Tailored discrete concepts for pde constrained optimization
M. Hinze
1
Tailored discrete concepts for pde constrained optimization
M. Hinze
Fachbereich Mathematik
Bereich Optimierung und Approximation, Universität Hamburg
(joint work with Klaus Deckelnick, Andreas Günther, Michael Hintermüller, Christian Meyer, Ulrich
Matthes, Anton Schiela, Morten Vierling)
Warwick, May 20, 2010
Tailored discrete concepts for pde constrained optimization
M. Hinze
2
Topic of this talk
Optimal control of pdes with pointwise constraints:
(P)
min
u∈Uad ,y ∈Yad
J(y , u) s.t. PDE (y ) = B(u)
Discrete counterpart:
(Ph )
min
h ,y ∈Y h
uh ∈Uad
h
ad
Jh (yh , uh ) s.t. PDEh (yh ) = Bh (uh )
Questions:
h and Ansatz for u ?
Appropriate choice of Uad
h
h and Ansatz for y ?
Appropriate choice of Yad
h
Aim: Capture as much structure as possible of (P) on the discrete level.
Tailored discrete concepts for pde constrained optimization
M. Hinze
3
Model problem, general setting
(P)
1
u∈Uad ,y ∈Yad
2
subject to y = G(Bu).
min
Z
J(y , u) =
|y − z|2 +
D
α
kuk2U
2
Here, U denotes a Hilbert space, Uad ⊆ U closed and convex, B linear bounded
control operator, which maps abstract controls to feasible rhs. Furthermore,
Yad ⊆ Y with state space Y and y = G(Bu) iff
Ay = Bu in D, plus b.c. (plus i.c.)
P
P
elliptic case: D = Ω and Ay := − di,j =1 ∂xj aij yxi + di=1 bi yxi + cy
uniformly elliptic operator, Y = H 1 (Ω)
parabolic case: D = (0, T ] × Ω and
P
P
Ay := yt − di,j =1 ∂xj aij yxi + di=1 bi yxi + cy with strongly elliptic leading
part, Y = W (0, T ).
Typical choices:
Uad = {a ≤ u ≤ b} and Yad = {y ≤ c}, or {|∇y | ≤ δ}.
Tailored discrete concepts for pde constrained optimization
M. Hinze
4
Existence and uniqueness of solutions, derivatives, Yad = {y ≤ c}
For every u we have a unique y (u) = G(Bu). So we may minimize the
reduced functional
J(u) := J(G(Bu), u)
instead.
Problem (P) admits a unique solution u with corresponding state
y = G(Bu).
J 0 (u) = αu + RB ∗ p with p denoting the adjoint state determined by
A∗ p = y (u) − z,
and R :
U∗
→ U denoting the Riesz isomorphism.
Tailored discrete concepts for pde constrained optimization
M. Hinze
5
Optimality conditions, Yad = {y ≤ c}(Casas 86,93)
Let u ∈ Uad denote the unique optimal control with associated state y = G(Bu)
and let the
Slater condition: ∃ũ ∈ Uad such that G(B ũ) < b in D̄
be satisfied. Then there exist µ ∈ M(D̄) and some p such that there holds
A∗ p = y − z + µ
1
u = PUad − RB ∗ p ,
α
Z
µ ≥ 0, y ≤ c in D and
(c − y )d µ = 0,
D̄
where
elliptic case: p ∈ W 1,s (Ω) for all s < d /(d − 1).
parabolic case: p ∈ Ls (W 1,σ ) for all s, σ ∈ [1, 2) with 2/s + d /σ > d + 1.
Tailored discrete concepts for pde constrained optimization
M. Hinze
6
Only control constraints, Yad ≡ Y
Optimality conditions:
Ay = Bu
A∗ p = y − z
1
u = PUad − RB ∗ p ,
α
The adjoint p in general is smoother than the state y .
The optimal control and the adjoint state p are coupled via an algebraic
relation ⇒ a discretization of p induces a discretization of u.
Reduction to p, say delivers
AA∗ p − BPUad
−
1
RB ∗ p
α
= Az,
which in the parabolic case is a bvp for p in space-time.
Tailored discrete concepts for pde constrained optimization
M. Hinze
7
Only state constraints, Uad ≡ U
Optimality conditions:
Ay = Bu
A∗ p = y − z + µ
1
u = − RB ∗ p,
α Z
µ ≥ 0, y ≤ c in D and
(c − y )d µ = 0,
D̄
The state y in general is smoother than the adjoint p.
The optimal control and the adjoint state p are coupled via an algebraic
relation ⇒ a discretization of p induces a discretization of u.
A discretization of y ideally should deliver feasible discrete states.
Tailored discrete concepts for pde constrained optimization
M. Hinze
8
Discretization – a variational concept (H., COAP 2005)
Discrete optimal control problem:
Z
1
α
|yh − z|2 + kuk2U
u∈Uad
2 D
2
subject to yh = Gh (Bu) and yh ≤ Ih c.
min Jh (u) :=
Here, yh (u) = Gh (Bu) denotes the
p.l. and continuous fe approximation to y (u) (elliptic case),
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!
Tailored discrete concepts for pde constrained optimization
M. Hinze
9
Variational versus conventional discretization
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−1
−1
−0.8
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
−1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Tailored discrete concepts for pde constrained optimization
M. Hinze
10
Variational discretization for time-dependent problems
Movie time-dependent problems
Tailored discrete concepts for pde constrained optimization
M. Hinze
11
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
P
µh = nv
j =1 µj δxj (elliptic case, xi fe nodes, k = nv ),
R
Pm Pnv
µh = i =1 j =1 µij δxj ◦ |I1 | I •dt (parabolic case, xi fe nodes, Ii dg
i
i
intervals, k = nv + m),
we have
Z
Z
(yh − z)vh +
a(vh , ph ) =
D
vh d µh
∀vh ∈ Xh ,
D̄
hB ∗ ph + αuh , v − uh iU ∗ ,U ≥ 0
∀v ∈ Uad ,
Z
µj ≥ 0, yh ≤ Ih c, and
Ih c − yh d µh = 0.
D̄
Here, δx denotes the Dirac measure concentrated at x and Ih is the usual
Lagrange interpolation operator.
Tailored discrete concepts for pde constrained optimization
M. Hinze
12
Algorithms for variational discretization
Define
Gh (u) = PUad
−
1
RB ∗ ph (yh (u)) .
α
The optimality condition reads Gh (u) = 0 and motivates the fix–point iteration
u given, do until convergence
1
u + = PUad − RB ∗ ph (yh (u)) , u = u + .
α
Two questions immediately arise.
1
Is this algorithm numerically implementable?
2
Does this algorithm converge?
Answers:
1. Yes, whenever for given u it is possible to numerically evaluate the expression
1
PUad − RB ∗ ph (yh (u))
α
in the i − th iteration, with an numerical overhead which is independent of the
iteration counter of the algorithm.
Tailored discrete concepts for pde constrained optimization
M. Hinze
13
Algorithms for variational discretization, cont.
2. Yes, if α > kRB ∗ Sh∗ Sh BkL(U) , since PUad is non–expansive.
Condition too restrictive → semi–smooth Newton method applied to Gh (u) = 0:
u given, solve until convergence
Gh0 (u)u + = −Gh (u) + Gh0 (u)u,
u = u+.
1. This algorithm is implementable whenever the fix–point iteration is, since
− Gh (u) + Gh0 (u)u =
1 0
1
1
∗
−
RB
p
(u)
RB ∗ Sh∗ Sh Bu.
= −PUad − RB ∗ ph (u) − PU
h
α
α ad
α
2. For every α > 0 this algorithm is locally fast convergent (H. (COAP 2005),
Vierling).
Tailored discrete concepts for pde constrained optimization
M. Hinze
14
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 B ũ.
Tailored discrete concepts for pde constrained optimization
M. Hinze
15
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∞ + ky h (uh ) − yh k∞ +
n
o
+ C (kuk, kuh k) ky − yh (u)k + ky h (uh ) − yh k .
Here, yh (u) = Gh (Bu), y h (uh ) = G(Buh ).
We need uniform estimates for discrete approximations.
Tailored discrete concepts for pde constrained optimization
M. Hinze
16
Error estimates, parabolic case
Deckelnick, H. (JCM 2010)
Controls u ∈ L2 (0, T )m , and fi ∈ H 1 (Ω) given actuations.
Bu :=
m
X
ui (t)fi (x),
y0 ∈ H 2 (Ω).
i =1
Then y = G(Bu) ∈ {v ∈ L∞ (H 2 ), vt ∈ L2 (H 1 )} and we have with yh = Gh (Bu)
and time stepping δt ∼ h 2
ky − yh k∞ ≤ C
p
h | log h|,
√
h,
(d = 2)
(d = 3)
This is not an off-the-shelf result! It yields
αku − uh k2 + ky − yh k2 ≤ C
p
h | log h|,
√
h,
(d = 2)
(d = 3).
Tailored discrete concepts for pde constrained optimization
M. Hinze
17
Error estimates, elliptic case
Deckelnick, H. (SINUM 2007, ENUMATH 2007)
Bu ∈ L2 (Ω):
(
ku − uh kU , ky − yh kH 1 =
1
if d = 2,
if d = 3,
O(h 2 ),
1
O(h 4 ),
Bu ∈ W 1,s (Ω):
3
d
ku − uh kU , ky − yh kH 1 ≤ Ch 2 − 2s
q
| log h|.
Bu ∈ L∞ (Ω):
ku − uh kU , ky − yh kH 1 ≤ Ch| log h|.
U = L2 (Ω), Uad = {u ≤ d }, uh p.c.:
ku − uh kU , ky − yh kH 1 ≤ C
h| log h|,
√
h,
Similar results obtained by C. Meyer for discrete controls.
if d = 2,
if d = 3.
Tailored discrete concepts for pde constrained optimization
M. Hinze
18
Numerical experiment 1
Ω := B1 (0), α > 0,
z(x) := 4 +
and u0 (x) := 4 +
1
1
1
−
|x|2 +
log |x|, b(x) := |x|2 + 4,
π
4π
2π
1
|x|2
4απ
−
J(u) :=
1
2απ
1
2
Z
log |x|.
|y − z|2 +
Ω
α
2
Z
|u − u0 |2 ,
Ω
where y = G(u).
Unique solution u ≡ 4 with corresponding state y ≡ 4 and multipliers
p(x) =
1
1
|x|2 −
log |x| and µ = δ0 .
4π
2π
Tailored discrete concepts for pde constrained optimization
M. Hinze
19
Experimental order of convergence
RL
1
2
3
4
5
ku − uh k
0.788985
0.759556
0.919917
0.966078
0.986686
ky − yh k
0.536461
1.147861
1.389378
1.518381
1.598421
Thank you very much for coming, and for your attention
Tailored discrete concepts for pde constrained optimization
M. Hinze
20
Relaxing constraints – Lavrentiev (H., Meyer COAP 2008)
Lavrentiev Regularization: relax y ≤ c to λu + y ≤ c (λ > 0). Numerical
analysis yields
Bu λ ∈ L2 (Ω) uniformly:
ku − uhλ k ∼ ku − u λ k + ku λ − uhλ k ∼
Bu λ ∈ W 1,s (Ω) uniformly for all s ∈ (1,
√
λ + h 1−d /4 ,
d
):
d −1
ku − uhλ k ∼ ku − u λ k + ku λ − uhλ k ∼
√
Bu λ ∈ L∞ (Ω), Buhλ ∈ L∞ (Ω) uniformly:
ku − uhλ k ∼ ku − u λ k + ku λ − uhλ k ∼
λ + h 2−d /2− ,
√
λ + h| log h|.
Tailored discrete concepts for pde constrained optimization
M. Hinze
21
Relaxing constraints – penalization (Hintermüller, H., SINUM 2009)
Relax y ≤ c with
γ
2
R
Ω
|(y − c)+ |2 dx in cost functional.
Bu γ ∈ L2 (Ω) uniformly:
γ
γ
ku − uh k ∼ ku − u γ k + ku γ − uh k ∼
1/2
1
+ h 1−d /4 ,
∼ h 1−d /p + √ h −d /2
γ
Bu γ ∈ W 1,s (Ω) for all s ∈ (1,
γ
d
)
d −1
uniformly:
γ
ku − uh k ∼ ku − u γ k + ku γ − uh k ∼
1/2
1
∼ h 1−d /p + √ h −d /2
+ h 2−d /2− ,
γ
γ
Bu γ ∈ L∞ (Ω), Buh ∈ L∞ (Ω) uniformly:
γ
γ
ku − uh k ∼ ku − u γ k + ku γ − uh k ∼
1/2
1
∼ h 1−d /p + √ h −d /2
+ h| log h|.
γ
Tailored discrete concepts for pde constrained optimization
M. Hinze
22
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
10
γ
10
15
10
Tailored discrete concepts for pde constrained optimization
M. Hinze
23
Relaxing constraints – barriers (H., Schiela, COAP 2009)
Barriers: relax y ≤ c by adding −µ
Numerical analysis yields
R
Ω
log (c − y )dx to cost functional (µ > 0).
Bu µ ∈ L2 (Ω) uniformly:
µ
µ
ku − uh k ∼ ku − u µ k + ku µ − uh k ∼
Bu µ ∈ W 1,s (Ω) for all s ∈ (1,
ku −
µ
uh k
d
)
d −1
√
µ + h 1−d /4 ,
uniformly:
µ
∼ ku − u µ k + ku µ − uh k ∼
√
µ + h 2−d /2− ,
µ
Bu µ ∈ L∞ (Ω), Buh ∈ L∞ (Ω) uniformly:
µ
µ
ku − uh k ∼ ku − u µ k + ku µ − uh k ∼
√
µ + h| log h|.
Tailored discrete concepts for pde constrained optimization
M. Hinze
24
Relaxing constraints – barriers, numerical results
0
10
−1
overall error
10
−2
10
1/2
−3
10
−7
10
−6
10
−5
10
−4
10
µ
−3
10
−2
10
−1
10
Tailored discrete concepts for pde constrained optimization
M. Hinze
25
Consequence: Grid size h and parameters (λ, γ, µ) should be coupled;
√
Lavrentiev:
λ ∼ h 2−d /2 ,
√
Barriers:
µ ∼ h 2−d /2 ,
Penalization (p = ∞):
√1
γ
∼ h 1+d /2 (optimal ?).
Tailored discrete concepts for pde constrained optimization
M. Hinze
26
Constraints on the gradient
Consider
min J(u) =
u∈Uad
1
2
Z
|y − z|2 +
Ω
α
r
Z
|u|r
Ω
+
α
2
Z
|u|2
Ω
where y = G(u), i.e. solves the pde, and ∇y ∈ Yad .
Here
Yad = {z ∈ C 0 (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 ∈ C 0 (Ω̄)d .
Slater condition:
∃û ∈ Uad |∇ŷ (x)| < δ, x ∈ Ω̄, where ŷ solves the pde with u = û.
Tailored discrete concepts for pde constrained optimization
M. Hinze
27
Optimality conditions (Casas & Fernandez)
An element u ∈ Uad is a solution if and only if there exist µ
~ ∈ M(Ω̄)d and
d
p ∈ Lt (Ω) (t < d −1
) such that
0
1,t 0
R
pAz
R − Ω (y − z)z
~
Ω̄ (z − ∇y ) · d µ
R
= Ω̄ ∇z · d µ
~
≤0
∀z ∈ W 2,t (Ω) ∩ W0
∀z ∈ Yad ,
R
+ αu)(ũ − u)
≥0
∀ũ ∈ Uad for r = 2, or
p + α((u+)|u|r −2 u)
=0
R
Ω
Ω (p
Structure of multiplier: µ
~ =
{x ∈ Ω̄ | |∇y (x)| = δ}.
1
∇y
δ
(Ω)
in Ω for r > d .
µ, where µ ∈ M(Ω̄) ≥ 0 is concentrated on
Tailored discrete concepts for pde constrained optimization
M. Hinze
28
FE discretization, conventional
Piecewise linear, continuous Ansatz for the state yh = Gh (u) ∈ Xh .
The discrete control problem reads
Z
Z
Z
1
α
α
min Jh (u) :=
|yh − z|2 +
|u|r +
|u|2
u∈Uad
2 Ω
r
2 Ω
1 Z Ω
h
subject to yh = Gh (u) and
∇yh
∈ Yad
,
T ∈Th
|T | T
where
h
Yad
:= {ch : Ω̄ → Rd | ch|T is constant and |ch|T | ≤ δ, T ∈ Th }.
Tailored discrete concepts for pde constrained optimization
M. Hinze
29
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.
Tailored discrete concepts for pde constrained optimization
M. Hinze
30
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
kyh k, kuh kLr , kph k
1
r
L r −1
,
d
P
T ∈Th,X
|T | |µT | ≤ C ,
1
d
ky − yh k ≤ Ch 2 (1− r ) , ku − uh kLr ≤ Ch r (1− r ) , and
ku − uh kL2 ≤ Ch
1 (1− d
2
r
)
.
These results are also valid for a piecewise constant Ansatz of the control.
Tailored discrete concepts for pde constrained optimization
M. Hinze
31
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
A−1 vh · wh +
yh divwh
Ω
Ω
Z
Z
Z
zh divvh −
a0 yh zh +
u zh
Ω
Ω
Ω
=
0
∀wh ∈ Vh
=
0
∀zh ∈ Yh .
Tailored discrete concepts for pde constrained optimization
M. Hinze
32
FE discretization, cont.
The discrete control problem reads
1
2
Z
α
|u|2
u∈Uad
2
Ω
1Ω Z
h
subject to (yh , vh ) = Gh (u) and
A−1 vh
∈ Yad
,
T ∈Th
|T | T
min Jh (u) :=
Z
|yh − z|2 +
where
h
Yad
:= {ch : Ω̄ → Rd | ch|T is constant and |ch|T | ≤ δ, T ∈ Th }.
Tailored discrete concepts for pde constrained optimization
M. Hinze
33
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
A−1 χh · wh +
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
Ω
Z
(ph + αuh )(ũ − uh )
Ω
Z
X
µ
~ T · ch|T − − A−1 vh
=
0∀zh ∈ Yh .
≥
0∀ũ ∈ Uad
≤
h
0 ∀ch ∈ Yad
.
Ω
T ∈Th
T
R
Structure of the multiplier: µ
~ T = µT δ1 −T A−1 vh , where µT ∈ R. Furthermore,
R
µT ≥ 0 and µT > 0 only if −T A−1 vh = δ.
Tailored discrete concepts for pde constrained optimization
M. Hinze
34
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
kyh k,
P
T ∈Th
|~
µT | ≤ C , and
1
1
ku − uh k + ky − yh k ≤ Ch 2 | log h| 2 .
Tailored discrete concepts for pde constrained optimization
M. Hinze
35
Constraints on the gradient, example
We take Ω = B2 (0) and consider
1
1
ky − zk2L2 (Ω) + kuk2L2 (Ω)
2
2
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
min J(u) =
−∆y = f + u in Ω,
y = 0 on ∂Ω.
Data:
z(x) :=
1
4
1
2
+ 12 ln 2 − 14 |x|2
ln 2 − 21 ln |x|
, 0 ≤ |x| ≤ 1
, 1 < |x| ≤ 2
Solution:
y (x) ≡ z(x) and u(x) =
−1
0
f (x) :=
2
0
, 0 ≤ |x| ≤ 1
, 1 < |x| ≤ 2
, 0 ≤ |x| ≤ 1
, 1 < |x| ≤ 2
Tailored discrete concepts for pde constrained optimization
M. Hinze
36
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
Tailored discrete concepts for pde constrained optimization
M. Hinze
37
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
Results show the predicted behaviour, since r = ∞.
ky − yh k
1.90217
1.25741
1.23233
1.16576
Tailored discrete concepts for pde constrained optimization
M. Hinze
38
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
−1
0.1
0.05
−0.8
−0.7
−1
2
2
1
0
0
−1
−2
−2
−2
−2
−0.8
−0.9
Tailored discrete concepts for pde constrained optimization
M. Hinze
39
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
ky P − 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.
Tailored discrete concepts for pde constrained optimization
M. Hinze
40
Thank you very much for your attention
Tailored discrete concepts for pde constrained optimization
M. Hinze
41
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 +
Ω
α
ku − u0 k2U ,
2
Jh (yh , uh ) =
1
2
Z
|yh − z|2 +
Ω
α
kuh − u0 k2U .
2
Aim: Extend DWR method of Becker, Kapp, Rannacher,...) to construct ideal
meshes w.r.t. the error J(yh , uh ) − J(y , u).
Tailored discrete concepts for pde constrained optimization
M. Hinze
42
Goal oriented adaptivity, error representation
Let
ρp (·) :=
ρu (·) :=
ρy (·) :=
Jy (yh , uh )(·) − a(·, ph ) + hµh , ·i,
Ju (yh , uh )(·) − (·, ph ) and
−a(yh , ·) + (uh , ·).
Then for b = Ih b (Günther, H. (J. Numer. Math. 2008))
J(y , u) − J(yh , uh ) =
1 p
1
ρ (y − ih y ) + ρy (p − ih p)+
2
2
1
+ {hµ + µh , yh − y i + hλ + λh , uh − ui}
2
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 − y i → h−div(~
µ + µ~h ), yh − y i
Tailored discrete concepts for pde constrained optimization
M. Hinze
43
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
x1
0.8
1
0
0
0.2
0.4
0.6
x
1
0.8
1
Tailored discrete concepts for pde constrained optimization
M. Hinze
44
Goal oriented adaptivity, error
global refinement
local refinement
−1
h
h
|J* − J(y , u )|
10
−2
10
−3
10
Tailored discrete concepts for pde constrained optimization
M. Hinze
45
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
Work in progress, collaboration with O. Benedix and B. Vexler
Ieff
2.0
0.5
1.2
1.9
1.4
0.3
Tailored discrete concepts for pde constrained optimization
M. Hinze
46
Thank you very much for your attention