Institute for Mathematics and Scientific Computing University of Graz, Austria PhD program in Mathematics for Technology Catania, May 22, 2007
Motivation Optimal control of evolution problems: min J(y, u) s.t. ẏ(t) = F(y(t), u(t)) for t > 0, y(0) = y, u U Optimization methods: First-order methods: gradient type methods per iteration nonlinear state and linear adjoint equations Second-order methods: SQP or Newton methods per iteration coupled linear state and linear adjoint equations Spatial discretization by FE or FD large-scale problems and feedback-strategies not feasible Model reduction by POD
Outline Suboptimal control of a nonlinear heat equation Suboptimal control of the Navier-Stokes equation POD for parameter estimation
Nonlinear heat equation [Diwoky/V.] Model problem: minj(y, u) = 2 subject to Ω y(t, x) z(x) 2 dx + β 2 T 0 Γ u(t, s) 2 dsdt y t (t, x) = k y(t, x) for (t, x) Q = (0, T) Ω y (t, s) = b(y(t, s)) + u(t, s) n for (t, s) Σ = (0, T) Γ y(0, x) = y (x) for x Ω R 2 Assumptions: T, β, k > 0, z, y C(Ω), b C 2, (R) with b 0
Infinite-dimensional problem Optimization variables: z = (y, u) Z, Z function space Equality constraints: e = (e, e 2 ) e (z), ϕ = T y t (t, x)ϕ(t, x) + k y(t, x) ϕ(t, x)dxdt 0 Ω T 0 Γ ( b(y(t, s)) + u(t, s) ) ϕ(t, s)dsdt e 2 (z) = y(0, ) y Infinite-dimensional optimization in function spaces: min J(z) subject to e(z) = 0 Lagrange function: L(z, p) = J(z) + e(z), p Optimality conditions: L(z, p) =! 0 (Fréchet-derivatives)
First-order optimality conditions y L(y, u, p)! = 0: adjoint equation p t (t, x) = k p(t, x) p n (t, s) = b (y(t, s))p(t, s) p(t, x) = (y(t, x) z(x)) for (t, x) Q = (0, T) Ω for (t, s) Σ = (0, T) Γ for x Ω u L(z, p)! = 0: optimality condition βu = kp on Σ p L(z, p)! = 0: state equation y t (t, x) = k y(t, x) y (t, s) = b(y(t, s)) + u(t, s) n y(0, x) = y (x) for (t, x) Q for (t, s) Σ for x Ω
SQP methods SQP: sequentiel quadratic programming Quadratic programming problem: L(z, p) = J(z) + e(z), p min L(z n, p n ) + L z (z n, p n )δz + 2 L zz(z n, p n )(δz, δz) subject to e(z n ) + e (z n )δz = 0 First-order optimality conditions for (QP n ): KKT system ( ) ( ) ( Lzz (z n, p n ) e (z n ) δz Lz (z e (z n = n, p n ) ) 0 δp e(z n ) ) (QP n ) Convergence: locally quadratic rate in (z n, p n ) (infinite-dimensional) Globalization: modification of the Hessian and line-search methods Alternative: trust-region methods
POD model reduction Goal: POD Galerkin ansatz using l POD basis functions Snapshot POD: solve of heat equation for 0 t <... < t n T Problems: unknown optimal control good snapshot set? u = k βp depends on p POD approximation for p? Strategy: iterate basis computation and include adjoint information in the snapshot ensemble
Dynamic POD strategy [Hinze et al./sachs et al.] () Choose estimate u 0 ; compute snapshots by solving state equation with u = u 0 and adjoint equation with y = y(u 0 ); i := 0 (2) Determine l POD basis functions and associated ROM of infinite-dimensional optimization problem (3) Compute solution u i+ of optimization problem (e.g., by SQP) (4) If Ψ(i) = ui+ u i u i+ TOL then stop (stopping criterium) (5) i := i + ; compute snapshots by solving state equation with control u = u i and adjoint equation with y = y(u i ); go back to (2)
Numerical results Data: y (x, x 2 ) = 0x x 2, z(x, x 2 ) = 2 + 2 2x x 2, b(y) = arctan(y), k = β = 0, T =, 85 FEs Recall: Ψ(i) = ui+ u i u i+ stopping criterium for dynamic POD strategy i relative L 2 error for y relative L 2 error for u J(y, u) Ψ(i) 0 4.4 2.0 0.358.00.0 8. 0.360 0.3 2 0.9 6.8 0.36 0.08 POD opt 0.5 5.7 0.358 FE 0.358 POD Compute snapshots M-flops 8 CPU time in s 3.3 Compute POD basis M-flops 0.44 CPU time in s 0.0 Solve with SQP M-flops 84 CPU time in s 22 total M-flops.0 0 2.9 0 5 FE CPU time in s 2.5 0 6.6 0 3
Suboptimal control PDE: y t = y in (0, T) Ω y n = 0 on (0, ) Γ y n = u(t)q on (0, ) Γ 2 y(0) = 0 on Ω R 3 Boundary condition at z = 0.5: ( ) 2 q(t, x) = e x 0.7 cos(2πt) ( ) 2 e y 0.7 sin(2πt) Cost functional: J(y, u) = 2 Ω y(t) 2 dx + σ T 2 0 u(t) 2 dt
POD computation Snapshot ensembles: { V = span {ȳ h (t j )} j, { V 2 = span { p h (t j )} j, V 3 = V V 2 E(l) = l i= λ i 00%: { ȳh (t j) ȳ h (t j ) t { p h (t j) p h (t j ) t } } j } } l E(l) for V E(l) for V 2 E(l) for V 3 l = 45.89 % 70.44 % 48.20 % l = 3 87.65 % 97.4 % 84.39 % l = 7 99.37 % 00.00 % 98.06 % l = 99.78 % 00.00 % 99.82 % l = 5 99.80 % 00.00 % 99.90 % j
Approximation of the dual variable Theoretical POD estimate for the dual: ) p l p L 2 (0,T;V) ( y C l y + p P l p 0 0 p P l p L 2 (0,T;H (Ω)) 0 2 p P l p L 2 (0,T;L 2 (Ω)) 0 2 0 4 0 4 0 6 0 6 0 8 0 0 0 8 {ψ } {ψ 2 } {ψ 3 } 0 0 3 5 7 9 3 5 Number l of POD basis functions {ψ } {ψ 2 } {ψ 3 } 3 5 7 9 3 5 Number l of POD basis functions
Approximation of the control variable 2 Comparison of the optimal controls Comparison of u h u 5.5 0.2 0 0.5 0 0.5 u h u 5 for ψ i u 5 for ψ i 2 u 5 for ψ i 3 0.2 0.4 0.6 0.8 t axis 0.2 u h u 5 for ψ i u h u 5 for ψ i 2 u h u 5 for ψ i 3 0.2 0.4 0.6 0.8 t axis l u h u l for {ψi }l i= u h u l for {ψi 2 }l i= u h u l for {ψi 3 }l i= l = 0.500 0.5437 0.4672 l = 3 0.3792 0.200 0.869 l = 5 0.3506 0.0588 0.20 l = 9 0.303 0.0585 0.0566 l = 3 0.2057 0.0596 0.0555 u h u l for different POD basis {ψ j i }l i= corresponding to the ensembles V j, j =, 2, 3
Control of the Navier-Stokes equation [Hinze] Optimal control problem: min J(y, u) = Z T Z y z 2 dxdt + α Z T Z u 2 dxdt (y,u) 2 0 Ω 2 0 Ω subject to y t + (y )y ν y + p = Bu div y = 0 in Q = (0, T) Ω in Q y(t, ) = y d on (0, T) Γ d ν ηy(t, ) = pη on (0, T) Γ outlet y(0, ) = y in Ω Domain Ω:
Outline Nonlinear heat equation Navier-Stokes Parameter Id. Computation of the POD basis Uncontrolled flow: POD basis functions: ψ, ψ2, ψ3 and ψ4 Vorticity 0 9.04762 8.09524 7.4286 6.9048 5.238 4.2857 3.33333 2.38095.42857 0.4769-0.4769 -.42857-2.38095-3.33333-4.2857-5.238-6.9048-7.4286-8.09524-9.04762-0 Y 0 - -2 2 3 4 5 6 7 2 Vorticity 0 9.04762 8.09524 7.4286 6.9048 5.238 4.2857 3.33333 2.38095.42857 0.4769-0.4769 -.42857-2.38095-3.33333-4.2857-5.238-6.9048-7.4286-8.09524-9.04762-0 Y 2 0 - -2 8 2 3 4 X Vorticity 0 9.04762 8.09524 7.4286 6.9048 5.238 4.2857 3.33333 2.38095.42857 0.4769-0.4769 -.42857-2.38095-3.33333-4.2857-5.238-6.9048-7.4286-8.09524-9.04762-0 Y 0-2 3 4 5 6 7 8 6 7 8 X 2 Vorticity 0 9.04762 8.09524 7.4286 6.9048 5.238 4.2857 3.33333 2.38095.42857 0.4769-0.4769 -.42857-2.38095-3.33333-4.2857-5.238-6.9048-7.4286-8.09524-9.04762-0 Y 2-2 5 X 0 - -2 2 3 4 5 6 7 8 X
projections predictions POD properties POD ansatz: y l = y m + l i= α i(t)ψ i with y m = m m i= y(t i) Projection onto the POD subspace: α i (t) and ( y(t) y m ) Tψi 4 2 amplitude 0 2 4 0 2 4 6 8 time Different number of snapshots: 4.0 2.0 N=4 N=8 N=0 N=6 N=20 amplitude 0.0-2.0-4.0 34.0 36.0 38.0 40.0 42.0 44.0 time
Optimal solution Goal: mean flow field y m Starting value for optimizer: uncontrolled flow, i.e., y for u = 0 Optimal solution: flow field y l
Parameter estimation [Kahlbacher/V.] Model equations: g(x) = x 3 4 u + ( ) u + au = g in Ω = (0, ) (0, ) ( ) 3 u 4 n + 3 2 u = on Γ Data: choose a id 0 and compute (FE) solution u(a id ) to ( ) Reconstruction: estimate a 0 from u d = ( + εδ)u(a id ) Γ with random ε and factor δ = 5% Constrained optimization: min J(a, u) = α u u d 2 ds+κ a 2 s.t. (a, u) solves ( ) and a 0 Relaxation of the inequality: Γ min J λ (a, u) = J(a, u)+ max{ 0, λ + (0 a) } 2 s.t. (a, u) solves ( )
Global convergent optimization method Outer loop: augmented Lagrangian method control of k and λ k Inner loop: globalized SQP algorithm with fixed ( k, λ k ) for { min J k 3 (a, u) s.t. 4 u + ( ) u + au = g in Ω λ k 3 u 4 n + 3 2 u = on Γ Numerical results: α = 5000, κ = 0.0005, a id = 25, l = 7 0 0 Decay of the first eigenvalues 0 2 relative errors: 0 4 u l u(a id ) u(a id ).87 0 5 0 6 0 8 a l a id a id 6 0 3 0 0 2 3 4 5 6 7
Parameter identification Model equations: a(x) is piecewise constant on 4 subdomains of Ω 3 4 u + ( 2 ) u + au = in Ω = (0, ) (0, ) (0, ) 3 u 4 n + 3 2 u = on Γ ( ) Data: choose a id 0 and compute (FE) solution u(a id ) to ( ) Reconstruction: estimate a 0 from u d = ( + εδ)u(a id ) Γ with random ε and factor δ = 5% 0 0 Decay of the first eigenvalues 0 0 2 0 3 0 4 2 3 4 5 6 7
Global convergent optimization method Outer loop: augmented Lagrangian method control of k and λ k Inner loop: globalized SQP algorithm with fixed ( k, λ k ) for min J k λ k (a, u) s.t. 3 4 u + ( 2 ) u + au = in Ω 3 4 u n + 3 2 u = on Γ Numerical results: α = 0000, κ = 0 3 ( 2, 4, 4, 8 )T, a id = (0, 20, 5, 7) T, l = 7 POD FE a a id a id.0 0 2 6. 0 3 u u(a id ) u(a id ) 5.6 0 3.9 0 3 CPU time in s 30.4 504.2
Parameter identification of the diffusion coefficient Model equations: g(x) = x c u + 25u = in Ω = (0, ) (0, ) c u n + 3 2 u = g on Γ ( ) Data: choose c id 0 and compute (FE) solution u(a id ) to ( ) Reconstruction: estimate c 0 from u d = ( + εδ)u(c id ) Γ with random ε and factor δ = 5% Optimal state and decay of the first eigenvalues: FE solution 0 0 Decay of the first eigenvalues 0.25 0 0.2 0.5 0 2 0. 0 3 0.05 0.5 y axis 0 0 0.2 0.4 x axis 0.6 0.8 0 4.5 2 2.5 3 3.5 4 4.5 5
Global convergent optimization method Outer loop: augmented Lagrangian method control of k and λ k Inner loop: globalized SQP algorithm with fixed ( k, λ k ) for min J k λ k (c, u) s.t. { c u + 25u = in Ω c u n + 3 2 u = g on Γ Numerical results: α = 00000, κ = 0.0000, c id = 3, l = 5 POD FE c c id c id 4.7 0 3 4.2 0 3 u u(c id ) u(c id ) 6.74 0 4 6.72 0 4 CPU time in s 30.4 262.2