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Variational discretization of PDE constrained optimal control problems with measure controls Evelyn Herberg joint work with Michael Hinze and Henrik Schumacher Workshop: New trends in PDE constrained optimization October 14, 2019 - RICAM Linz,

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Variational discretization of PDE constrained

  • ptimal control problems with measure controls

Evelyn Herberg joint work with Michael Hinze and Henrik Schumacher Workshop: New trends in PDE constrained optimization October 14, 2019 - RICAM Linz, Austria

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Motivation

Goal: preserve sparsity

  • n the discrete level

Herberg Variational discretization October 14, 2019 1

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Problem formulation

Parabolic control problem (from [2])

min

(u0,u)∈M( ¯ Ωc)×M( ¯ Qc)

J(u0, u) = 1 q y − ydq

Lq(Q)+α uM( ¯ Qc)+β u0M( ¯ Ωc) ,

(P) where y solves the state equation:

      

∂ty − △y = u in Q = Ω × (0, T) y(x, 0) = u0 in Ω ⊂ Rd y(x, t) = 0

  • n Σ = Γ × (0, T)

Theorem ([2, Theorem 2.7]) Problem (P) has at least one solution (¯ u, ¯ u0) ∈ M(¯ Qc) × M( ¯ Ωc) for every 1 ≤ q < min

  • 2, d+2

d

  • . If q > 1 the solution is unique.

———————————————————————————————

[2] Casas, E., & Kunisch, K. (2016)

Herberg Variational discretization October 14, 2019 2

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Problem formulation

Optimality conditions

Theorem ([2, Theorem 3.1]) Let (¯ u, ¯ u0) denote a solution to (P) with associated state ¯

  • y. Then there

exists an element w ∈ L2(0, T; H1

0(Ω)) ∩ C(¯

Q) satisfying

      

−∂tw − △w = ¯ g in Q, w(x, T) = 0 in Ω, w(x, t) = 0

  • n Σ,

where ¯ g

  

= |¯ y − yd|q−2 (¯ y − yd) , if 1 < q < min

  • 2, d+2

d

  • ,

∈ sign(¯ y − yd) , if q = 1. ———————————————————————————————

[2] Casas, E., & Kunisch, K. (2016)

Herberg Variational discretization October 14, 2019 3

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Problem formulation

Optimality conditions

Theorem (continued) w also fulfills:

      

  • ¯

Qc w d¯

u + α ¯ uM( ¯

Qc) = 0,

wC( ¯

Qc)

  • = α

, if ¯ u = 0, ≤ α , if ¯ u = 0

      

  • ¯

Ωc w(0)d¯

u0 + α ¯ u0M( ¯

Ωc) = 0,

w(0)C( ¯

Ωc)

  • = β

, if ¯ u0 = 0, ≤ β , if ¯ u0 = 0 Furthermore w is unique if q > 1. ———————————————————————————————

[2] Casas, E., & Kunisch, K. (2016)

Herberg Variational discretization October 14, 2019 4

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Problem formulation

Sparsity structure

Remark ([2, Corollary 3.2]) Let (¯ u, ¯ u0) denote a solution to (P) with associated state ¯

  • y. Then we

have the following sparsity structure: Supp(¯ u+) ⊂

  • (x, t) ∈ ¯

Qc : w(x, t) = −α

  • Supp(¯

u−) ⊂

  • (x, t) ∈ ¯

Qc : w(x, t) = +α

  • Supp(¯

u+

0 ) ⊂

  • x ∈ ¯

Ωc : w(x, 0) = −β

  • Supp(¯

u−

0 ) ⊂

  • x ∈ ¯

Ωc : w(x, 0) = +β

  • where ¯

u = ¯ u+ − ¯ u− and ¯ u0 = ¯ u+

0 − ¯

u−

0 are the Jordan decompositions.

———————————————————————————————

[2] Casas, E., & Kunisch, K. (2016)

Herberg Variational discretization October 14, 2019 5

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Fenchel predual (based on [3])

  • W := {w ∈ Lp(Q) : ∂tw, ∂xw, ∂2

xw ∈ Lp(Q), w| ¯ Ω×{T} = w|Γ ×[0,T] = 0}

  • W ֒

→ C(¯ Q), Φ : W → C( ¯ Ωc) × C(¯ Qc) embedding and restriction, 1

q + 1 p = 1,

  • L := −(∂t + ∆) : W → Lp(Q) the adjoint operator of the state equation

The Fenchel predual of (P) is: min

w∈Wq K(w) := 1 pLwp Lp(Q) + Lw, ydLp(Q),Lq(Q) + ℓα,β(Φw),

(P∗) with ℓα,β : C( ¯ Ωc) × C(¯ Qc) → ¯ R ℓα,β(f0, f ) :=

  • 0,

if f C( ¯

Qc) ≤ α and f0C( ¯ Ωc) ≤ β,

∞, else. − → existence and uniqueness (for q > 1) of solutions ———————————————————————————————

[3] Clason, C., & Kunisch, K. (2011)

Herberg Variational discretization October 14, 2019 6

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Variational discretization (based on [6])

min

(u0,u) ∈ M( ¯ Ωc)×M( ¯ Qc)

Jσ(u0, u) := 1

qyσ(u0, u) − ydq Lq(Qh)

(Pσ) + αuM( ¯

Qc) + βu0M( ¯ Ωc)

Petrov-Galerkin method

  • discrete state space:
  • piecewise linear and continuous finite elements in space
  • piecewise constant functions w.r.t. time
  • discrete test space:
  • piecewise linear and continuous finite elements in space
  • piecewise linear and continuous functions w.r.t. time
  • control space is not discretized
  • yields Crank-Nicholson scheme with implicit Euler step[4],[5]

———————————————————————————————

[4] Daniels, N. von, Hinze, M., & Vierling, M. (2015) [5] Goll, C., Rannacher, R., & Wollner, W. (2015) [6] Hinze, M. (2005)

Herberg Variational discretization October 14, 2019 7

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Variational discretization

discrete optimality system delivers: ¯ wσ∞ ≤ α, ¯ w0,h∞ ≤ β, supp(¯ u+) ⊂

  • (x, t) ∈ ¯

Qc : ¯ wσ(x, t) = −α

  • ,

supp(¯ u−) ⊂

  • (x, t) ∈ ¯

Qc : ¯ wσ(x, t) = +α

  • ,

supp(¯ u+

0 ) ⊂

  • x ∈ ¯

Ωc : ¯ w0,h(x) = −β

  • ,

supp(¯ u−

0 ) ⊂

  • x ∈ ¯

Ωc : ¯ w0,h(x) = +β

  • .

Herberg Variational discretization October 14, 2019 8

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Variational discretization

discrete optimality system delivers: ¯ wσ∞ ≤ α, ¯ w0,h∞ ≤ β, supp(¯ u+) ⊂

  • (x, t) ∈ ¯

Qc : ¯ wσ(x, t) = −α

  • ,

supp(¯ u−) ⊂

  • (x, t) ∈ ¯

Qc : ¯ wσ(x, t) = +α

  • ,

supp(¯ u+

0 ) ⊂

  • x ∈ ¯

Ωc : ¯ w0,h(x) = −β

  • ,

supp(¯ u−

0 ) ⊂

  • x ∈ ¯

Ωc : ¯ w0,h(x) = +β

  • .

in the generic setting: supp(¯ u) ⊂ {(xj, tk)} and supp(¯ u0) ⊂ {(xj)}. induced discrete control space: dirac measures in space and time Uh × Uσ

Herberg Variational discretization October 14, 2019 8

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Variational discretization

min

(u0,u) ∈ M( ¯ Ωc)×M( ¯ Qc)

Jσ(u0, u) := 1

qyσ(u0, u) − ydq Lq(Qh)

(Pσ) + αuM( ¯

Qc) + βu0M( ¯ Ωc),

where yσ(u0, u) = L−∗

σ (Φ∗ hΥhu0 + Φ∗ σΥσu). (Υh ⊕ Υσ) : M( ¯ Ωc) × M( ¯ Qc) → Uh × Uσ, (Φh ⊕ Φσ)∗ : Uh × Uσ → W∗

σ,

L−∗

σ

: W∗

σ → Yσ Herberg Variational discretization October 14, 2019 9

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Variational discretization

min

(u0,u) ∈ M( ¯ Ωc)×M( ¯ Qc)

Jσ(u0, u) := 1

qyσ(u0, u) − ydq Lq(Qh)

(Pσ) + αuM( ¯

Qc) + βu0M( ¯ Ωc),

where yσ(u0, u) = L−∗

σ (Φ∗ hΥhu0 + Φ∗ σΥσu). (Υh ⊕ Υσ) : M( ¯ Ωc) × M( ¯ Qc) → Uh × Uσ, (Φh ⊕ Φσ)∗ : Uh × Uσ → W∗

σ,

L−∗

σ

: W∗

σ → Yσ

  • multiple solutions (ˆ

u0, ˆ u) ∈ M( ¯ Ωc) × M(¯ Qc)

  • unique solution (¯

u0,h, ¯ uσ) ∈ Uh × Uσ for q > 1

  • (Υhˆ

u0, Υσˆ u) = (¯ u0,h, ¯ uσ) for all solutions (ˆ u0, ˆ u) ∈ M( ¯ Ωc) × M(¯ Qc) (similar to [1]) ———————————————————————————————

[1] Casas, E., Clason, C., & Kunisch, K. (2013)

Herberg Variational discretization October 14, 2019 9

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Variational discretization - Convergence

Theorem Let (¯ u0,h, ¯ uσ) be the unique solution of (Pσ) belonging to Uh × Uσ, Γ of class C1,1, 1 < q < min{2, d+2

d } and σ = (τ, h) the discretization

parameter consisting of temporal (τ) and spatial (h) gridsizes . If {(¯ u0,h, ¯ uσ)}σ is a sequence of such solutions with associated states {¯ yσ}σ the following convergence properties hold: lim

|σ|→0 ¯

y − ¯ yσLq(Q) = 0, (¯ u0,h, ¯ uσ) ∗ ⇀ (¯ u0, ¯ u) as |σ| → 0 in M( ¯ Ωc) × M(¯ Qc), lim

|σ|→0(¯

u0hM( ¯

Ωc), ¯

uσM( ¯

Qc)) = (¯

u0M( ¯

Ωc), ¯

uM( ¯

Qc)),

where (¯ u0, ¯ u) is the unique solution of (Pσ) and ¯ y associated state. proof similar to the convergence proof in [2] ———————————————————————————————

[2] Casas, E., & Kunisch, K. (2016)

Herberg Variational discretization October 14, 2019 10

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Full Discretization (from [2])

min

(u0h,uσ)∈ Uσ×Uh

JDG(u0h, uσ) := 1

qyσ(u0h, uσ) − ydq Lq(Qh)

(Pσ) + αuσM( ¯

Qc) + βu0hM( ¯ Ωc)

Discontinuous Galerkin method

  • discrete state and test space:
  • piecewise linear and continuous finite elements in space
  • piecewise constant functions w.r.t. time
  • discrete control space:
  • dirac measures concentrated in the finite element nodes in space
  • piecewise constant functions w.r.t. time
  • yields Euler time stepping scheme

− → similar convergence properties ———————————————————————————————

[2] Casas, E., & Kunisch, K. (2016)

Herberg Variational discretization October 14, 2019 11

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Computational Results

Numerical setup

  • Ω = (0, 1) and T = 1, 5
  • Qc = (1

4, 3 4) × (1 4, 5 4)

  • u0 = 0, q = 4

3 and p = 4

min

u∈M( ¯ Qc)

Jσ(u) = 1 q yσ − ydq

Lq(Qh) + α uM( ¯ Qc) ,

(Pσ) where yσ solves the state equation:

      

∂tyσ − △yσ = u in Q = Ω × (0, T) yσ(x, 0) = 0 in Ω ⊂ R yσ(x, t) = 0

  • n Σ = Γ × (0, T)

Herberg Variational discretization October 14, 2019 12

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Computational Results

Source Location

Control (x = 0.5, t = 0.5) Desired state (solution of PDE) Discrete desired state (4 × 12 grid, τ = h

2)

Herberg Variational discretization October 14, 2019 13

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Computational Results

Source Location - Variational Discretization

Desired state yd Optimal control for α = 1

3

Associated control (α = 0) Associated state for α = 1

3

Herberg Variational discretization October 14, 2019 14

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Computational Results

Source Location - Discontinuous Galerkin

Desired state yd Optimal control for α = 1

3

Associated control (α = 0) Associated state for α = 1

3

Herberg Variational discretization October 14, 2019 15

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Computational Results

Source Location

Optimal control ¯ uσ for α = 1

3

(variational discretization) ¯ uσM( ¯

Qc) = 0.6518 and

supp(¯ uσ) = {(0.5, 0.5)} Optimal control ¯ uDG for α = 1

3

(discontinuous Galerkin) ¯ uDGM( ¯

Qc) = 0.6845 and

supp(¯ uDG) = {0.5} × (0.5, 0.625]

Herberg Variational discretization October 14, 2019 16

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Computational Results

Convergence

Fenchel duality : yd = L−∗Φ∗u − |Lw|p−2Lw

control u = δ¯

x,¯ t

interpolated associated state y(u) = L−∗Φ∗u adjoint state −w(¯ x,¯ t) = α, |w| < α else desired state yd (¯ x,¯ t) = (0.5, 0.5), α = 0.25

Herberg Variational discretization October 14, 2019 17

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Computational Results

Convergence - solutions on 4 × 48 grid

Variational discretization Discontinuous Galerkin

Herberg Variational discretization October 14, 2019 18

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Computational Results

Convergence - state convergence

lim

|σ|→0 ¯

y − ¯ yσLq(Q) = 0

10-2 10-1 10-3 10-2 10-1 VD: h^1.0904 DG: h^0.99592

τ = h

2

10-2 10-1 10-2 10-1 VD: h^1.1131 DG: h^1.0989

τ = h2

2

Herberg Variational discretization October 14, 2019 19

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Computational Results

Convergence - convergence of measure norms

lim

|σ|→0 ¯

uσM( ¯

Qc) = ¯

uM( ¯

Qc)

10-2 10-1 10-2 10-1 VD: h^0.78638 DG: h^0.77196

τ = h

2

10-2 10-1 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 VD: h^0.70051 DG: h^0.68172

τ = h2

2

Herberg Variational discretization October 14, 2019 20

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References

[1] Casas, E., Clason, C., & Kunisch, K. (2013). Parabolic control problems in measure spaces with sparse solutions. SIAM Journal on Control and Optimization, 51(1), 28-63. [2] Casas, E., & Kunisch, K. (2016). Parabolic control problems in space-time measure

  • spaces. ESAIM: Control, Optimisation and Calculus of Variations, 22(2), 355-370.

[3] Clason, C., & Kunisch, K. (2011). A duality-based approach to elliptic control problems in non-reflexive Banach spaces. ESAIM: Control, Optimisation and Calculus of Variations, 17(1), 243-266. [4] Daniels, N. von, Hinze, M., & Vierling, M. (2015). Crank–Nicolson Time Stepping and Variational Discretization of Control-Constrained Parabolic Optimal Control Problems. SIAM Journal on Control and Optimization, 53(3), 1182-1198. [5] Goll, C., Rannacher, R., & Wollner, W. (2015). The damped Crank–Nicolson time-marching scheme for the adaptive solution of the Black–Scholes equation. Journal of Computational Finance, 18(4), 1-37. [6] Hinze, M. (2005). A variational discretization concept in control constrained optimization: the linear-quadratic case. Computational Optimization and Applications, 30(1), 45-61.

  • H., E., Hinze, M., & Schumacher, H. (2018). Maximal discrete sparsity in parabolic
  • ptimal control with measures. arXiv: 1804.10549

Herberg Variational discretization October 14, 2019 21