Optimal Control of
PDEs
Fredi Tröltzsch
Primal-dual active
set strategy
The continuous case
Theory and Numerics for the Optimal
Control of PDEs
Fredi Tröltzsch
Technische Universität Berlin
Summer school Elgersburg
The system to be solved
The discrete case
FEM discretization
Primal-dual active set
strategy
Optimal Control of
PDEs
Fredi Tröltzsch
Primal-dual active
set strategy
The continuous case
The system to be solved
The discrete case
Lecture 3
Numerical methods
FEM and primal dual active set strategy
FEM discretization
Primal-dual active set
strategy
Optimal control problem
Optimal Control of
PDEs
Fredi Tröltzsch
min f (u) :=
1
λ
kS u − yΩ k2L2 (Ω) + kuk2L2 (Ω)
2
2
subject to
Primal-dual active
set strategy
The continuous case
The system to be solved
The discrete case
FEM discretization
Primal-dual active set
strategy
u ∈ Uad
2
= u ∈ L (Ω) : ua (x) ≤ u(x) ≤ ub (x) a.e. in Ω .
S : L2 (Ω) → L2 (Ω) is the solution operator to
−∆y
y |Γ
= u
= 0
The optimal control has to satisfy
ū(x) = IP[ua (x),ub (x)] − λ−1 p(x) .
Optimal Control of
PDEs
Fredi Tröltzsch
−1
ū(x) = IP[ua (x),ub (x)] − λ
p(x) .
Primal-dual active
set strategy
The continuous case
Define
The system to be solved
The discrete case
µ = −(λ−1 p + ū) = −λ−1 f 0 (ū).
FEM discretization
Primal-dual active set
strategy
Discussion of the projection formula yields

if −λ−1 p(x) < ua (x)
(⇔ µ(x) < 0)

 ua (x),
−1
−1
ū(x) =
−λ p(x), if −λ p(x) ∈ [ua (x), ub (x)] (⇔ µ(x) = 0)


ub (x),
if −λ−1 p(x) > ub (x)
(⇔ µ(x) > 0).
(1)
Upper case: by Definition of µ and ū = ua ⇒ µ(x) < 0, hence ū(x) + µ(x) < ua (x).
Lower case: Analogous ū(x) + µ(x) > ub (x).
Middle case: µ(x) = 0, hence ū(x) + µ(x) = −λ−1 p(x) ∈ [ua (x), ub (x)].
Optimal Control of
PDEs
Fredi Tröltzsch
Primal-dual active
set strategy
Conclusion: The optimal control u = ū satisfies
The continuous case
The system to be solved



ua (x),
u(x) =
−λ−1 p(x),


ub (x),
if
if
if
u(x) + µ(x) < ua (x)
u(x) + µ(x) ∈ [ua (x), ub (x)]
u(x) + µ(x) > ub (x).
The discrete case
(2)
Also the converse is valid, as one can easily discuss: If u ∈ Uad
satisfies (2), then u is optimal.
Conclusion: u(x) + µ(x) is an indicator for activity or inactivity
of the inequality constraints.
This motivates the primal-dual active set strategy:
FEM discretization
Primal-dual active set
strategy
Primal-dual active set strategy
Optimal Control of
PDEs
Fredi Tröltzsch
2
Initialization: Fix arbitrary u0 , µ0 in L (Ω); u0 need not be
admissible.
Let un−1 and µn−1 be determined.
Computation of un : Perform the following steps
S1
(New active and inactive sets)
Abn =
x : un−1 (x) + µn−1 (x) > ub (x)
Aan =
x : un−1 (x) + µn−1 (x) < ua (x)
In = Ω \ (Abn ∪ Aan ).
If Aan = Aan−1 and Abn = Abn−1 , then STOP; optimality.
If not,
Primal-dual active
set strategy
The continuous case
The system to be solved
The discrete case
FEM discretization
Primal-dual active set
strategy
Optimal Control of
PDEs
Primal-dual active set strategy
Fredi Tröltzsch
Primal-dual active
set strategy
The continuous case
The system to be solved
S2
(New control)
The discrete case
FEM discretization
Solve the following linear system for u ∈ L2 (Ω),

 ua
−∆y = u
−λ−1 p
u=

−∆p = y − yΩ ,
ub
y , p ∈ H01 (Ω):
on
in
on
Aan
In
Abn .
Define un := u, pn := p, µn := −(λ−1 pn + un ), n := n + 1
GOTO S1
Primal-dual active set
strategy
Optimal Control of
PDEs
Linear system in practice
Fredi Tröltzsch
Practicable form of the system in S2:
Characteristic functions
Relations for u:
χan
and
χbn
of
Aan
bzw.
Primal-dual active
set strategy
Abn .
The continuous case
The system to be solved
The discrete case
FEM discretization
u + (1 −
χan
−
χbn ) λ−1 p
=
χan
ua +
χbn
Primal-dual active set
strategy
ub .
System for y and p:
−∆p
(1 − χan − χbn ) λ−1 p
−∆y
−y
−u
+u
= 0
= −yΩ
= χan ua + χbn ub .
References: Ito and Kunisch 2000, Bergounioux/Ito/Kunisch
1999, Book Ito/Kunisch 2008
Semismooth Newton method, superlinear convergence.
(3)
Optimal Control of
PDEs
Finite Elements
Fredi Tröltzsch
Assume
Primal-dual active
set strategy
I
Ω 2D polygonal
I
Regular triangulation
I
piecewise linear and continuous ansatz functions
{Φ1 , ..., Φ` } ⊂ H01 (Ω).
I
piecewise constant control functions, i.e. u is constant on
each triangle of the triangulation.
I
ansatz functions ei , i = 1, ..., m: 1 on triangle i, zero in
the remaining ones.
I
Therefore
y (x) =
The continuous case
`
X
i=1
yi Φi (x),
The system to be solved
The discrete case
u(x) =
m
X
ui ei (x)
i=1
with real unknowns yi , uj , i = 1, . . . , `, j = 1, . . . , m.
FEM discretization
Primal-dual active set
strategy
Insert the ansatz in the Poisson equation, use Φj as test
function. Then
Z X
`
yi ∇Φi · ∇Φj dx =
Ω i=1
Z X
m
Optimal Control of
PDEs
Fredi Tröltzsch
Primal-dual active
set strategy
ui ei Φj dx
Ω i=1
The continuous case
The system to be solved
The discrete case
FEM discretization
Obtain linear system for ~y = (y1 , . . . , y` )> and
~u = (u1 , . . . , um )> :
Kh ~y = Bh ~u
with stiffness matrix Kh and some matrix Bh ,
Z
Z
∇Φi · ∇Φj dx,
kij =
bij =
Ω
h is the mesh size of the triangulation.
Φi ej dx.
Ω
Primal-dual active set
strategy
Optimal Control of
PDEs
Fredi Tröltzsch
After an easy calculation
Primal-dual active
set strategy
The continuous case
The system to be solved
1
discrete case
y −yΩ 22 + λ kuk22 = 1 ~y > Mh ~y −a~h >~y + λ ~u > Dh ~u + 1 kyΩ k22 The
FEM discretization
L (Ω)
L (Ω)
L (Ω) 2
2
2
2
2
Primal-dual active set
strategy
with mass matrices Mh , Dh and a vector ~ah ,
Z
mh,ij =
Z
Φi Φj dx,
dh,ij =
Ω
Dh is a diagonal matrix.
Z
ei ej dx,
Ω
ah,i =
Φi yΩ dx.
Ω
Optimal Control of
PDEs
Fredi Tröltzsch
Primal-dual active
set strategy
We obtain a finite-dimensional quadratic optimization problem:
The continuous case
The system to be solved
The discrete case
min
FEM discretization
1 >
λ
~y Mh ~y − a~h >~y + ~u > Dh ~u ,
2
2
Kh ~y = Bh ~u ,
~ua ≤ ~u ≤ ~ub
with bounds ~ua = (ua , . . . , ua )> , ~ub = (ub , . . . , ub )> .
Primal-dual active set
strategy
(4)
Finite-dimensional optimality system
Optimal Control of
PDEs
Fredi Tröltzsch
Kh ~y = Bh ~u ,
Primal-dual active
set strategy
~ua ≤ ~u ≤ ~ub ,
The continuous case
Kh ~p = Mh ~y − ~ah ,
(λ Dh ~u + Bh>~p )> (~v − ~u ) ≥ 0
The system to be solved
The discrete case
∀ ~ua ≤ ~v ≤ ~ub
Since Dh is diagonal, define
µ
~ = − (λDh )−1 Bh>~p + ~u
For optimal ~u

 ua , if ui + µi < ua
µi , if ui + µi ∈ [ua , ub ]
ui =

ub , if ui + µi > ub ,
i = 1, . . . , m.
⇒ active set strategy
FEM discretization
Primal-dual active set
strategy
Finite-dimensional active set strategy
Optimal Control of
PDEs
Fredi Tröltzsch
Initial vectors u~0 , µ~0
In the n-th step:
Primal-dual active
set strategy
The continuous case
The system to be solved
Abn
Aan
In
=
=
=
i ∈ {1, . . . , m} : un−1,i + µn−1,i > ub,i ,
i ∈ {1, . . . , m} : un−1,i + µn−1,i < ua,i ,
{1, . . . , m} \ (Abn ∪ Aan )
Define ”characteristic” diagonal matrices Xna , Xnb with
1, if i ∈ Aan
1, if i ∈ Abn
a
b
Xn,ii =
Xn,ii
=
0, else,
0, else
Define Eh := (λDh )−1 (I − Xna − Xnb ).
Notice that dh,ii = 0 iff i ∈ Aan ∪ Abn .
The discrete case
FEM discretization
Primal-dual active set
strategy
Optimal Control of
PDEs
Linear system to solve in practice
Fredi Tröltzsch
Primal-dual active
set strategy
The continuous case
Next, solve the following system for ~p, ~y and ~u :
The system to be solved
The discrete case
FEM discretization
Primal-dual active set
strategy

0

 Kh
Eh Bh>
Kh
−Mh
0




~p
0
−Bh
 ~  

−~ah
0  y =

a~
b~
~
u
I
Xn ua + Xn ub .
Result: ~un := ~u und µn := − (λDh )−1 Bh>~pn + ~un .
Converges in finitely many steps. Stops, if the solution is
feasible the first time.
(5)