Regularization of optimal control problems Daniel Wachsmuth (RICAM - - PowerPoint PPT Presentation

Regularization of optimal control problems

Daniel Wachsmuth (RICAM Linz) joint work with Gerd Wachsmuth (TU Chemnitz)

• 1. Control-constrained problems
• 2. Tychonov regularization and regularization error

estimates

• 3. Necessary conditions for convergence rates
• 4. Parameter choice: a discrepancy principle
• 5. Parameter choice & discretization

RICAM special semester Daniel Wachsmuth

Optimization problem

1 Optimization problem: min Su − z2

Y + βuL1(Ω)

subject to u ∈ Uad. Here: U = L2(Ω), Y Hilbert spaces, S : U → Y linear and compact

• perator. Uad ⊂ U convex, closed, and non-empty, β ≥ 0 fixed.

Motivation: Actuator placement problem

RICAM special semester Daniel Wachsmuth

Optimization problem

1 Optimization problem: min Su − z2

Y + βuL1(Ω)

subject to u ∈ Uad. Here: U = L2(Ω), Y Hilbert spaces, S : U → Y linear and compact

• perator. Uad ⊂ U convex, closed, and non-empty, β ≥ 0 fixed.

Motivation: Actuator placement problem Solvability: If z ∈ R(S) or Uad is bounded then problem is solvable. Uniqueness of solutions: If in addition S is injective then solution is unique.

RICAM special semester Daniel Wachsmuth

Ill-posedness, difference to inverse problems

2 Ill-posedness: Solutions are not stable with respect to perturbations, i.e. in case where only zδ ≈ z is available. Problem is difficult to solve numerically. (β > 0 does not help) Difference to inverse problems: Non-attainability: z ∈ S(Uad) The constraint u ∈ Uad is not extra information about solution, but a serious constraint: it may hinder us to reach the goal z.

RICAM special semester Daniel Wachsmuth

Model problem

3 Elliptic PDE: S is the mapping from u → y, where y is the weak solution of −∆y = u in Ω y = 0

• n ∂Ω

Optimal control problem: Minimize 1 2Su − z2

L2(Ω) + βuL1(Ω)

subject to ua ≤ u ≤ ub a.e. on Ω. Here: ua, ub ∈ L2(Ω), ua ≤ 0 ≤ ub, Y = L2(Ω). Standing assumption on S: S : U = L2(Ω) → Y and S∗ : Y ∗ → L∞(Ω) linear and continuous.

RICAM special semester Daniel Wachsmuth

Ill-posedness

4 Recall standing assumption on S: S : L2(Ω) → Y and S∗ : Y ∗ → L∞(Ω) linear and continuous. Theorem: Under this assumptions the range of S is closed if and only if it is finite-dimensional. Proof by closed-range theorem and a result of [Grothendieck ’54].

RICAM special semester Daniel Wachsmuth

• 1. Control-constrained problems
• 2. Tychonov regularization and regularization error

estimates

• 3. Necessary conditions for convergence rates
• 4. Parameter choice: a discrepancy principle
• 5. Parameter choice & discretization

RICAM special semester Daniel Wachsmuth

Regularized problem

5 Noisy data: zδ with z − zδY ≤ δ (prototype for discretization error) Take α > 0. Regularized problem: Minimize min 1 2Su − zδ2

Y + βuL1(Ω) + α

2 u2

L2

subject to ua ≤ u ≤ ub. Unique solution in case α > 0: uδ

α, y δ α := Suδ α.

Solution of original problem denoted by u0, y0 := Su0. Questions: Convergence for (α, δ) → 0 ? Choice α = α(δ)?

RICAM special semester Daniel Wachsmuth

Optimality condition

6 Necessary optimality condition: There exists λδ

α ∈ ∂ · L1(Ω)(uδ α)

such that with pδ

α := S∗(zδ − y δ α) it holds

(αuδ

α − pδ α + βλδ α, u − uδ α) ≥ 0

∀u ∈ Uad. Consequence 1: y0 − y δ

α2 Y + αu0 − uδ α2 L2 ≤ α(u0, u0 − uδ α)L2 + δy0 − y δ αY

Consequence 2: Noise error estimate yα − y δ

αY ≤ δ

uα − uδ

αY ≤ 1

2 δ √α

RICAM special semester Daniel Wachsmuth

Structure of solutions

7 Bang-bang solutions in case β = 0: Control u0 a.e. on the bounds. In general: u0 discontinuous with jumps across {x ∈ Ω : |p0| = β}.

RICAM special semester Daniel Wachsmuth

Source conditions

8 Observed convergence rates: u0 − uαL2 ∼ α1/2 Explanation wanted! Source conditions:

e.g. [Neubauer][Engl, Hanke, Neubauer]

u0 ∈ R(S∗), u0 can be in R((S∗S)ν/2) only for small ν. Source conditions alone cannot explain these convergence rates. Idea: Exploit the structure (discontinuity).

RICAM special semester Daniel Wachsmuth

Regularity condition

9 Assumption: There exist a set I ⊂ Ω, a function w ∈ Y , and positive constants κ, c such that it holds:

• 1. (source condition) I ⊃ {x ∈ Ω : |p0(x)| = β} and

u0 =    Proj[ua,0] (S∗w)

• n {x ∈ Ω : p0(x) = −β}

Proj[0,ub] (S∗w)

• n {x ∈ Ω : p0(x) = +β}
• 2. (structure of active set) A = Ω \ I and for all ǫ > 0

meas

• {x ∈ A : 0 <
• |p0(x)| − β
• < ǫ}
• ≤ c ǫκ.

[Wachsmuth2 2010] RICAM special semester Daniel Wachsmuth

Regularity condition - comments

10 Source condition:

• Gives smoothness of u0 on I.
• Weaker than u0 = ProjUad (S∗w)

[Wachsmuth2 2010]

Structure of active set:

• Allows to control the size of almost-active sets.
• If p0 ∈ C1( ¯

Ω) and ∇p0 = 0 on {x : |p0(x)| = β} then Assumption is fulfilled with κ = 1.

[Deckelnick, Hinze 2010]

References:

• Case I = ∅: [Felgenhauer 2003][Deckelnick, Hinze 2010][Wachsmuth2 2009]
• Case A = ∅: (p0 = 0) with stronger source condition

u0 = ProjUad (S∗w)

[Neubauer 1988][Lorenz, R¨

• sch 2010]

RICAM special semester Daniel Wachsmuth

Regularization error

11 Regularization error estimate: Let the assumption be satisfied. Then we have y0 − yαY ≤ c αd, u0 − uαL2 ≤ c αd− 1

2

with d =       

1 2−κ

if κ ≤ 1, 1 if κ > 1 and A = Ω and w = 0,

κ+1 2

if κ > 1 and A = Ω or w = 0. [For κ = 1 we get u0 − uαL2 = O(α1/2).] Question: What are necessary conditions to obtain these convergence rates?

RICAM special semester Daniel Wachsmuth

Regularization error

12 Optimal a-priori choice: With α ∼ δ1/d we get y0 − y δ

αY ≤ c δ,

u0 − uδ

αL2 ≤ c δ1− 1

2d .

Problem: How to choose α if κ is unknown?

RICAM special semester Daniel Wachsmuth

Regularization error - Lp-case

13 Weaker assumption: Let S∗ : Y → Lp, 2 < p < ∞. Regularization error estimate: Let the assumption be satisfied. Then we have y0 − yαY ≤ c αd, u0 − uαL2 ≤ c αd− 1

2 ,

with d = min 1 + κ 2 − 1 p , d1, d2

• ,

where d1, d2 are given by d1 =    1 if A = Ω +∞ if A = Ω , d2 =   

p+κ 2 p+4 κ−κ p

if p ≤ 4 or κ ≤

2 p p−4,

+∞

• therwise.

Consistency: no rate for p ց 2, rates for p → ∞ coincide with p = ∞.

RICAM special semester Daniel Wachsmuth

• 1. Control-constrained problems
• 2. Tychonov regularization and regularization error

estimates

• 3. Necessary conditions for convergence rates
• 4. Parameter choice: a discrepancy principle
• 5. Parameter choice & discretization

RICAM special semester Daniel Wachsmuth

Convergence rate d = 1

14 Suppose y0 − yαY + p0 − pαL∞ ≤ c α. Difference quotients y0−yα

α

are bounded in Y . With a special test function ˜ u ∈        {u0}

• n {x ∈ Ω : |p0| = β}

[ua, 0]

• n {x ∈ Ω : p0 = −β}

[0, ub]

• n {x ∈ Ω : p0 = β}

it holds (αuα − (pα − p0), ˜ u − uα) ≥ 0. Hence (u0 + S∗ ˙ y0, ˜ u − u0) ≥ 0 for each weak accumulation point ˙ y0 of y0−yα

α

.

RICAM special semester Daniel Wachsmuth

Convergence rate d = 1

15 Due to construction of ˜ u, we have u0 =    Proj[ua,0] (S∗ ˙ y0)

• n {x ∈ Ω : p0(x) = −β}

Proj[0,ub] (S∗ ˙ y0)

• n {x ∈ Ω : p0(x) = +β}

Necessary condition to obtain d ≥ 1: If y0−yαY ≤ c αd, d ≥ 1, then there is ˙ y0 ∈ Y such that u0 =    Proj[ua,0] (S∗ ˙ y0)

• n {x ∈ Ω : p0(x) = −β}

Proj[0,ub] (S∗ ˙ y0)

• n {x ∈ Ω : p0(x) = +β}

In addition, y0 − yαY = o(α) implies ˙ y0 = 0 and u0 = 0 on {x ∈ Ω : |p0(x)| = β}.

• resembles source-condition part of our assumption

see also [Neubauer 1988]

• source condition realized by derivative of α → yα at α = 0.

RICAM special semester Daniel Wachsmuth

Convergence rate d > 1

16 Abbreviation: µ(ǫ) := meas

• {x ∈ A : 0 <
• |p0(x)| − β
• < ǫ}
• Define auxiliary function ˜

uα as solution of min

u∈Uad −(p0, u) + βuL1(Ω) + α

2 u2

L2.

Suppose there exists M, σ > 0 such that −M ≤ ua ≤ −σ ≤ 0 ≤ σ ≤ ub ≤ M. Then σ 2 p µ σ 2 α

• ≤ u0 − ˜

uαp

Lp(Ω) ≤ Mp µ(Mα).

Hence we have the implication u0 − ˜ uα2

L2(Ω) ≤ Cαd−1/2

⇒ µ(ǫ) ≤ ˜ C ǫ2d−1

RICAM special semester Daniel Wachsmuth

Convergence rate d > 1

17 Necessary condition to obtain d > 1: If y0 − yαY ≤ c αd for all α > 0, d > 1, then there are positive constants C, ǫ0 such that meas

• {x ∈ Ω : 0 <
• |p0(x)| − β
• < ǫ}
• ≤ C ǫ2d−1

∀ǫ ∈ (0, ǫ0)

RICAM special semester Daniel Wachsmuth

Convergence rate d > 1

17 Necessary condition to obtain d > 1: If y0 − yαY ≤ c αd for all α > 0, d > 1, then there are positive constants C, ǫ0 such that meas

• {x ∈ Ω : 0 <
• |p0(x)| − β
• < ǫ}
• ≤ C ǫ2d−1

∀ǫ ∈ (0, ǫ0) Summary:

• Convergence rate d ≥ 1 equivalent to our assumption with

choice I = {x ∈ Ω : |p0(x)| = β}.

• Convergence rate d = 1: source-condition part on I is necessary

and sufficient

• Convergence rate d < 1: necessary conditions completely open.

[Neubauer 1988] RICAM special semester Daniel Wachsmuth

• 1. Control-constrained problems
• 2. Tychonov regularization and regularization error

estimates

• 3. Necessary conditions for convergence rates
• 4. Parameter choice: a discrepancy principle
• 5. Parameter choice & discretization

RICAM special semester Daniel Wachsmuth

18 In the sequel: β = 0 (no L1-cost) for simplicity.

RICAM special semester Daniel Wachsmuth

Discrepancy measure

19 Morozov’s discrepancy principle for linear inverse problems: α(δ) := sup{α ≥ 0 : Suδ

α − zδY ≤ τδ}

Problem: What is a discrepancy measure?

• 1. Suδ

α − zδY : does not vanish if (α, δ) → 0.

• 2. Suδ

α − zδ2 Y − Su0 − z2 Y : not computable

Idea: Use the expansion 1 2Su0 − Suδ

α2 Y ≤ (u0 − uδ α, pδ α)L2 + (zδ − z, Suδ α − Su0)Y

. . . and replace u0 by suitable control bound (u0 − uδ

α, pδ α)L2 ≤

• {pδ

α>0}

pδ

α(ub − uδ α) +

• {pδ

α<0}

pδ

α(ua − uδ α) =: Dδ α.

RICAM special semester Daniel Wachsmuth

Discrepancy principle

20 Discrepancy measure: Dδ

α =

• {pδ

α>0}

pδ

α(ub − uδ α) +

• {pδ

α<0}

pδ

α(ua − uδ α)

Parameter choice rule:

[Wachsmuth2, 2011]

α(δ) := sup{α ≥ 0 : Dδ

α ≤ τδ2},

τ > 0. Properties:

• Dδ

α ≥ 0.

• Under regularity assumption: Dδ

α → 0 for α → 0.

• If S∗z = 0, ua ≤ −σ < σ ≤ ub, and δ sufficiently small

then α(δ) < +∞.

RICAM special semester Daniel Wachsmuth

Convergence rates

21 Estimate of Dδ

α: Under the regularity assumption it holds

Dδ

α ≤ c α(pδ α − p0κ L∞ + ακ).

Error estimates for the states and adjoints: y δ

α(δ) − y0Y + yα(δ) − y0Y ≤ c δ,

pδ

α(δ) − p0L∞ + pα(δ) − p0L∞ ≤ c δ.

Lower bound of α(δ): (Take α > α(δ)) δ2−κ ≤ c α(δ), δ2 α(δ) ≤ c δκ Error in controls: If regularity assumption holds then u0 − uδ

α2 L2 ≤ c (1 + δ

κ(1−κ) κ+1 )δκ.

.. resembles optimal a-priori rate δκ/2.

[Wachsmuth2, 2011] RICAM special semester Daniel Wachsmuth

• 1. Control-constrained problems
• 2. Tychonov regularization and regularization error

estimates

• 3. Necessary conditions for convergence rates
• 4. Parameter choice: a discrepancy principle
• 5. Parameter choice & discretization

RICAM special semester Daniel Wachsmuth

Discretization - uniform estimates

22 Inaccuracies: exact data δ = 0, but inexact operators Sh ≈ S Challenge: Discretization introduces not only perturbation of z in the functional but also perturbation of S. Discretization: Mesh size h > 0, discrete operators Sh ≈ S, S∗

h ≈ S∗

with S − ShL(L2,Y ) ≤ c δy(h) S∗ − S∗

hL(Y ∗,L∞) ≤ c δp(h)

Goal: Choose α = α(h) optimally.

RICAM special semester Daniel Wachsmuth

A-priori error estimate

23 A-priori error estimate: If assumption is satisfied with A = Ω (bang-bang controls), then

[Deckelnick, Hinze 2010]

y0 − y0,hY ≤ c

• δy(h) + δp(h)

1 2−κ

• u0 − u0,hL1 ≤ c
• δy(h)κ + δp(h)

κ 2−κ

• Goal: Choose α = α(h) such that

y0 − yα(h),hY ≤ c

• δy(h) + δp(h)

1 2−κ

• u0 − uα(h),hL1 ≤ c
• δy(h)κ + δp(h)

κ 2−κ

• RICAM special semester

Daniel Wachsmuth

Discrepancy measure

24 Discrete error estimate: y0,h − yα,h2

Y ≤ (u0,h − uα,h, pα,h) ≤ Dα,h

with computable Dα,h :=

• {pα,h<0}

(ua − uα,h)pα,h +

• {pα,h>0}

(ub − uα,h)pα,h. Idea: In order to get y0 − yα(h),hY ∼ δy(h) + δp(h)

1 2−κ ,

ensure y0,h − yα,hY ∼ δy(h) + δp(h)

1 2−κ

RICAM special semester Daniel Wachsmuth

Discrepancy principle

25 Parameter choice rule:

[D. Wachsmuth, 2011]

α(δ) := sup

• α ≥ 0 : Dα,h ≤ τ
• δy(h)2 + δp(h)2

, τ > 0. Under our Assumption, we can prove: Convergence rates:

[D. Wachsmuth, 2011]

y0 − yα(h),hY ≤ c

• δy(h) + δp(h)

1 2−κ

• u0 − uα(h),hL1 ≤ c
• δy(h)κ + δp(h)

κ 2−κ

• Regularization with α(h) gives same error rates as optimal a-priori

error estimate

• κ is not needed a-priori
• Regularized problems are easier to solve

RICAM special semester Daniel Wachsmuth

Towards full adaptive algorithm

26 Discretization: was assumed to be uniform Question: What about adaptive discretizations in combination with adaptive regularization? Challenge: Proof of convergence of adaptive algorithm relies convergence of adaptive discretization for fixed α > 0. Open.

RICAM special semester Daniel Wachsmuth

27 Open problems:

• Necessary conditions for convergence rates d ≤ 1.
• Discrepancy principle if the set {p0 = 0} has positive measure?
• Nonlinear problems? Role of second-order sufficient optimality

conditions.

• What if state constraints are present?

Thank you very much!

RICAM special semester Daniel Wachsmuth