Optimization with PDEs in the presence of
constraints − tailored discrete concepts and error
analysis
M. Hinze
Department Mathematik
Bereich Optimierung und Approximation, Universität Hamburg
(joint with Klaus Deckelnick and Andreas Günther)
Warwick, January 14, 2009
M. Hinze
Optimization with pdes − tailored discrete concepts
Topic of this talk
Optimal control of pdes with pointwise constraints:
min
u∈Uad ,y∈Yad
J(y, u) s.t. PDE(y) = B(u)
Analysis: Casas 85,93 (pointwise state constraints), Casas &
Fernandez 93 (pointwise constraints on gradient)
Numerical analysis (pointwise state constraints):
A priori:
Original problem: Casas & Mateos; Deckelnick & H.;
Meyer;....
Ralaxation: Group of Rösch; Group of Tröltzsch;
Hintermüller & H.; H. & Meyer; H. & Schiela; ...
A posteriori: Benedix, Vexler & Wollner; Günther & H.;
Hintermüller, Hoppe & Kieweg.
Numerical analysis (pointwise constraints on gradient):
Deckelnick, Günther, & H.
M. Hinze
Optimization with pdes − tailored discrete concepts
State and control constraints
Model problem
min J(u) =
1
Z
|y − z|2 +
α
kuk2U
2 D
2
subject to y = G(Bu) and y ≤ b in D.
u∈Uad
Here, Uad ⊆ U closed and convex, α > 0, z, b, sufficiently
smooth, and y = G(Bu) iff
Ay = Bu in D, plus b.c. (plus i.c.)
I
I
elliptic case:
Pd D = Ω and Pd
Ay := − i,j=1 ∂xj aij yxi + i=1 bi yxi + cy uniformly
elliptic operator,
parabolic case:
P D = (0, T]× ΩPand
Ay := yt − di,j=1 ∂xj aij yxi + di=1 bi yxi + cy with
strongly elliptic leading part.
Slater condition: ∃ũ ∈ Uad such that G(Bũ) < b in D̄.
M. Hinze
Optimization with pdes − tailored discrete concepts
Optimality conditions (Casas 86,93)
Let u ∈ Uad denote the unique optimal control and
y = G(Bu). Then there exist µ ∈ M(D̄) and some p such
that there holds
Z
Z
Z
pAv =
(y − z)v +
vdµ
∀v ∈ X,
D
D
D̄
∗
hB p + αu, v − uiU∗ ,U ≥ 0
∀v ∈ Uad ,
Z
µ ≥ 0, y ≤ b in D and
(b − y)dµ = 0,
D̄
where
I
elliptic case: p ∈ P
W1,s (Ω) for all s < d/(d − 1) and
2
X = H (Ω) with di,j=1 aij vxi νj = 0 on ∂Ω,
I
parabolic case: p ∈ Ls (W1,σ ) for all s, σ ∈ [1, 2) with
2/s + d/σ > d + 1 and X = {v ∈ C0 (Q̄); v(0, ·) =
0} ∩ {v ∈ L2 (H2 ), vt ∈ L2 (H1 )}.
M. Hinze
Optimization with pdes − tailored discrete concepts
Discretization – a variational concept
Discrete optimal control problem:
Z
α
1
|yh − z|2 + kuk2U
min Jh (u) :=
u∈Uad
2 D
2
subject to yh = Gh (Bu) and yh ≤ Ih b.
Here, yh (u) = Gh (Bu) denotes the
I
p.l. and continuous fe approximation to y(u) (elliptic
case),
I
dg(0) in time and p.l. and continuous fe in space
approximation to y(u) (parabolic case), i.e.
a(yh , vh ) = hBu, vh i for all vh ∈ Xh .
We do not discretize the control!
M. Hinze
Optimization with pdes − tailored discrete concepts
Variational versus conventional discretization
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Movie time dependent problems
M. Hinze
Optimization with pdes − tailored discrete concepts
Discrete optimality conditions
Let uh ∈ Uad denote the unique variational–discrete optimal
control, yh = G(Buh ). There exist µ ∈ Rk and ph ∈ Xh such
that with
Pnv
I µh =
j=1 µj δxj (elliptic case, xi fe nodes, k = nv),
R
Pm Pnv
1
I µh =
i=1
j=1 µij δxj ◦ |Ii | Ii •dt (parabolic case, xi fe
nodes, Ii dg intervals, k = nv + m),
we have
Z
Z
(yh − z)vh +
a(vh , ph ) =
D
vh dµh
∀vh ∈ Xh ,
D̄
hB∗ ph + αuh , v − uh iU∗ ,U ≥ 0
∀v ∈ Uad ,
Z
µj ≥ 0, yh ≤ Ih b, and
Ih b − yh dµh = 0.
D̄
Here, δx denotes the Dirac measure concentrated at x and Ih
is the usual Lagrange interpolation operator.
M. Hinze
Optimization with pdes − tailored discrete concepts
Results
Let uh ∈ Uad denote the variational–discrete optimal solution
with corresponding state yh ∈ Xh and µh ∈ M(D̄). Then for
h small enough
kyh k, kuh kU , kµh kM(D̄) ≤ C.
For the proof a discrete counterpart to the Slater condition
is needed, which is deduced from uniform convergence of the
discrete states associated to the Slater point Bũ.
M. Hinze
Optimization with pdes − tailored discrete concepts
Results, cont.
Let u denote the solution of the continuous problem and uh
the variational discrete optimal control. Then
αku − uh k2 + ky − yh k2 ≤
n
o
≤ C(kµkM(D̄) , kµh kM(D̄) ) ky − yh (u)k∞ + kyh (uh ) − yh k∞ +
n
o
h
+ C(kuk, kuh k) ky − yh (u)k + ky (uh ) − yh k .
Here, yh (u) = Gh (Bu), yh (uh ) = G(Buh ).
We need uniform estimates for discrete approximations.
M. Hinze
Optimization with pdes − tailored discrete concepts
Error estimates, parabolic case
Controls u ∈ L2 (0, T)m , and fi ∈ H1 (Ω) given actuations.
Bu :=
m
X
ui (t)fi (x),
y0 ∈ H2 (Ω).
i=1
Then y = G(Bu) ∈ {v ∈ L∞ (H2 ), vt ∈ L2 (H1 )} and we have
with yh = Gh (Bu) and time stepping δt ∼ h2
ky − yh k∞ ≤ C
p
h
√ | log h|, (d = 2)
h,
(d = 3)
This is not an off-the-shelf result! It yields
p
h | log h|, (d = 2)
2
2
αku − uh k + ky − yh k ≤ C √
h,
(d = 3).
M. Hinze
Optimization with pdes − tailored discrete concepts
Error estimates, elliptic case
Deckelnick, H. (SINUM 2007, ENUMATH 2007):
I Bu ∈ L2 (Ω):
(
1
O(h 2 ), if d = 2,
ku − uh kU , ky − yh kH1 =
1
O(h 4 ), if d = 3,
I
Bu ∈ W1,s (Ω):
3
d
ku − uh kU , ky − yh kH1 ≤ Ch 2 − 2s
I
p
| log h|.
Bu ∈ L∞ (Ω):
ku − uh kU , ky − yh kH1 ≤ Ch| log h|.
I
U = L2 (Ω), Uad = {u ≤ d}, uh p.c.:
h| log h|,
ku − uh kU , ky − yh kH1 ≤ C √
h,
if d = 2,
if d = 3.
Similar results obtained by C. Meyer for discrete controls.
M. Hinze
Optimization with pdes − tailored discrete concepts
Numerical experiment 1
Ω := B1 (0), α > 0,
z(x) := 4 +
1
π
−
1
4π
|x|2 +
1
|x|2
4απ
2π
log |x|, b(x) := |x|2 + 4,
1
2απ
log |x|.
Z
Z
1
α
J(u) :=
|y − z|2 +
|u − u0 |2 ,
2 Ω
2 Ω
and u0 (x) := 4 +
−
1
where y = G(u).
Unique solution u ≡ 4 with corresponding state y ≡ 4 and
multipliers
p(x) =
1
4π
|x|2 −
1
2π
M. Hinze
log |x| and µ = δ0 .
Optimization with pdes − tailored discrete concepts
Experimental order of convergence
RL
1
2
3
4
5
ku − uh k
0.788985
0.759556
0.919917
0.966078
0.986686
M. Hinze
ky − yh k
0.536461
1.147861
1.389378
1.518381
1.598421
Optimization with pdes − tailored discrete concepts
Relaxing constraints – Lavrentiev (H., Meyer COAP 2008)
Lavrentiev Regularization: relax y ≤ b to λu + y ≤ b
(λ > 0). Numerical analysis yields
I
Buλ ∈ L2 (Ω) uniformly:
λ
λ
λ
ku − uλ
h k ∼ ku − u k + ku − uh k ∼
I
I
λ + h1−d/4 ,
d
):
d−1
Buλ ∈ W1,s (Ω) uniformly for all s ∈ (1,
λ
λ
λ
ku − uλ
h k ∼ ku − u k + ku − uh k ∼
√
√
λ + h2−d/2− ,
∞
Buλ ∈ L∞ (Ω), Buλ
h ∈ L (Ω) uniformly:
λ
λ
λ
ku − uλ
h k ∼ ku − u k + ku − uh k ∼
M. Hinze
√
λ + h| log h|.
Optimization with pdes − tailored discrete concepts
Relaxing constraints R– penalization (Hintermüller, H.)
Relax y ≤ b with
I
γ
2
Ω
|(y − b)+ |2 dx in cost functional.
Buγ ∈ L2 (Ω) uniformly:
ku − uγh k ∼ ku − uγ k + kuγ − uγh k ∼
1 −d/2 1/2
1−d/p
+ h1−d/4 ,
∼ h
+√ h
γ
I
Buγ ∈ W1,s (Ω) for all s ∈ (1,
d
)
d−1
uniformly:
ku − uγh k ∼ ku − uγ k + kuγ − uγh k ∼
1 −d/2 1/2
1−d/p
∼ h
+√ h
+ h2−d/2− ,
γ
I
Buγ ∈ L∞ (Ω), Buγh ∈ L∞ (Ω) uniformly:
ku − uγh k ∼ ku − uγ k + kuγ − uγh k ∼
1 −d/2 1/2
1−d/p
∼ h
+√ h
+ h| log h|.
γ
M. Hinze
Optimization with pdes − tailored discrete concepts
Relaxing constraints – penalization, numerical results
Convergence of |u−uγ |
2
hL
2
10
h = 1/16
h = 1/32
h = 1/64
h = 1/128
h = 1/256
h = 1/512
O(γ−1/2)
0
|u−uγh|L2
10
−2
10
−4
10
−6
10
0
10
5
10
M. Hinze
10
γ
10
15
10
Optimization with pdes − tailored discrete concepts
Relaxing constraints – barriers
R
Barriers: relax y ≤ b by adding −µ Ω log (b − y)dx to cost
functional (µ > 0). Numerical analysis yields
I
Buµ ∈ L2 (Ω) uniformly:
µ
µ
µ
ku − uµ
h k ∼ ku − u k + ku − uh k ∼
I
Buµ ∈ W1,s (Ω) for all s ∈ (1,
d
)
d−1
µ + h1−d/4 ,
uniformly:
µ
µ
µ
ku − uµ
h k ∼ ku − u k + ku − uh k ∼
I
√
√
µ + h2−d/2− ,
∞
Buµ ∈ L∞ (Ω), Buµ
h ∈ L (Ω) uniformly:
µ
µ
µ
ku − uµ
h k ∼ ku − u k + ku − uh k ∼
√
µ + h| log h|.
This is work in progress with Anton Schiela.
M. Hinze
Optimization with pdes − tailored discrete concepts
Relaxing constraints – barriers, numerical results
0
10
−1
overall error
10
−2
10
1/2
−3
10
−7
10
−6
10
−5
−4
10
10
µ
M. Hinze
−3
10
−2
10
−1
10
Optimization with pdes − tailored discrete concepts
Consequence: Grid size h and parameters (λ, γ, µ) should be
coupled;
√
Lavrentiev: λ ∼ h2−d/2 ,
√
Barriers: µ ∼ h2−d/2 ,
Penalization (p = ∞):
√1
γ
M. Hinze
∼ h1+d/2 (optimal ?).
Optimization with pdes − tailored discrete concepts
Constraints on the gradient
Consider
min J(u) =
u∈Uad
1
2
Z
2
|y − z| +
Ω
α
r
Z
|u|
r
+
α
Ω
2
Z
|u|
2
Ω
where y = G(u), i.e. solves the pde, and ∇y ∈ Yad .
Here
Yad = {z ∈ C0 (Ω̄)d | |z(x)| ≤ δ, x ∈ Ω̄},
and
r=2:
r>d:
Uad = {u ∈ L2 (Ω) | a ≤ u ≤ b a.e. in Ω}(a, b ∈ L∞ ),
Uad = Lr (Ω).
Then Uad ⊂ Lr (Ω) for r > d ⇒ ∇y ∈ C0 (Ω̄)d .
Slater condition:
∃û ∈ Uad |∇ŷ(x)| < δ, x ∈ Ω̄, where ŷ solves the pde with u = û.
M. Hinze
Optimization with pdes − tailored discrete concepts
Optimality conditions (Casas & Fernandez)
An element u ∈ Uad is a solution if and only if there exist
d
) such that
µ
~ ∈ M(Ω̄)d and p ∈ Lt (Ω) (t < d−1
R
R
R
0
0
pAz
µ ∀z ∈ W2,t (Ω) ∩ W01,t (Ω)
R − Ω (y − z)z = Ω̄ ∇z · d~
µ ≤0
∀z ∈ Yad ,
Ω̄ (z − ∇y) · d~
R
∀ũ ∈ Uad for r = 2, or
Ω (p + αu)(ũ − u) ≥ 0
Ω
p + α((u+)|u|r−2 u) = 0
in Ω for r > d.
Structure of multiplier: µ
~ = δ1 ∇y µ, where µ ∈ M(Ω̄) ≥ 0
is concentrated on {x ∈ Ω̄ | |∇y(x)| = δ}.
M. Hinze
Optimization with pdes − tailored discrete concepts
FE discretization, conventional
Piecewise linear, continuous Ansatz for the state
yh = Gh (u) ∈ Xh .
The discrete control problem reads
Z
Z
Z
α
α
1
2
r
2
|yh − z| +
|u| +
|u|
min Jh (u) :=
u∈Uad
2 Ω
rZ Ω
2 Ω
1
h
∈ Yad
,
subject to yh = Gh (u) and
∇yh
T∈Th
|T| T
where
h
Yad
:= {ch : Ω̄ → Rd | ch|T is constant and |ch|T | ≤ δ, T ∈ Th }.
M. Hinze
Optimization with pdes − tailored discrete concepts
FE discretization, conventional, optimality conditions
The variational discrete problem has a unique solution
uh ∈ Uad . There exist µT ∈ Rd , T ∈ Th,X and ph ∈ Xh such
that with yh = Gh (uh ) we have
Z
X
a(vh , ph ) =
(yh − z)vh +
|T| ∇vh|T · µT
∀vh ∈ Xh ,
Ω
T∈Th,X
X
|T| chT − ∇yh|T · µT ≤ 0
∀ch ∈ Ch ,
T∈Th,X
ph + α((uh +)|uh |r−2 uh ) = 0
in Ω.
Structure of the multiplier: µ
~ T = µT δ1 ∇yhT , where µT ∈ R.
Furthermore, µT ≥ 0 and µT > 0 only if |∇yhT | = δ.
M. Hinze
Optimization with pdes − tailored discrete concepts
Results
Deckelnick, Günther, H. (Oberwolfach Report 2008): Let
uh ∈ Uad be the variational discrete optimal solution with
corresponding state yh ∈ Xh and adjoint variables ph ∈ Xh ,
µ
~ T (T ∈ Th ).
Then for h small enough
P
|T| |µT | ≤ C,
I
kyh k, kuh kLr , kph k
I
ky − yh k ≤ Ch 2 (1− r ) , ku − uh kLr ≤ Ch r (1− r ) , and
1
d
ku − uh kL2 ≤ Ch 2 (1− r ) .
r
L r−1
1
,
d
T∈Th,X
1
d
These results are also valid for a piecewise constant Ansatz
of the control.
M. Hinze
Optimization with pdes − tailored discrete concepts
FE discretization, Raviart Thomas
Mixed fe approximation of the state with lowest order
Raviart–Thomas element, i.e.
(yh , vh ) = Gh (u) ∈ Yh × Vh
denotes the solution of
Z
Z
−1
v h · wh +
yh divwh = 0
Ω Z
Z
zh divvh −
a0 yh zh +
u zh = 0
A
∀wh ∈ Vh
Ω
Z
Ω
Ω
∀zh ∈ Yh .
Ω
M. Hinze
Optimization with pdes − tailored discrete concepts
FE discretization, cont.
The discrete control problem reads
1
Z
u∈Uad
2
|yh − z|2 +
α
Z
|u|2
2
Ω
Ω1 Z
h
A−1 vh
subject to (yh , vh ) = Gh (u) and
∈ Yad
,
T∈Th
|T| T
min Jh (u) :=
where
h
Yad
:= {ch : Ω̄ → Rd | ch|T is constant and |ch|T | ≤ δ, T ∈ Th }.
M. Hinze
Optimization with pdes − tailored discrete concepts
FE discretization, optimality conditions
The discrete problem has a unique solution uh ∈ Uad .
Furthermore, there are µ
~ T ∈ Rd and (ph , χh ) ∈ Yh × Vh such
that with (yh , vh ) = Gh (uh ) we have
Z
Z
Z
X
−1
A χh · w h +
ph divwh +
µ
~ T · − A−1 wh = 0∀wh ∈ Vh
Ω
Ω
T
T∈Th
Z
Z
Z
zh divχh −
Ω
a0 ph zh + (yh − z) zh = 0∀zh ∈ Yh .
Ω
Z
(ph + αuh )(ũ − uh ) ≥ 0∀ũ ∈ Uad
Ω
Z
X
h
µ
~ T · ch|T − − A−1 vh ≤ 0 ∀ch ∈ Yad
Ω
T∈Th
T
R
Structure of the multiplier: µ
~ T = µT δ1 −T A−1 vh , where
µ
R. Furthermore,
µT ≥ 0 and µT > 0 only if
RT ∈−1
− A vh = δ.
T
M. Hinze
Optimization with pdes − tailored discrete concepts
Results
Deckelnick, Günther, H. (Numer. Math 2008): Let uh ∈ Uad
be the optimal solution of the discrete problem with
corresponding state (yh , vh ) ∈ Yh × Vh and adjoint variables
(ph , χh ) ∈ Yh × Vh , µ
~ T , T ∈ Th .
Then for h small enough
P
|~
µT | ≤ C, and
I
kyh k,
I
ku − uh k + ky − yh k ≤ Ch 2 | log h| 2 .
T∈Th
1
M. Hinze
1
Optimization with pdes − tailored discrete concepts
Constraints on the gradient, example
We take Ω = B2 (0) and consider
min J(u) =
1
2
ky − zk2L2 (Ω) +
1
2
kuk2L2 (Ω)
with pointwise bounds on the constraints, i.e. {a ≤ u ≤ b},
where a, b ∈ L∞ (Ω), and pointwise bounds on the gradient,
i.e. |∇y(x)| ≤ δ := 1/2. State and control satisfy
−∆y = f + u in Ω,
y = 0 on ∂Ω.
Data:
z(x) :=
1
4
1
2
+ 12 ln 2 − 41 |x|2 , 0 ≤ |x| ≤ 1
ln 2 − 12 ln |x|
, 1 < |x| ≤ 2
f(x) :=
2
0
Solution:
y(x) ≡ z(x) and u(x) =
M. Hinze
−1 , 0 ≤ |x| ≤ 1
0 , 1 < |x| ≤ 2
Optimization with pdes − tailored discrete concepts
Numerical experiment, piecewise constant control Ansatz
0.2
0
−0.2
−0.4
0.05
−0.6
0
2
−0.8
−1.2
2
2
1
−1
1
0
1
0
−1
−2
−2
−1
0
1
0
2
−1
−1
−2 −2
M. Hinze
Optimization with pdes − tailored discrete concepts
Experimental order of convergence
RL
1
2
3
4
ku − uh kL4
0.76678
0.33044
0.27542
0.28570
ku − uh k
0.72339
0.64248
0.54054
0.53442
ky − yh k
1.90217
1.25741
1.23233
1.16576
Results show the predicted behaviour, since r = ∞.
M. Hinze
Optimization with pdes − tailored discrete concepts
Numerical solution, mixed finite elements
0.1
0.55
0
0.7
0.5
0.4
0.6
0.45
0.2
0.5
0.4
0.35
0.4
−0.1
−0.2
0
−0.3
−0.2
−0.4
0.3
0.3
−0.4
−0.5
−0.6
−0.6
0.25
0.2
0.2
0.1
0.15
0
2
2
1
0
0
0.1
0.05
−1
−0.8
−0.7
−1
2
2
1
0
0
−0.8
−0.9
−1
−2
−2
−2
M. Hinze
−2
Optimization with pdes − tailored discrete concepts
Experimental order of convergence
RL
1
2
3
ku − uh k
0.98576
0.51814
0.50034
ky − yh k
1.06726
1.02547
1.01442
kyP − yhP k
1.08949
1.09918
1.08141
Superscript P denotes post–processed piecewise linear state.
It attains the same order of convergence but yields
significantly smaller approximation error.
M. Hinze
Optimization with pdes − tailored discrete concepts
Thank you very much for your attention
M. Hinze
Optimization with pdes − tailored discrete concepts
Goal oriented adaptivity
Let aij = δij , bi = 0, and c = 1. Further let
Uad = {c ≤ u ≤ d},
Yad = {a ≤ y ≤ b},
and, for example,
J(y, u) =
1
2
Z
|y − z|2 +
α
ku − u0 k2U ,
2
Ω
Z
1
α
Jh (yh , uh ) =
|yh − z|2 + kuh − u0 k2U .
2 Ω
2
Aim: Extend DWR method of Becker, Kapp, Rannacher,...)
to construct ideal meshes w.r.t. the error J(yh , uh ) − J(y, u).
M. Hinze
Optimization with pdes − tailored discrete concepts
Goal oriented adaptivity, error representation
Let
ρp (·) := Jy (yh , uh )(·) − a(·, ph ) + hµh , ·i,
ρu (·) := Ju (yh , uh )(·) − (·, ph ) and
ρy (·) := −a(yh , ·) + (uh , ·).
Then for b = Ih b (Günther, H. (J. Numer. Math. 2008))
J(y, u) − J(yh , uh ) =
+
I
I
I
1
2
1
2
ρp (y − ih y) +
1
2
ρy (p − ih p)+
{hµ + µh , yh − yi + hλ + λh , uh − ui}
ρu (·) does not appear in this representation.
No differences of the multipliers µ, µh , λ, λh appear in
this representation.
Constraints on gradient:
ρp (·) = Jy (yh , uh )(·) − a(·, ph ) − hdivµ~h , ·i and
hµ + µh , yh − yi → h−div(~
µ + µ~h ), yh − yi
M. Hinze
Optimization with pdes − tailored discrete concepts
Goal oriented adaptivity, multiplier support and mesh
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.5
x2
x2
0.6
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0
0
0.1
0.2
0.4
0.6
0.8
1
x1
0
0
0.2
0.4
0.6
0.8
1
x
1
M. Hinze
Optimization with pdes − tailored discrete concepts
Goal oriented adaptivity, error
global refinement
local refinement
−1
h
h
|J* − J(y , u )|
10
−2
10
−3
10
1000
10000
100000
m
M. Hinze
Optimization with pdes − tailored discrete concepts
Goal oriented adaptivity, efficiency
m
484
1013
2730
8038
23216
69645
h
0.0673
0.0673
0.0673
0.0673
0.0673
0.0673
hmin
0.0476
0.0238
0.0119
0.0060
0.0030
0.0015
|J∗ − J(yh , uh )|
0.43855
0.04606
0.03477
0.01992
0.00516
0.00033
Ieff
2.0
0.5
1.2
1.9
1.4
0.3
Work in progress, collaboration with O. Benedix and B.
Vexler
M. Hinze
Optimization with pdes − tailored discrete concepts
Thank you very much for your attention
M. Hinze
Optimization with pdes − tailored discrete concepts