Proper Orthogonal Decomposition for Optimal Control Problems with Mixed Control-State Constraints

Transcription

1 Proper Orthogonal Decomposition for Optimal Control Problems with Mixed Control-State Constraints Technische Universität Berlin Martin Gubisch, Stefan Volkwein University of Konstanz March, 3 Martin Gubisch, Stefan Volkwein (University of Konstanz) TU Berlin (3) March, 3 / 5

2 The optimal control problem Problem formulation Problem formulation min (y,u) Y U J(y, u) = y y Q L (,T;H) + κ u U (OCP) subject to ẏ(t), ϕ V,V + y(t), ϕ H = (Bu + f )(t), ϕ V,V y(), ϕ H = y, ϕ H (ϕ V = H (Ω)) (ϕ H = L (Ω)) and εu(t) + (Iy)(t) u b (t) with U = L (, T; R m ) and B : U L (, T; H), I : L (, T; H) U given by (Bu)(t, x) = m ( u i (t)χ i (x), (Iy)(t) = i= supp(χ i) ) y(t, x)dx. Martin Gubisch, Stefan Volkwein (University of Konstanz) TU Berlin (3) March, 3 / 5

3 The optimal control problem Lagrangian functional Lagrange functional Define the Lagrange functional L : Y U L (, T; V) H L (, T; R m ) by L (y, u, p, λ) = J(y, u) + E(y, u), p L (,T;V ),L (,T;V) + y() y, p H + (Iy + εu) u b, λ L (,T;R m ) where (E(y, u)ϕ)(t) = ẏ(t), ϕ V,V + y(t), ϕ H (Bu + f )(t), ϕ V,V and the set of admissible points X ad = {(y, u) Y U E(y, u) = & y() = y & εu + Iy u b }. Martin Gubisch, Stefan Volkwein (University of Konstanz) TU Berlin (3) March, 3 3 / 5

4 The optimal control problem Existence and uniqueness Existence and uniqueness THEOREM. Assume that the closed, convex and bounded set X ad is nonempty. Then there exists a unique solution (ȳ, ū) to (OCP). Further, there is a unique pair of Lagrange multipliers ( p, λ) such that ȳ(t) + ȳ(t), H (Bū + f )(t) = ȳ() y = p(t) + p(t), H + (I λ)(t) + (ȳ yq )(t) = p(t) = (OS) λ(t) χ Ab(ȳ, p) κū(t) (B p)(t) + ε λ(t) = ( ε B p + κ ε Iȳ κ ) ε u b = where the active set A b (ȳ, p) is { } A b (y, p) = t [, T] ε κ (B p + Iy)(t) > u b (t). Martin Gubisch, Stefan Volkwein (University of Konstanz) TU Berlin (3) March, 3 4 / 5

5 Solution methods Primal-dual active set strategy Primal-dual active set strategy (PDASS) Algorithm (Primal-dual active set strategy) Require: Initial state-adjoint state pair (y, p ) Y Y. : Set k = : repeat 3: Calculate active set A b (y k, p k ) 4: Solve the primal-dual system (OS) to get (y k+, p k+ ) 5: Set k = k + 6: until A b (y k, p k ) = A b (y k, p k ) 7: Return ū = u(y k, p k ). Martin Gubisch, Stefan Volkwein (University of Konstanz) TU Berlin (3) March, 3 5 / 5

6 Solution methods Proper orthogonal decomposition Proper orthogonal decomposition (POD) Problem: After elimination of u, λ, the discrete linear system (OS) is still of the dimension N t N x. Idea: For l N x, find an optimal finite element basis (ψ,..., ψ l ) such that the corresponding Galerkin solution y l is preferably close to ȳ: min i=,...,l φ i,φ j V =δ ij ȳ l ȳ, φ i H φ i i= V }{{} =y l. (POD) Realization: Perform an eigenvalue decomposition of a compact, self-adjoint, non-negative operator K : V V which includes the dynamics of the state solution. Define the canonical mappings Y : L (, T; R) V and Y : V L (, T; R)), Yφ = φ, y L (,T;R) V, Y φ = φ, y( ) V L (, T; R) and choose K = YY. Martin Gubisch, Stefan Volkwein (University of Konstanz) TU Berlin (3) March, 3 6 / 5

7 Solution methods Proper orthogonal decomposition Proper orthogonal decomposition (POD) THEOREM. (Continuous version) Let y C(, T; V) an arbitrary state and let (λ i, ψ i ) i N an eigenvalue decomposition of K = K(y) with λ i λ i+ for all i N. Then (ψ i ) i N is a complete orthonormal system in V, the rank-l POD basis ψ l = (ψ,..., ψ l ) is a solution to (POD) and the residual of y l = y(ψ l ) sums up to y y l L (,T;V) = i=l+ λ i. Remark. POD can also be provided in H instead of V. Problem: Good approximation properties of the POD basis are only guaranteed if the POD elements belong to K(ȳ). However, ȳ is not known, of course. Hence, an adaptive strategy is applied to construct an appropriate POD basis. Martin Gubisch, Stefan Volkwein (University of Konstanz) TU Berlin (3) March, 3 7 / 5

8 Solution methods Proper orthogonal decomposition Proper orthogonal decomposition (POD) Discrete POD. Let Y R Nx Nt a discrete approximation of y such as y(t j, x i ) Y ij (FDM) or y(t j, x) Nx Y ij φ i (x) (FEM). i= Further, let U l R Nx l be the matrix of the first l eigenvectors to WYΘY T W R Nx Nx where Θ = diag( t) R Nt Nt provides the discrete L (, T) scalar product and W = diag( x) R Nx Nx (FDM) or W = ( φ i, φ j H ) R Nx Nx (FEM) provides the discrete L (Ω) scalar product. Then a discrete POD basis is given by ψ l = W U l. Remark. If N x N t or W cannot easily be calculated, the eigenvalue decomposition should be provided for the transposed problem ΘY T WY Θ R Nt Nt. Martin Gubisch, Stefan Volkwein (University of Konstanz) TU Berlin (3) March, 3 8 / 5

9 Solution methods Reduced order modeling (ROM) Proper orthogonal decomposition Algorithm (Reduced order modeling) Require: l N x, initial control guess u l U, desired exactness ɛ : repeat : Calculate full-order state solution y = y(u l ) 3: Calculate POD basis ψ = ψ(y) 4: Apply PDASS on the reduced system (OS) to get u l = u(ψ) 5: until Aposti(u l ) < ɛ 6: Return ū = u l Martin Gubisch, Stefan Volkwein (University of Konstanz) TU Berlin (3) March, 3 9 / 5

10 Numerical experiments Numerical example Numerical example desired state y Q & state constraint u b 3.5 yq = yq(t, x) direction x.5 time t 3 Fig.. The desired state y Q L (, T; H) and the upper mixed control-state bound u b, interpreted as a pure pointwise state constraint. Martin Gubisch, Stefan Volkwein (University of Konstanz) TU Berlin (3) March, 3 / 5

11 Numerical experiments Numerical example Numerical example optimal state ȳ.5 y = y(t, x) direction x.5 time t 3 Fig.. The optimal state solution ȳ Y satisfying the state constraints. Martin Gubisch, Stefan Volkwein (University of Konstanz) TU Berlin (3) March, 3 / 5

12 Numerical experiments Numerical example Numerical example optimal control ū 5 u = u(t, x) direction x.5 time t 3 Fig. 3. The optimal control term Bū L (, T; H) controlling the state equation. Martin Gubisch, Stefan Volkwein (University of Konstanz) TU Berlin (3) March, 3 / 5

13 Numerical results Numerical experiments Numerical results 5 Rom error and a posteriori error bound Aposti Rom error SingVals error ū u l U Pod-rank l Fig. 4. The ROM errors of the control u l for different POD basis ranks l, the error bound provided by the a posteriori error estimator and the singular values (squares of the eigenvalues) of the operator YY which can be used as a costless error indicator. Martin Gubisch, Stefan Volkwein (University of Konstanz) TU Berlin (3) March, 3 3 / 5

14 Numerical effort Numerical experiments Numerical effort Numerical effort for the reduced order model 5 Calculate error bound Solve linear system Assemble linear system Solve eigenvalue problem Calculate Snapshots calculation time in seconds 5 5 Process Time # Total Assemble full system.66 sec sec Solve full system.7 sec 9.43 sec Total 6.4 sec Solve full snapshots equations. sec. sec Solve eigenvalue problem.4 sec.84 sec Assemble ROM system.53 sec 7 9. sec Solve ROM system.45 sec sec Evaluate error estimator. sec.3 sec Total 8. sec Tab.. The calculation times for solving the optimization problem with and without model reduction. With 5 POD elements, the ROM problem has to be solved two times; solvings of two eigenvalue problems are required in addition to update the POD basis. Nevertheless, 9.7% of the calculation time is spared in total. Fig. 5. The numerical effort of the single algorithm processes for solving the ROM problem on an Intel(R) Core(TM) i5.4ghz processor. Martin Gubisch, Stefan Volkwein (University of Konstanz) TU Berlin (3) March, 3 4 / 5

15 References References Hintermüller, M., Ito, K. & Kunisch, K.: The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim., vol. 3, no. : pp , 3. Hintermüller, M., Kopacka, I. & Volkwein, S.: Mesh-independence and preconditioning for solving parabolic control problems with mixed control-state constraints. ESAIM: COVC, vol. 5: pp , 9. Hinze, M. & Volkwein, S.: Error estimates for abstract linear-quadratic optimal control problems using proper orthogonal decomposition. Comput. Optim. Appl., vol. 39: pp , 8. Tröltzsch, F. & Volkwein, S.: POD a-posteriori error estimates for linear-quadratic optimal control problems. Comput. Optim. Appl., vol. 44: pp , 8. Volkwein, S.: Model Reduction using Proper Orthogonal Decomposition. Vorlesungsskript Universität Konstanz, p. 4,. Martin Gubisch, Stefan Volkwein (University of Konstanz) TU Berlin (3) March, 3 5 / 5