Download Sparse Optimal Control in Measure Spaces for Elliptic and Parabolic

Faculty of Mathematics
Technische Universität München
Sparse Optimal Control in Measure Spaces
for Elliptic and Parabolic Systems
Boris Vexler1
joint work with
Karl Kunisch (KFU Graz) and Konstantin Pieper1
1 Faculty of Mathematics
Chair of Optimal Control
Technische Universität München
June 2014
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
1
Faculty of Mathematics
Technische Universität München
Outline
1. Motivation
2. Functional analytic setting
3. Optimality system and regularity
4. Discretization and error estimates
5. Parabolic case: Directional Sparsity
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
2
Faculty of Mathematics
Technische Universität München
Outline
1. Motivation
2. Functional analytic setting
3. Optimality system and regularity
4. Discretization and error estimates
5. Parabolic case: Directional Sparsity
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
3
Faculty of Mathematics
Technische Universität München
Motivation: Sparse control of PDEs
I
L1 -norm regularization leads to sparse solutions
min
Boris Vexler
1
kAx − bk22 + αkxk1 ,
2
Sparse Control Problems in Measure Spaces
x ∈ RN
June 2014
4
Faculty of Mathematics
Technische Universität München
Motivation: Sparse control of PDEs
I
L1 -norm regularization leads to sparse solutions
min
1
kAx − bk22 + αkxk1 ,
2
x ∈ RN
→ Solution x̄ has typically only a handful ( N) of non-zero entries
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
4
Faculty of Mathematics
Technische Universität München
Motivation: Sparse control of PDEs
I
L1 -norm regularization leads to sparse solutions
min
1
kAx − bk22 + αkxk1 ,
2
x ∈ RN
→ Solution x̄ has typically only a handful ( N) of non-zero entries
→ For α ≥ α̂
Boris Vexler
⇒
x̄ = 0.
Sparse Control Problems in Measure Spaces
June 2014
4
Faculty of Mathematics
Technische Universität München
Motivation: Sparse control of PDEs
I
L1 -norm regularization leads to sparse solutions
min
1
kAx − bk22 + αkxk1 ,
2
x ∈ RN
→ Solution x̄ has typically only a handful ( N) of non-zero entries
→ For α ≥ α̂
I
⇒
x̄ = 0.
Extension to optimal control problems with PDEs?
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
4
Faculty of Mathematics
Technische Universität München
Motivation: Sparse control of PDEs
I
L1 -norm regularization leads to sparse solutions
min
1
kAx − bk22 + αkxk1 ,
2
x ∈ RN
→ Solution x̄ has typically only a handful ( N) of non-zero entries
→ For α ≥ α̂
⇒
x̄ = 0.
I
Extension to optimal control problems with PDEs?
I
Applications:
I
I
optimal actuator placement,
e.g., optimal distribution of light sources
Point sourse identi cation
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
4
Faculty of Mathematics
Technische Universität München
Motivation: Sparse control of PDEs
I
L1 -norm regularization leads to sparse solutions
min
1
kAx − bk22 + αkxk1 ,
2
x ∈ RN
→ Solution x̄ has typically only a handful ( N) of non-zero entries
→ For α ≥ α̂
⇒
x̄ = 0.
I
Extension to optimal control problems with PDEs?
I
Applications:
I
I
I
optimal actuator placement,
e.g., optimal distribution of light sources
Point sourse identi cation
Right choice of function spaces?
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
4
Faculty of Mathematics
Technische Universität München
Motivation: Sparse control of PDEs
I
L1 -norm regularization leads to sparse solutions
min
1
kAx − bk22 + αkxk1 ,
2
x ∈ RN
→ Solution x̄ has typically only a handful ( N) of non-zero entries
→ For α ≥ α̂
⇒
x̄ = 0.
I
Extension to optimal control problems with PDEs?
I
Applications:
I
I
optimal actuator placement,
e.g., optimal distribution of light sources
Point sourse identi cation
I
Right choice of function spaces?
I
Discretization concept / error estimates / Optimization algorithms
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
4
Faculty of Mathematics
Technische Universität München
Motivation: Sparse control of PDEs
Optimal control problem
1
Minimize J(q, u) = ku − ud k2L2 (Ω) + αkqkX ,
2
(
−∆u = q
in Ω,
u=0
I
q ∈ X, u ∈ V
on ∂Ω,
the standard setting X = L2 (Ω) does not lead to a sparse solution
→ for typical optimal control supp q̄ = Ω!
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
5
Faculty of Mathematics
Technische Universität München
Motivation: Sparse control of PDEs
Optimal control problem
1
Minimize J(q, u) = ku − ud k2L2 (Ω) + αkqkX ,
2
(
−∆u = q
in Ω,
u=0
I
q ∈ X, u ∈ V
on ∂Ω,
the standard setting X = L2 (Ω) does not lead to a sparse solution
→ for typical optimal control supp q̄ = Ω!
I
Desired structure
−∆u =
N
X
qi δxi
i=1
with unknown N and unknown positions xi of Diracs
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
5
Faculty of Mathematics
Technische Universität München
Motivation: Sparse control of parabolic PDEs
Optimal control problem
Minimize J(q, u) =
1
ku − ud k2L2 (I×Ω) + αkqkX ,
2
∂t u − ∆u = q in I × Ω,
Boris Vexler
q ∈ X, u ∈ V
u(0) = u0 in Ω,
Sparse Control Problems in Measure Spaces
June 2014
6
Faculty of Mathematics
Technische Universität München
Motivation: Sparse control of parabolic PDEs
Optimal control problem
Minimize J(q, u) =
1
ku − ud k2L2 (I×Ω) + αkqkX ,
2
∂t u − ∆u = q in I × Ω,
I
q ∈ X, u ∈ V
u(0) = u0 in Ω,
Desired structure I
∂t u − ∆u =
N
X
qi (t)δxi
i=1
with unknown N and unknown (but xed) positions xi of Diracs
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
6
Faculty of Mathematics
Technische Universität München
Motivation: Sparse control of parabolic PDEs
Optimal control problem
Minimize J(q, u) =
1
ku − ud k2L2 (I×Ω) + αkqkX ,
2
∂t u − ∆u = q in I × Ω,
I
q ∈ X, u ∈ V
u(0) = u0 in Ω,
Desired structure I
∂t u − ∆u =
N
X
qi (t)δxi
i=1
I
with unknown N and unknown (but xed) positions xi of Diracs
Desired structure II
N
X
∂t u − ∆u =
qi (t)δxi (t)
i=1
with unknown N and unknown trajectories xi (t) of Diracs
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
6
Faculty of Mathematics
Technische Universität München
Outline
1. Motivation
2. Functional analytic setting
3. Optimality system and regularity
4. Discretization and error estimates
5. Parabolic case: Directional Sparsity
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
7
Faculty of Mathematics
Technische Universität München
Functional analytic setting
Cost functional
Minimize J(q, u) =
1
ku − ud k2L2 (Ω) + αkqkX ,
2
q ∈ X, u ∈ V
State equation
−∆u = q
u=0
Boris Vexler
in Ω,
on ∂Ω,
Sparse Control Problems in Measure Spaces
June 2014
8
Faculty of Mathematics
Technische Universität München
Functional analytic setting
Cost functional
Minimize J(q, u) =
1
ku − ud k2L2 (Ω) + αkqkX ,
2
q ∈ X, u ∈ V
State equation
−∆u = q
u=0
I
in Ω,
on ∂Ω,
Choices for X?
I
X = L1 (Ω)
I
boundedness of a seq. {qn } in L1 (Ω)
Boris Vexler
⇒
existence can not be guaranteed
6⇒
Sparse Control Problems in Measure Spaces
qnk * q̄
June 2014
8
Faculty of Mathematics
Technische Universität München
Functional analytic setting
Cost functional
Minimize J(q, u) =
1
ku − ud k2L2 (Ω) + αkqkX ,
2
q ∈ X, u ∈ V
State equation
−∆u = q
u=0
I
I
in Ω,
on ∂Ω,
Choices for X?
I
X = L1 (Ω)
I
boundedness of a seq. {qn } in L1 (Ω)
⇒
existence can not be guaranteed
6⇒
qnk * q̄
choose X = M(Ω), the space of regular Borel measures on Ω
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
8
Faculty of Mathematics
Technische Universität München
Functional analytic setting
Cost functional
Minimize J(q, u) =
1
ku − ud k2L2 (Ω) + αkqkX ,
2
q ∈ X, u ∈ V
State equation
−∆u = q
u=0
I
I
in Ω,
on ∂Ω,
Choices for X?
I
X = L1 (Ω)
I
boundedness of a seq. {qn } in L1 (Ω)
⇒
existence can not be guaranteed
6⇒
qnk * q̄
choose X = M(Ω), the space of regular Borel measures on Ω
I
C. Clason and K. Kunisch
A duality based approach to control problems in non-re exive Banach spaces
ESAIM COCV 17, pp. 243-266, 2011
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
8
Faculty of Mathematics
Technische Universität München
Example I: Sparse controls
Minimize J(q, u) =
1
ku − ud k2L2 (Ω0 ) + αkqkM(Ωc )
2
subject to
−∆u − b · ∇u = q in Ω,
(a) ud on Ωo
Boris Vexler
(b) optimal state ū
u = 0 on ∂Ω
(c) optimal control q̄ on Ωc
Sparse Control Problems in Measure Spaces
June 2014
9
Faculty of Mathematics
Technische Universität München
Example II: Inverse point source reconstruction
I
state equation: ∂t u − ∆u − b · ∇u = q & boundary and initial cond.
Γin
b(x)
Ωc
x0
Ω
b(x)
Γout
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
10
Faculty of Mathematics
Technische Universität München
Literature
I Measure valued controls in elliptic equations
I G. Buttazzo, N. Varchon and H. Zoubairi
Optimal measures for elliptic problems
Ann. Mat. Pura Appl. 185, pp. 207 221, 2006.
I
C. Clason and K. Kunisch
A duality based approach to control problems in non-re exive Banach spaces
ESAIM COCV 17, pp. 243-266, 2011
I
E. Casas, C. Clason and K. Kunisch
Approximation of elliptic control problems in measure spaces with sparse solutions
SIAM J. Control Optim. 50, pp. 1735 1752, 2012
I
K. Pieper and B. Vexler
A priori error analysis for discretization of sparse elliptic optimal control problems in measure spase
SIAM J. Control Optim. 51, pp. 2788-2808, 2013
I
E. Casas and K. Kunisch
Optimal control of semilinear elliptic equations in measure spaces
SIAM J. Control Optim. 52 , pp. 339-364, 2013
I Measure valued controls in parabolic equations
I E. Casas and E. Zuazua
Spike controls for elliptic and parabolic PDE
Systems and Control letters 63, pp. 311-318, 2013
I
E. Casas, C. Clason and K. Kunisch
Parabolic control problems in measure spaces with sparse solutions
SIAM J. Control Optim. 51, pp. 28-63, 2013
I
K. Kunisch, K. Pieper and B. Vexler
Measure valued directional sparsity for parabolic optimal control problems
SIAM J. Control Optim., submitted
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
11
Faculty of Mathematics
Technische Universität München
Functional analytic setting
I
Choose X = M(Ω), the space of regular Borel measures on Ω
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
12
Faculty of Mathematics
Technische Universität München
Functional analytic setting
I
I
Choose X = M(Ω), the space of regular Borel measures on Ω
Identify M(Ω) = (C0 (Ω))∗ with
Z
hq, ϕi =
ϕ dq, q ∈ M(Ω), ϕ ∈ C0 (Ω)
Ω
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
12
Faculty of Mathematics
Technische Universität München
Functional analytic setting
I
I
Choose X = M(Ω), the space of regular Borel measures on Ω
Identify M(Ω) = (C0 (Ω))∗ with
Z
hq, ϕi =
ϕ dq, q ∈ M(Ω), ϕ ∈ C0 (Ω)
Ω
I
Norm for q ∈ M(Ω)
kqkM(Ω) =
sup
hq, ϕi
ϕ∈C0 (Ω)
kϕkC0 (Ω) ≤1
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
12
Faculty of Mathematics
Technische Universität München
Functional analytic setting
I
I
Choose X = M(Ω), the space of regular Borel measures on Ω
Identify M(Ω) = (C0 (Ω))∗ with
Z
hq, ϕi =
ϕ dq, q ∈ M(Ω), ϕ ∈ C0 (Ω)
Ω
I
Norm for q ∈ M(Ω)
kqkM(Ω) =
sup
hq, ϕi
ϕ∈C0 (Ω)
kϕkC0 (Ω) ≤1
→ Other possibility X = L1 (Ω) with an additional L2 -regularization
and/or control constraints (G. Stadler, G. Wachsmuth, R. Herzog)
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
12
Faculty of Mathematics
Technische Universität München
Functional analytic setting: State equation
State equation: very weak formulation for q ∈ M(Ω)
u ∈ L2 (Ω) : (u, −∆ϕ) = hq, ϕi
Boris Vexler
∀ϕ ∈ H2 (Ω) ∩ H10 (Ω)
Sparse Control Problems in Measure Spaces
June 2014
13
Faculty of Mathematics
Technische Universität München
Functional analytic setting: State equation
State equation: very weak formulation for q ∈ M(Ω)
u ∈ L2 (Ω) : (u, −∆ϕ) = hq, ϕi
∀ϕ ∈ H2 (Ω) ∩ H10 (Ω)
Regularity
u ∈ W01,sε (Ω),
Boris Vexler
sε =
d
− ε,
d−1
for all 0 < ε <
Sparse Control Problems in Measure Spaces
1
d−1
June 2014
13
Faculty of Mathematics
Technische Universität München
Functional analytic setting: State equation
State equation: very weak formulation for q ∈ M(Ω)
u ∈ L2 (Ω) : (u, −∆ϕ) = hq, ϕi
∀ϕ ∈ H2 (Ω) ∩ H10 (Ω)
Regularity
u ∈ W01,sε (Ω),
sε =
d
− ε,
d−1
for all 0 < ε <
1
d−1
Lemma: Stability estimate
k∇ukLsε (Ω) ≤
Boris Vexler
c
kqkM(Ω)
ε
Sparse Control Problems in Measure Spaces
June 2014
13
Faculty of Mathematics
Technische Universität München
Functional analytic setting: State equation
State equation: very weak formulation for q ∈ M(Ω)
u ∈ L2 (Ω) : (u, −∆ϕ) = hq, ϕi
∀ϕ ∈ H2 (Ω) ∩ H10 (Ω)
Regularity
u ∈ W01,sε (Ω),
sε =
d
− ε,
d−1
for all 0 < ε <
1
d−1
Lemma: Stability estimate
k∇ukLsε (Ω) ≤
c
kqkM(Ω)
ε
Weak formulation for q ∈ M(Ω)
u ∈ W01,sε (Ω) : (∇u, ∇ϕ) = hq, ϕi
Boris Vexler
1,s0ε
∀ϕ ∈ W0
Sparse Control Problems in Measure Spaces
(Ω) ,→ C0 (Ω)
June 2014
13
Faculty of Mathematics
Technische Universität München
Functional analytic setting
Optimal control problem
Minimize J(q, u) =
1
ku − ud k2L2 (Ω ) + αkqkM(Ω ) ,
2
q ∈ M(Ω),
subject to
u ∈ W01,sε (Ω) : (∇u, ∇ϕ) = hq, ϕi
Boris Vexler
Sparse Control Problems in Measure Spaces
1,s0ε
∀ϕ ∈ W0
(Ω)
June 2014
14
Faculty of Mathematics
Technische Universität München
Functional analytic setting
Optimal control problem
Minimize J(q, u) =
1
ku − ud k2L2 (Ω ) + αkqkM(Ω ) ,
2
q ∈ M(Ω),
subject to
u ∈ W01,sε (Ω) : (∇u, ∇ϕ) = hq, ϕi
I
1,s0ε
∀ϕ ∈ W0
(Ω)
Existence of (q̄, ū)
I
I
minimizing sequence {qn }, bounded in M(Ω) = (C0 (Ω))∗
Banach-Alaoglu theorem: {qnk } converging weakly∗ to q̄ ∈ M(Ω)
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
14
Faculty of Mathematics
Technische Universität München
Functional analytic setting
Optimal control problem
Minimize J(q, u) =
1
ku − ud k2L2 (Ω ) + αkqkM(Ω ) ,
2
q ∈ M(Ω),
subject to
u ∈ W01,sε (Ω) : (∇u, ∇ϕ) = hq, ϕi
I
(Ω)
Existence of (q̄, ū)
I
I
I
1,s0ε
∀ϕ ∈ W0
minimizing sequence {qn }, bounded in M(Ω) = (C0 (Ω))∗
Banach-Alaoglu theorem: {qnk } converging weakly∗ to q̄ ∈ M(Ω)
Uniqueness of (q̄, ū)
I
unique ū, injectivity of the solution operator ⇒ unique q̄
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
14
Faculty of Mathematics
Technische Universität München
Functional analytic setting
Optimal control problem
Minimize J(q, u) =
1
ku − ud k2L2 (Ωo ) + αkqkM(Ωc ) ,
2
q ∈ M(Ω),
subject to
u ∈ W01,sε (Ω) : (∇u, ∇ϕ) = hq, ϕi
I
(Ω)
Existence of (q̄, ū)
I
I
I
1,s0ε
∀ϕ ∈ W0
minimizing sequence {qn }, bounded in M(Ω) = (C0 (Ω))∗
Banach-Alaoglu theorem: {qnk } converging weakly∗ to q̄ ∈ M(Ω)
Uniqueness of (q̄, ū)
I
I
unique ū, injectivity of the solution operator ⇒ unique q̄
disjoint control and observation domains ⇒ multiple solutions!
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
14
Faculty of Mathematics
Technische Universität München
Outline
1. Motivation
2. Functional analytic setting
3. Optimality system and regularity
4. Discretization and error estimates
5. Parabolic case: Directional Sparsity
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
15
Faculty of Mathematics
Technische Universität München
Optimality conditions
Reduced formulation
Minimize j(q) = f(q) + αkqkM(Ω) ,
Boris Vexler
with f(q) =
Sparse Control Problems in Measure Spaces
1
ku(q) − ud k2L2 (Ω)
2
June 2014
16
Faculty of Mathematics
Technische Universität München
Optimality conditions
Reduced formulation
Minimize j(q) = f(q) + αkqkM(Ω) ,
with f(q) =
1
ku(q) − ud k2L2 (Ω)
2
Optimality condition
0 ∈ ∂j(q̄)
Boris Vexler
⇔
α kqkM(Ω) − kq̄kM(Ω) ≥ −f 0 (q̄)(q−q̄)
Sparse Control Problems in Measure Spaces
∀q ∈ M(Ω)
June 2014
16
Faculty of Mathematics
Technische Universität München
Optimality conditions
Reduced formulation
Minimize j(q) = f(q) + αkqkM(Ω) ,
with f(q) =
1
ku(q) − ud k2L2 (Ω)
2
Optimality condition
0 ∈ ∂j(q̄)
⇔
α kqkM(Ω) − kq̄kM(Ω) ≥ −f 0 (q̄)(q−q̄)
∀q ∈ M(Ω)
Optimality condition
(u(q) − ū, ū − ud ) + α kqkM(Ω) − kq̄kM(Ω) ≥ 0 ∀q ∈ M(Ω)
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
16
Faculty of Mathematics
Technische Universität München
Optimality system
State equation for ū ∈ W01,s (Ω)
−∆ū = q̄
ū = 0
in Ω,
on ∂Ω,
Adjoint equation for z̄ ∈ H2 (Ω) ∩ H10 (Ω)
−∆z̄ = ū − ud
z̄ = 0
in Ω,
on ∂Ω,
Optimality condition for q̄ ∈ M(Ω)
−hq − q̄, z̄i + αkq̄kM(Ω) ≤ αkqkM(Ω)
Boris Vexler
Sparse Control Problems in Measure Spaces
∀q ∈ M(Ω)
June 2014
17
Faculty of Mathematics
Technische Universität München
Consequences from the optimality system
I
kz̄kC0 (Ω) ≤ α
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
18
Faculty of Mathematics
Technische Universität München
Consequences from the optimality system
I
kz̄kC0 (Ω) ≤ α
I
Jordan decomposition q̄ = q̄+ − q̄−
supp q̄+ ⊂ { x ∈ Ω | z̄(x) = −α } ,
supp q̄− ⊂ { x ∈ Ω | z̄(x) = α }
→ sparsity!
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
18
Faculty of Mathematics
Technische Universität München
Consequences from the optimality system
I
kz̄kC0 (Ω) ≤ α
I
Jordan decomposition q̄ = q̄+ − q̄−
supp q̄+ ⊂ { x ∈ Ω | z̄(x) = −α } ,
supp q̄− ⊂ { x ∈ Ω | z̄(x) = α }
→ sparsity!
I
z̄ ∈ H2 (Ω) ∩ H10 (Ω) ,→ C0,β (Ω̄)
supp q̄ ⊂ Ω η = { x ∈ Ω | dist (x, ∂Ω) > η } ,
Boris Vexler
Sparse Control Problems in Measure Spaces
η>0
June 2014
18
Faculty of Mathematics
Technische Universität München
Consequences from the optimality system
I
kz̄kC0 (Ω) ≤ α
I
Jordan decomposition q̄ = q̄+ − q̄−
supp q̄+ ⊂ { x ∈ Ω | z̄(x) = −α } ,
supp q̄− ⊂ { x ∈ Ω | z̄(x) = α }
→ sparsity!
I
z̄ ∈ H2 (Ω) ∩ H10 (Ω) ,→ C0,β (Ω̄)
supp q̄ ⊂ Ω η = { x ∈ Ω | dist (x, ∂Ω) > η } ,
I
η>0
dist (supp q̄+ , supp q̄− ) > η
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
18
Faculty of Mathematics
Technische Universität München
Improved regularity
I
examples with q̄ = δx0 (ud behaves like a Green's function)
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
19
Faculty of Mathematics
Technische Universität München
Improved regularity
I
examples with q̄ = δx0 (ud behaves like a Green's function)
I
numerical observation: if Ωo = Ωc = Ω
ud ∈ L∞ (Ω)
Boris Vexler
⇒
ū ∈ L∞ (Ω)
Sparse Control Problems in Measure Spaces
June 2014
19
Faculty of Mathematics
Technische Universität München
Improved regularity
I
examples with q̄ = δx0 (ud behaves like a Green's function)
I
numerical observation: if Ωo = Ωc = Ω
ud ∈ L∞ (Ω)
I
⇒
ū ∈ L∞ (Ω)
proof?
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
19
Faculty of Mathematics
Technische Universität München
Improved regularity
I
examples with q̄ = δx0 (ud behaves like a Green's function)
I
numerical observation: if Ωo = Ωc = Ω
ud ∈ L∞ (Ω)
I
proof?
I
consequences:
⇒
ū ∈ L∞ (Ω)
I
excludes Diracs
I
ū ∈ H10 (Ω) ∩ L∞ (Ω)
I
q̄ ∈ H−1 (Ω) ∩ M(Ω)
I
typical solutions: line measures in 2d and surface measures in 3d
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
19
Faculty of Mathematics
Technische Universität München
Improved regularity
I
examples with q̄ = δx0 (ud behaves like a Green's function)
I
numerical observation: if Ωo = Ωc = Ω
ud ∈ L∞ (Ω)
I
proof?
I
consequences:
I
⇒
ū ∈ L∞ (Ω)
I
excludes Diracs
I
ū ∈ H10 (Ω) ∩ L∞ (Ω)
I
q̄ ∈ H−1 (Ω) ∩ M(Ω)
I
typical solutions: line measures in 2d and surface measures in 3d
for Ω0 ∩ Ωc = ∅ typical solution is a lin. combination of Diracs
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
19
Faculty of Mathematics
Technische Universität München
Improved regularity
Theorem (K. Pieper and B.V., SICON 2013)
Let Ωo = Ωc = Ω and ud ∈ L∞ (Ω). Then there holds
(a) ū ∈ L∞ (Ω) with
kūkL∞ (Ω) ≤ kud kL∞ (Ω)
(b) ū ∈ H10 (Ω) ∩ L∞ (Ω)
(c) q̄ ∈ H−1 (Ω) ∩ M(Ω)
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
20
Faculty of Mathematics
Technische Universität München
Improved regularity
Theorem (K. Pieper and B.V., SICON 2013)
Let Ωo = Ωc = Ω and ud ∈ L∞ (Ω). Then there holds
(a) ū ∈ L∞ (Ω) with
kūkL∞ (Ω) ≤ kud kL∞ (Ω)
(b) ū ∈ H10 (Ω) ∩ L∞ (Ω)
(c) q̄ ∈ H−1 (Ω) ∩ M(Ω)
I
is not true for Ωc ∩ Ωo = ∅
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
20
Faculty of Mathematics
Technische Universität München
Improved regularity
Theorem (K. Pieper and B.V., SICON 2013)
Let Ωo = Ωc = Ω and ud ∈ L∞ (Ω). Then there holds
(a) ū ∈ L∞ (Ω) with
kūkL∞ (Ω) ≤ kud kL∞ (Ω)
(b) ū ∈ H10 (Ω) ∩ L∞ (Ω)
(c) q̄ ∈ H−1 (Ω) ∩ M(Ω)
I
I
is not true for Ωc ∩ Ωo = ∅
Proof uses
I
I
I
potential theory
maximum principle (with measures as rhs)
optimality system
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
20
Faculty of Mathematics
Technische Universität München
Outline
1. Motivation
2. Functional analytic setting
3. Optimality system and regularity
4. Discretization and error estimates
5. Parabolic case: Directional Sparsity
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
21
Faculty of Mathematics
Technische Universität München
FEM-Discretization and error estimates
I
I
Discretization concept following Casas/Clason/Kunisch
[CCK2011]
E. Casas, C. Clason and K. Kunisch
Approximation of elliptic control problems in measure spaces with sparse solutions
SIAM J. Control Optim. 50 (2012), pp. 1735-1752
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
22
Faculty of Mathematics
Technische Universität München
FEM-Discretization and error estimates
I
I
I
Discretization concept following Casas/Clason/Kunisch
[CCK2011]
error estimates from [CCK2012]
d
kū − ūh kL2 (Ω) = O h1− 4
E. Casas, C. Clason and K. Kunisch
Approximation of elliptic control problems in measure spaces with sparse solutions
SIAM J. Control Optim. 50 (2012), pp. 1735-1752
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
22
Faculty of Mathematics
Technische Universität München
FEM-Discretization and error estimates
I
I
I
Discretization concept following Casas/Clason/Kunisch
[CCK2011]
error estimates from [CCK2012]
d
kū − ūh kL2 (Ω) = O h1− 4
estimate in [PV2013] in the general case
d
kū − ūh kL2 (Ω) = O h2− 2 |ln h|
I
E. Casas, C. Clason and K. Kunisch
Approximation of elliptic control problems in measure spaces with sparse solutions
SIAM J. Control Optim. 50 (2012), pp. 1735-1752
I
K. Pieper and B. Vexler
A priori error analysis for discretization of sparse elliptic optimal control problems in measure spase
SIAM J. Control Optim. 51, pp. 2788-2808, 2013
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
22
Faculty of Mathematics
Technische Universität München
FEM-Discretization and error estimates
I
I
Discretization concept following Casas/Clason/Kunisch
[CCK2011]
error estimates from [CCK2012]
d
kū − ūh kL2 (Ω) = O h1− 4
I
estimate in [PV2013] in the general case
d
kū − ūh kL2 (Ω) = O h2− 2 |ln h|
I
estimate in [PV2013] for ud ∈ L∞ (Ω) and Ωo = Ωc = Ω
(improved regularity)
kū − ūh kL2 (Ω) = O (h |ln h|)
I
E. Casas, C. Clason and K. Kunisch
Approximation of elliptic control problems in measure spaces with sparse solutions
SIAM J. Control Optim. 50 (2012), pp. 1735-1752
I
K. Pieper and B. Vexler
A priori error analysis for discretization of sparse elliptic optimal control problems in measure spase
SIAM J. Control Optim. 51, pp. 2788-2808, 2013
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
22
Faculty of Mathematics
Technische Universität München
Discretization concept
I
discretize the state variable with linear nite elements,
( rst) don't discretize the control
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
23
Faculty of Mathematics
Technische Universität München
Discretization concept
I
discretize the state variable with linear nite elements,
( rst) don't discretize the control
Discretized optimal control problem
Minimize J(qh , uh ) =
1
ku − ud k2L2 (Ω) + αkqh kM(Ω) ,
2 h
qh ∈ M(Ω),
subject to
uh ∈ Vh : (∇uh , ∇ϕh ) = hqh , ϕh i
Boris Vexler
Sparse Control Problems in Measure Spaces
∀ϕh ∈ Vh
June 2014
23
Faculty of Mathematics
Technische Universität München
Discretization concept
I
discretize the state variable with linear nite elements,
( rst) don't discretize the control
Discretized optimal control problem
Minimize J(qh , uh ) =
1
ku − ud k2L2 (Ω) + αkqh kM(Ω) ,
2 h
qh ∈ M(Ω),
subject to
uh ∈ Vh : (∇uh , ∇ϕh ) = hqh , ϕh i
I
∀ϕh ∈ Vh
existence as on the continuous level, ūh is unique
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
23
Faculty of Mathematics
Technische Universität München
Discretization concept
I
discretize the state variable with linear nite elements,
( rst) don't discretize the control
Discretized optimal control problem
Minimize J(qh , uh ) =
1
ku − ud k2L2 (Ω) + αkqh kM(Ω) ,
2 h
qh ∈ M(Ω),
subject to
uh ∈ Vh : (∇uh , ∇ϕh ) = hqh , ϕh i
I
I
∀ϕh ∈ Vh
existence as on the continuous level, ūh is unique
discrete solution operator Sh : q 7→ uh (q) is not injective
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
23
Faculty of Mathematics
Technische Universität München
Discretization concept
I
discretize the state variable with linear nite elements,
( rst) don't discretize the control
Discretized optimal control problem
Minimize J(qh , uh ) =
1
ku − ud k2L2 (Ω) + αkqh kM(Ω) ,
2 h
qh ∈ M(Ω),
subject to
uh ∈ Vh : (∇uh , ∇ϕh ) = hqh , ϕh i
I
I
I
∀ϕh ∈ Vh
existence as on the continuous level, ūh is unique
discrete solution operator Sh : q 7→ uh (q) is not injective
discrete optimal control is not unique!
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
23
Faculty of Mathematics
Technische Universität München
Discretization concept
I
with interior nodes {xi } of the mesh Th and basis functions ϕi ∈ Vh
X
Mh = { qh ∈ M(Ω) | qh =
βi δxi , βi ∈ R } ⊂ M(Ω)
i
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
24
Faculty of Mathematics
Technische Universität München
Discretization concept
I
with interior nodes {xi } of the mesh Th and basis functions ϕi ∈ Vh
X
Mh = { qh ∈ M(Ω) | qh =
βi δxi , βi ∈ R } ⊂ M(Ω)
i
I
Λh : M(Ω) → Mh
Λh q =
X
hq, ϕi iδxi
i
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
24
Faculty of Mathematics
Technische Universität München
Discretization concept
I
with interior nodes {xi } of the mesh Th and basis functions ϕi ∈ Vh
X
Mh = { qh ∈ M(Ω) | qh =
βi δxi , βi ∈ R } ⊂ M(Ω)
i
I
Λh : M(Ω) → Mh
Λh q =
X
hq, ϕi iδxi
i
Theorem (Casas/Clason/Kunisch)
There exists a unique solution q̄h ∈ Mh . For any other solution
q̃h ∈ M(Ω) there holds Λh q̃h = q̄h . Moreover
∗
q̄h * q̄ in M(Ω)
Boris Vexler
and kq̄h kM(Ω) → kq̄kM(Ω)
Sparse Control Problems in Measure Spaces
June 2014
24
Faculty of Mathematics
Technische Universität München
Discrete optimality system
State equation for ūh ∈ Vh
ūh ∈ Vh : (∇ūh , ∇ϕh ) = hq̄h , ϕh i
∀ϕh ∈ Vh
Adjoint equation for z̄h ∈ Vh
z̄h ∈ Vh : (∇ϕh , ∇z̄h ) = (ūh − ud , ϕh ) ∀ϕh ∈ Vh
Optimality condition for q̄h ∈ Mh
−hq − q̄h , z̄h i + αkq̄h kM(Ω) ≤ αkqkM(Ω)
Boris Vexler
Sparse Control Problems in Measure Spaces
∀q ∈ M(Ω)
June 2014
25
Faculty of Mathematics
Technische Universität München
Error estimates for the state equation
I
estimate in L2 (Ω)-norm
d
ku(q) − uh (q)kL2 (Ω) ≤ ch2− 2 kqkM(Ω)
→ does not lead to optimal estimates
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
26
Faculty of Mathematics
Technische Universität München
Error estimates for the state equation
I
estimate in L2 (Ω)-norm
d
ku(q) − uh (q)kL2 (Ω) ≤ ch2− 2 kqkM(Ω)
→ does not lead to optimal estimates
I
estimate in Lp (Ω)-norm, 1 < p <
ku(q) − uh (q)kLp (Ω) ≤ cp h
Boris Vexler
d
d−2
2− pd0
kqkM(Ω) ,
Sparse Control Problems in Measure Spaces
1
1
+ 0 =1
p p
June 2014
26
Faculty of Mathematics
Technische Universität München
Error estimates for the state equation
I
estimate in L2 (Ω)-norm
d
ku(q) − uh (q)kL2 (Ω) ≤ ch2− 2 kqkM(Ω)
→ does not lead to optimal estimates
I
estimate in Lp (Ω)-norm, 1 < p <
ku(q) − uh (q)kLp (Ω) ≤ cp h
I
d
d−2
2− pd0
kqkM(Ω) ,
1
1
+ 0 =1
p p
estimate in L1 (Ω)-norm
ku(q) − uh (q)kL1 (Ω) ≤ ch2 |ln h|2 kqkM(Ω)
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
26
Faculty of Mathematics
Technische Universität München
Error estimates for the state equation
I
estimate in L2 (Ω)-norm
d
ku(q) − uh (q)kL2 (Ω) ≤ ch2− 2 kqkM(Ω)
→ does not lead to optimal estimates
I
estimate in Lp (Ω)-norm, 1 < p <
ku(q) − uh (q)kLp (Ω) ≤ cp h
I
d
d−2
2− pd0
kqkM(Ω) ,
1
1
+ 0 =1
p p
estimate in L1 (Ω)-norm
ku(q) − uh (q)kL1 (Ω) ≤ ch2 |ln h|2 kqkM(Ω)
I
proof uses a duality argument and appropriate L∞ -estimates
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
26
Faculty of Mathematics
Technische Universität München
Error estimates for the optimal control problem
I
general assumption on ud
ud ∈ L∞ (Ω) in 2d and ud ∈ L3 (Ω) in 3d
I
general regularity: q̄ ∈ M(Ω) and ū ∈ W01,s (Ω)
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
27
Faculty of Mathematics
Technische Universität München
Error estimates for the optimal control problem
I
general assumption on ud
ud ∈ L∞ (Ω) in 2d and ud ∈ L3 (Ω) in 3d
I
general regularity: q̄ ∈ M(Ω) and ū ∈ W01,s (Ω)
Theorem (K. Pieper and B.V.)
Let (q̄, ū) be the optimal solution and (q̄h , ūh ) ∈ Mh × Vh the discrete
optimal solution. There holds
d
kū − ūh kL2 (Ω) ≤ ch2− 2 |ln h|
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
27
Faculty of Mathematics
Technische Universität München
Error estimates for the optimal control problem
I
general assumption on ud
ud ∈ L∞ (Ω) in 2d and ud ∈ L3 (Ω) in 3d
I
general regularity: q̄ ∈ M(Ω) and ū ∈ W01,s (Ω)
Theorem (K. Pieper and B.V.)
Let (q̄, ū) be the optimal solution and (q̄h , ūh ) ∈ Mh × Vh the discrete
optimal solution. There holds
d
kū − ūh kL2 (Ω) ≤ ch2− 2 |ln h|
I
Corollary:
d
kq̄ − q̄h kH−2 (Ω) ≤ ch2− 2 |ln h|
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
27
Faculty of Mathematics
Technische Universität München
Error estimates for the optimal control problem
I
general assumption on ud
ud ∈ L∞ (Ω) in 2d and ud ∈ L3 (Ω) in 3d
I
general regularity: q̄ ∈ M(Ω) and ū ∈ W01,s (Ω)
Theorem (K. Pieper and B.V.)
Let (q̄, ū) be the optimal solution and (q̄h , ūh ) ∈ Mh × Vh the discrete
optimal solution. There holds
d
kū − ūh kL2 (Ω) ≤ ch2− 2 |ln h|
I
Corollary:
d
kq̄ − q̄h kH−2 (Ω) ≤ ch2− 2 |ln h|
I
in general one can not expect kq̄ − q̄h kM(Ω) → 0.
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
27
Faculty of Mathematics
Technische Universität München
Error estimate for improved regularity
I
assume ud ∈ L∞ (Ω) and Ωo = Ωc = Ω
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
28
Faculty of Mathematics
Technische Universität München
Error estimate for improved regularity
I
assume ud ∈ L∞ (Ω) and Ωo = Ωc = Ω
I
improved regularity:
q̄ ∈ M(Ω) ∩ H−1 (Ω)
Boris Vexler
and ū ∈ L∞ (Ω) ∩ H10 (Ω)
Sparse Control Problems in Measure Spaces
June 2014
28
Faculty of Mathematics
Technische Universität München
Error estimate for improved regularity
I
assume ud ∈ L∞ (Ω) and Ωo = Ωc = Ω
I
improved regularity:
q̄ ∈ M(Ω) ∩ H−1 (Ω)
and ū ∈ L∞ (Ω) ∩ H10 (Ω)
Theorem (K. Pieper and B.V.)
Let (q̄, ū) be the optimal solution and (q̄h , ūh ) ∈ Mh × Vh the discrete
optimal solution. There holds
kū − ūh kL2 (Ω) ≤ ch|ln h|
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
28
Faculty of Mathematics
Technische Universität München
Error estimate for improved regularity
I
assume ud ∈ L∞ (Ω) and Ωo = Ωc = Ω
I
improved regularity:
q̄ ∈ M(Ω) ∩ H−1 (Ω)
and ū ∈ L∞ (Ω) ∩ H10 (Ω)
Theorem (K. Pieper and B.V.)
Let (q̄, ū) be the optimal solution and (q̄h , ūh ) ∈ Mh × Vh the discrete
optimal solution. There holds
kū − ūh kL2 (Ω) ≤ ch|ln h|
I
proof uses optimality systems, z̄ ∈ W 2,p (Ω), p < ∞
I
applies L∞ error estimates for z̄ and L2 estimates for ū
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
28
Faculty of Mathematics
Technische Universität München
Numerical results: Dirac example (I)
I
construct a radially symmetric example on Ω = B1 (0)
q̄ = δ0 ,
I
ū = (−∆)−1 δ0 ,
z̄ with z̄(0) = −α
set ud = ū + ∆z̄
(d) State 2d
Boris Vexler
(e) Adjoint 2d
Sparse Control Problems in Measure Spaces
(f) Control 2d
June 2014
29
Faculty of Mathematics
Technische Universität München
Numerical results: Dirac example (II)
|J(q̄, ū) − J(q̄h , ūh )|
kū − ūh kL2 (Ω)
2
ε
2 kq̄h kL2
100
10−1
h
−2
|J(q̄, ū) − J(q̄h , ūh )|
kū − ūh kL2 (Ω)
2
ε
2 kq̄h kL2
100
10
10
−4
h
−2
10
10−3
10−6
10−4
10−5
10−8
10−6
1
2
3
4
5
6
7
8
9
1
(g) Errors in 2d
I
2
3
d
Sparse Control Problems in Measure Spaces
5
(h) Errors in 3d
numerical results match a priori rates O h2− 2
Boris Vexler
4
June 2014
30
Faculty of Mathematics
Technische Universität München
Numerical example: improved regularity (I)
I
take a bounded ud as below (in 2d, α = 0.001).
(i) Desired state ud
I
(j) Optimal state ūh
(k) Optimal control q̄h
the optimal state ū is bounded, optimal control q̄ is a line measure
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
31
Faculty of Mathematics
Technische Universität München
Numerical example: improved regularity (II)
100
|J(q̄, ū) − J(q̄h , ūh )|
kū − ūh kL2 (Ω)
2
ε
2 kq̄h kL2
10−2
|J(q̄, ū) − J(q̄h , ūh )|
kū − ūh kL2 (Ω)
2
ε
2 kq̄h kL2
10−1
10−2
h
10
10−4
h
−3
10−4
−6
10
10−5
10−8
10−6
10−10
10−7
1
2
3
4
5
6
7
8
9
1
2
(l) Errors in 2d
I
3
4
5
6
(m) Errors in 3d
same order of convergence O(h) in 2d and 3d
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
32
Faculty of Mathematics
Technische Universität München
Outline
1. Motivation
2. Functional analytic setting
3. Optimality system and regularity
4. Discretization and error estimates
5. Parabolic case: Directional Sparsity
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
33
Faculty of Mathematics
Technische Universität München
Functional analytic setting: Choice of the control space
Optimal control problem
Minimize J(q, u) =
1
ku − ud k2L2 (I×Ω) + αkqkX ,
2
∂t u − ∆u = q in I × Ω,
Boris Vexler
q ∈ X, u ∈ V
u(0) = u0 in Ω, b.c.
Sparse Control Problems in Measure Spaces
June 2014
34
Faculty of Mathematics
Technische Universität München
Functional analytic setting: Choice of the control space
Optimal control problem
Minimize J(q, u) =
1
ku − ud k2L2 (I×Ω) + αkqkX ,
2
∂t u − ∆u = q in I × Ω,
I
q ∈ X, u ∈ V
u(0) = u0 in Ω, b.c.
X = M(I × Ω) → sparsity in space and time, u 6∈ L2 (I × Ω)!
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
34
Faculty of Mathematics
Technische Universität München
Functional analytic setting: Choice of the control space
Optimal control problem
Minimize J(q, u) =
1
ku − ud k2L2 (I×Ω) + αkqkX ,
2
∂t u − ∆u = q in I × Ω,
q ∈ X, u ∈ V
u(0) = u0 in Ω, b.c.
I
X = M(I × Ω) → sparsity in space and time, u 6∈ L2 (I × Ω)!
I
X = L2 (I; M(Ω)) → sparsity in space, e.g. moving
sources [CCK13]
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
34
Faculty of Mathematics
Technische Universität München
Functional analytic setting: Choice of the control space
Optimal control problem
Minimize J(q, u) =
1
ku − ud k2L2 (I×Ω) + αkqkX ,
2
∂t u − ∆u = q in I × Ω,
q ∈ X, u ∈ V
u(0) = u0 in Ω, b.c.
I
X = M(I × Ω) → sparsity in space and time, u 6∈ L2 (I × Ω)!
I
X = L2 (I; M(Ω)) → sparsity in space, e.g. moving
sources [CCK13]
I
X = M(Ω; L2 (I)) → (directional) sparsity in space [PKV14]
→ sparsity pattern is constant in time!
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
34
Faculty of Mathematics
Technische Universität München
Control space
I
q ∈ X = M(Ωc ; L2 (I)) is a countably additive vector measure of
bounded total variation; q : B(Ωc ) → L2 (I)
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
35
Faculty of Mathematics
Technische Universität München
Control space
I
I
q ∈ X = M(Ωc ; L2 (I)) is a countably additive vector measure of
bounded total variation; q : B(Ωc ) → L2 (I)
total variation of q, |q| ∈ M(Ωc )
(∞
)
X
|q|(B) = sup
kq(Bn )kL2 (I) : {Bn } ⊂ B(Ωc ) disj. partition of B
n=1
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
35
Faculty of Mathematics
Technische Universität München
Control space
I
I
q ∈ X = M(Ωc ; L2 (I)) is a countably additive vector measure of
bounded total variation; q : B(Ωc ) → L2 (I)
total variation of q, |q| ∈ M(Ωc )
(∞
)
X
|q|(B) = sup
kq(Bn )kL2 (I) : {Bn } ⊂ B(Ωc ) disj. partition of B
n=1
I
norm on X
kqkM(Ωc ;L2 (I)) = k|q|kM(Ωc ) = |q|(Ωc )
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
35
Faculty of Mathematics
Technische Universität München
Control space
I
I
q ∈ X = M(Ωc ; L2 (I)) is a countably additive vector measure of
bounded total variation; q : B(Ωc ) → L2 (I)
total variation of q, |q| ∈ M(Ωc )
(∞
)
X
|q|(B) = sup
kq(Bn )kL2 (I) : {Bn } ⊂ B(Ωc ) disj. partition of B
n=1
I
norm on X
kqkM(Ωc ;L2 (I)) = k|q|kM(Ωc ) = |q|(Ωc )
I
Radon-Nikodym derivative / polar decomposition dq = q0 d|q|, i.e.
Z
Z
ϕ(x) dq =
ϕ(x)q0 (t, x) d|q| ∈ L2 (I), ∀ϕ ∈ C0 (Ωc ).
Ωc
Ωc
with q0 ∈ L∞ (Ωc , |q|, L2 (I)), and kq0 (x)kL2 (I) = 1 for |q|-a.a. x ∈ Ωc
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
35
Faculty of Mathematics
Technische Universität München
Control space
I
Identify X = M(Ωc ; L2 (I)) with the dual of C(Ωc , L2 (I))
(Singer's representation theorem)
Z
hq, vi =
(q0 (x), v(x))L2 (I) d|q|(x), q ∈ X, v ∈ C(Ωc , L2 (I))
Ωc
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
36
Faculty of Mathematics
Technische Universität München
Control space
I
Identify X = M(Ωc ; L2 (I)) with the dual of C(Ωc , L2 (I))
(Singer's representation theorem)
Z
hq, vi =
(q0 (x), v(x))L2 (I) d|q|(x), q ∈ X, v ∈ C(Ωc , L2 (I))
Ωc
I
dense embeddings
L2 (I; C(Ωc )) ,→ C(Ωc , L2 (I))
and for the dual spaces
M(Ωc ; L2 (I)) ,→ L2 (I; M(Ω))
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
36
Faculty of Mathematics
Technische Universität München
Control space
I
Identify X = M(Ωc ; L2 (I)) with the dual of C(Ωc , L2 (I))
(Singer's representation theorem)
Z
hq, vi =
(q0 (x), v(x))L2 (I) d|q|(x), q ∈ X, v ∈ C(Ωc , L2 (I))
Ωc
I
dense embeddings
L2 (I; C(Ωc )) ,→ C(Ωc , L2 (I))
and for the dual spaces
M(Ωc ; L2 (I)) ,→ L2 (I; M(Ω))
I
``moving Dirac'' example: I = Ω = (0, 1), q(t) = δ{x=t}
q ∈ L2 (I; M(Ω)) but q 6∈ M(Ωc ; L2 (I))
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
36
Faculty of Mathematics
Technische Universität München
State equation
State equation
∂t u − ∆u = q
u=0
u(0) = 0
Boris Vexler
in I × Ω,
on I × ∂Ω,
in Ω.
Sparse Control Problems in Measure Spaces
June 2014
37
Faculty of Mathematics
Technische Universität München
State equation
State equation
∂t u − ∆u = q
u=0
u(0) = 0
in I × Ω,
on I × ∂Ω,
in Ω.
Weak formulation for u ∈ L2 (I × Ω)
(u, −∂t φ − ∆φ) = hq, φi
Boris Vexler
∀φ ∈ L2 (I; H2 (Ω)) ∩ H1 (I; L2 (Ω))
Sparse Control Problems in Measure Spaces
June 2014
37
Faculty of Mathematics
Technische Universität München
State equation
State equation
∂t u − ∆u = q
u=0
u(0) = 0
in I × Ω,
on I × ∂Ω,
in Ω.
Weak formulation for u ∈ L2 (I × Ω)
(u, −∂t φ − ∆φ) = hq, φi
I
∀φ ∈ L2 (I; H2 (Ω)) ∩ H1 (I; L2 (Ω))
For q ∈ M(Ωc ; L2 (I)) there holds
u ∈ L2 (I; W01,s (Ω)) ∩ H1 (I; W −1,s (Ω)),
Boris Vexler
Sparse Control Problems in Measure Spaces
s<
d
d−1
June 2014
37
Faculty of Mathematics
Technische Universität München
State equation
State equation
∂t u − ∆u = q
u=0
u(0) = 0
in I × Ω,
on I × ∂Ω,
in Ω.
Weak formulation for u ∈ L2 (I × Ω)
(u, −∂t φ − ∆φ) = hq, φi
I
∀φ ∈ L2 (I; H2 (Ω)) ∩ H1 (I; L2 (Ω))
For q ∈ M(Ωc ; L2 (I)) there holds
u ∈ L2 (I; W01,s (Ω)) ∩ H1 (I; W −1,s (Ω)),
I
s<
d
d−1
u ∈ L∞ (Ī; Ls (Ω)), u ∈ C(Ī; W −ε,s (Ω))
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
37
Faculty of Mathematics
Technische Universität München
Optimal control problem
Cost functional
Minimize J(q, u) =
1
ku − ud k2L2 (I×Ωo ) + αkqkM(Ωc ;L2 (I))
2
State equation
∂t u − ∆u = q
u=0
u(0) = 0
Boris Vexler
in I × Ω,
on I × ∂Ω,
in Ω.
Sparse Control Problems in Measure Spaces
June 2014
38
Faculty of Mathematics
Technische Universität München
Optimal control problem
Cost functional
Minimize J(q, u) =
1
ku − ud k2L2 (I×Ωo ) + αkqkM(Ωc ;L2 (I))
2
State equation
∂t u − ∆u = q
u=0
u(0) = 0
I
in I × Ω,
on I × ∂Ω,
in Ω.
Existence by (sequentional) Banach-Alaoglu theorem
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
38
Faculty of Mathematics
Technische Universität München
Optimal control problem
Cost functional
Minimize J(q, u) =
1
ku − ud k2L2 (I×Ωo ) + αkqkM(Ωc ;L2 (I))
2
State equation
∂t u − ∆u = q
u=0
u(0) = 0
in I × Ω,
on I × ∂Ω,
in Ω.
I
Existence by (sequentional) Banach-Alaoglu theorem
I
Solution operator S : M(Ωc ; L2 (I)) → L2 (I × Ω) is compact
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
38
Faculty of Mathematics
Technische Universität München
Optimal control problem
Cost functional
Minimize J(q, u) =
1
ku − ud k2L2 (I×Ωo ) + αkqkM(Ωc ;L2 (I))
2
State equation
∂t u − ∆u = q
u=0
u(0) = 0
in I × Ω,
on I × ∂Ω,
in Ω.
I
Existence by (sequentional) Banach-Alaoglu theorem
I
Solution operator S : M(Ωc ; L2 (I)) → L2 (I × Ω) is compact
I
Uniqueness for Ωo = Ωc . For Ωo ∩ Ωc = ∅ in general no
uniqueness.
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
38
Faculty of Mathematics
Technische Universität München
Optimality system
State equation
∂t u − ∆u = q
u=0
u(0) = 0
in I × Ω,
on I × ∂Ω,
in Ω.
Adjoint equation
−∂t z − ∆z = (ū − ud )χΩo
z=0
z(T) = 0
in I × Ω,
on I × ∂Ω,
in Ω.
Optimality condition
−hq − q̄, z̄ χΩc i + αkq̄kM(Ω;L2 (I)) ≤ αkqkM(Ω;L2 (I))
Boris Vexler
Sparse Control Problems in Measure Spaces
∀q ∈ M(Ω; L2 (I))
June 2014
39
Faculty of Mathematics
Technische Universität München
Structure of the optimal control
Theorem
(a) kz̄(·, x)kL2 (I) ≤ α
(b) d q̄ =
∀x ∈ Ωc
q̄0 (t, x) d|q̄|
and
supp |q̄| ⊂ { x ∈ Ωc | kz̄(·, x)kL2 (I) = α }
(c) q̄0 = − α1 χΩc z̄ almost everywhere in L∞ (Ωc , |q|, L2 (I))
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
40
Faculty of Mathematics
Technische Universität München
Structure of the optimal control
Theorem
(a) kz̄(·, x)kL2 (I) ≤ α
(b) d q̄ =
∀x ∈ Ωc
q̄0 (t, x) d|q̄|
and
supp |q̄| ⊂ { x ∈ Ωc | kz̄(·, x)kL2 (I) = α }
(c) q̄0 = − α1 χΩc z̄ almost everywhere in L∞ (Ωc , |q|, L2 (I))
I
q̄ ∈ M(Ω; L2 (I)) ⊂ L2 (I; M(Ω)). We have q̄(t) ∈ M(Ω) and
supp |q̄(t)| ⊂ supp |q̄| ⊂ { x ∈ Ωc | kz̄(·, x)kL2 (I) = α } , almost all t ∈ I
I
sparsity pattern constant in time
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
40
Faculty of Mathematics
Technische Universität München
Structure of the optimal control
Theorem
(a) kz̄(·, x)kL2 (I) ≤ α
(b) d q̄ =
∀x ∈ Ωc
q̄0 (t, x) d|q̄|
and
supp |q̄| ⊂ { x ∈ Ωc | kz̄(·, x)kL2 (I) = α }
(c) q̄0 = − α1 χΩc z̄ almost everywhere in L∞ (Ωc , |q|, L2 (I))
I
q̄ ∈ M(Ω; L2 (I)) ⊂ L2 (I; M(Ω)). We have q̄(t) ∈ M(Ω) and
supp |q̄(t)| ⊂ supp |q̄| ⊂ { x ∈ Ωc | kz̄(·, x)kL2 (I) = α } , almost all t ∈ I
I
I
sparsity pattern constant in time
additional regularity of q̄ by (c)
q̄ ∈ C(Ī; M(Ω))
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
40
Faculty of Mathematics
Technische Universität München
Finite element discretization
I
Discretization concept
I
I
state variable: dG(0) in time (implicit Euler), P1 in space
control variable: dG(0) in time, Mh in space
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
41
Faculty of Mathematics
Technische Universität München
Finite element discretization
I
Discretization concept
I
I
state variable: dG(0) in time (implicit Euler), P1 in space
control variable: dG(0) in time, Mh in space
Theorem (K. Pieper and B.V.)
Let d = 2, let (q̄, ū) be the optimal solution and
(q̄h , ūh ) ∈ X0 (Mh ) × Vkh the discrete optimal solution. There holds
1
kū − ūh kL2 (I×Ω) ≤ c k 2 + h |ln h|2
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
41
Faculty of Mathematics
Technische Universität München
Finite element discretization
I
Discretization concept
I
I
state variable: dG(0) in time (implicit Euler), P1 in space
control variable: dG(0) in time, Mh in space
Theorem (K. Pieper and B.V.)
Let d = 2, let (q̄, ū) be the optimal solution and
(q̄h , ūh ) ∈ X0 (Mh ) × Vkh the discrete optimal solution. There holds
1
kū − ūh kL2 (I×Ω) ≤ c k 2 + h |ln h|2
I
I
based on an estimate w.r.t L∞ (Ω; L2 (I)) for the adjoint state
D. Leykekhman and B. Vexler
Optimal a Priori Error Estimates of Parabolic Optimal Control Problems with Pointwise Control
SIAM J. Num. Analysis 51, pp. 2797-2821, 2013
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
41
Faculty of Mathematics
Technische Universität München
Finite element discretization
I
Discretization concept
I
I
state variable: dG(0) in time (implicit Euler), P1 in space
control variable: dG(0) in time, Mh in space
Theorem (K. Pieper and B.V.)
Let d = 2, let (q̄, ū) be the optimal solution and
(q̄h , ūh ) ∈ X0 (Mh ) × Vkh the discrete optimal solution. There holds
1
kū − ūh kL2 (I×Ω) ≤ c k 2 + h |ln h|2
I
I
based on an estimate w.r.t L∞ (Ω; L2 (I)) for the adjoint state
I
k is independent on h
D. Leykekhman and B. Vexler
Optimal a Priori Error Estimates of Parabolic Optimal Control Problems with Pointwise Control
SIAM J. Num. Analysis 51, pp. 2797-2821, 2013
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
41
Faculty of Mathematics
Technische Universität München
Regularization, Optimization, Adaptivity
I
Regularization
jγ (q) = j(q) +
Boris Vexler
γ
kqk2L2
2
Sparse Control Problems in Measure Spaces
June 2014
42
Faculty of Mathematics
Technische Universität München
Regularization, Optimization, Adaptivity
I
Regularization
jγ (q) = j(q) +
I
γ
kqk2L2
2
Error analysis
kū − ūγ k ≤ cγ β
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
42
Faculty of Mathematics
Technische Universität München
Regularization, Optimization, Adaptivity
I
Regularization
jγ (q) = j(q) +
I
γ
kqk2L2
2
Error analysis
kū − ūγ k ≤ cγ β
I
Semi-smooth Newton method for xed γ > 0
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
42
Faculty of Mathematics
Technische Universität München
Regularization, Optimization, Adaptivity
I
Regularization
jγ (q) = j(q) +
I
γ
kqk2L2
2
Error analysis
kū − ūγ k ≤ cγ β
I
Semi-smooth Newton method for xed γ > 0
I
Adaptivity and path following γ → 0
I
a posteriori error estimate for the regularization error
I
a posteriori error estimate for the discretization error
I
adaptive algorithm for path following and mesh re nement
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
42
Faculty of Mathematics
Technische Universität München
Extensions to the wave equation by young scientists
Boris Vexler
Sparse Control Problems in Measure Spaces
June 2014
43