followingform:

A = u in Ω

A∗ = y − yd in Ω

u =− α−1p in Ω.

The last equation can be reformulated by the first two equationsto the fixed point equation

u = (−α−1A−∗A−1)u− (−α−1A−∗)yd

=Ku+ q




which we can rewrite it in the compact form as follows:

(I − K)u =q.

Based on this compact form, we formulate the following fixedpoint iteration:

un :=Kun−1 + q, n ∈ N ,




Part 4:Numerical results




Example

u =sin(3πx1) sin(3πx2)

y =sin(3πx1) sin(3πx2)

p =− α sin(3πx1) sin(3πx2)

We get the corresponding desired state as

yd =18π2α sin(3πx1) sin(3πx2) + sin(3πx1) sin(3πx2).




Table: Comparison of iterations to solve Example 1. with different valuesof α and different mesh sizes for tolerance of 1E − 12 for Multigrid withdifferent smoother (A Reduced KKT system)

smoother level \α 1 1E-02 1E-04 1E-08 1E-126 7 7 9 div. div.

Jacobi(0.5) 7 6 6 7 10 div.8 5 5 5 7 div.9 4 4 4 5 div.6 5 5 6 div. div.

Gaus Seidel 7 5 5 6 div. div.8 5 5 5 27 div.9 6 6 6 9 div.6 7 7 8 div. div.

SSOR(0.5) 7 6 6 7 div. div.8 6 6 6 10 div.9 6 6 6 8 div.




Table: Comparison of iterations to solve Example 1. with different valuesof α and different mesh sizes for tolerance of 1E − 12 for Multigrid withdifferent smoother and preconditioner (A Reduced KKT system)

smoother Preconditioner level \α 1 1E-02 1E-04 1E-08 1E-126 7 8 19 div. div.

SSOR(1.5) 7 8 10 10 163 div.8 8 8 9 28 div.9 8 8 9 26 div.6 4 5 7 39∗ div.

BICGSTAB SSOR(0.5) 7 5 5 6 16 div.8 5 5 6 10 419 5 5 5 8 296 4 4 7 9∗ 9∗

SSOR(0.5) 7 5 5 6 8 98 5 5 5 7 10

GMRES(30) 9 5 5 5 7 10




Table: Comparison of iterations to solve Example 1. with different valuesof α and different mesh sizes for tolerance of 1E − 12 for Multigrid withdifferent smoother and preconditioner ( KKT system)

smoother Preconditioner level \α 1 1E-02 1E-04 1E-08 1E-126 4 5 9 div div

SSOR(0.5) 7 5 6 8 137 div.8 5 5 6 13 div.

BICGSTAB 9 5 6 7 9 div.6 4 4 7 98∗ 104∗

SSOR(0.5) 7 5 5 6 35 7678 5 5 5 11 1106

GMRES(30) 9 5 5 5 7 23




Table: Comparison of iterations to solve Example 1. with different valuesof α at different grid level for tolerance of 1E − 12 with different solver(integral equation)

solver level \α 1 1E-02 1E-04 1E-08 1E-126 2 2 2 576∗ -

BICGSTAB 7 2 2 2 1281 -8 2 2 2 2318 -9 2 2 2 2429 -6 2 2 4 3∗ 3∗

GMRES 7 2 2 3 7 88 2 2 3 4 49 2 2 4 4 4




Figure: Example 1: Change in L2-error with level for different values of α






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