Proper Orthogonal Decomposition in PDE-Constrained Optimization

Transcription

1 Proper Orthogonal Decomposition in PDE-Constrained Optimization K. Kunisch Department of Mathematics and Computational Science University of Graz, Austria jointly with S. Volkwein

2 Dynamic Programming Principle { min J(y, u) = L(y(t), u(t))e µt dt ẏ(t) = f (y(t), u(t)), y() = y, u U ad. value function υ(y ) = min u Uad J(y, u). (DPP) υ(y ) = min u Uad { R T L(y(t; y, u), u(t)e µt dt + υ(y(t ; y, u))e µt } (HJB) µυ(y ) = min u U ad {( υ(y ), f (y, u)) + L(y, u)} replace y by y (t). u (t) = F(y (t)) where F(y (t)) argmin u Uad {( υ(y (t)), f (y (t), u)) + L(y (t), u)}

3 Dynamic Programming Principle { min J(y, u) = L(y(t), u(t))e µt dt ẏ(t) = f (y(t), u(t)), y() = y, u U ad. value function υ(y ) = min u Uad J(y, u). (DPP) υ(y ) = min u Uad { R T L(y(t; y, u), u(t)e µt dt + υ(y(t ; y, u))e µt } (HJB) µυ(y ) = min u U ad {( υ(y ), f (y, u)) + L(y, u)} replace y by y (t). u (t) = F(y (t)) where F(y (t)) argmin u Uad {( υ(y (t)), f (y (t), u)) + L(y (t), u)}

4 J(y, u) = ( Ω y(t, x) dx + β u(t) ) e µt dt, y t νy xx + yy x = in Q, νy x (, ) + σ y(, ) = u in (, ) νy x (, ) + σ y(, ) = g in (, ) y(, ) = y in Ω, jointly with Xie

5 Semi implicit solution of the Burgers equation (with opt. control) y(x) t axis t axis! "$# % &(' *),+ -. /) -* "43, ): & "; %) % <.- ),+ % -*) =?> "@"@+ A, <CB + "D E*)F "G>H I7 ),+;-. J"@) KL -M-* ;+ "@ %E.)N ' %E.&. I7IO y(x) Semi implicit solution of the Burgers equation (with opt. control) t axis t axis QP R % * S! " # T &U'C ),+ -. V) -. "W46,5 X.Y*7R )F & "$Z\[ ]^= > "_"D+ A, <CB + "@ %E.)N "G>H C7 *),+;-* `"_) Ka -8-. ;+ "@ %E.)4' E*&8 C7GO 3 4 5

6 Basics Construction of POD basis (X,, X ) HS, = t < t < < t n T {y j } n j= snap shots, e.g. velocity components of fluid V = span {y j } n j=, d = dim V; y j = d i= y j, ψ i X ψ i, j =,, n; {ψ i } ONB POD basis: for every l {,, d} min {ψ i } l i= n l y j y j, ψ i X ψ j X j= i= subject to ψ i, ψ j = δ i,j, i, j l

7 Y n : R n X, Y n υ = n j= β j υ j y(t j ), Y nz = ( z, y(t j ) X,.., z, y(t n ) X ) R n = Y n Y n = n j= β jy(t j ) y(t j ), X L(X) spatial correlation op. K n = Y ny n = y(t i ), β j y(t j ) X R n n temporal correlation op. R n... {σ j }, {ψ j }; K n... {σ j }, {ϕ j }; ψ k = σk Y n ϕ k

8 continuous POD Y : L (, T ; R) X, Yv = T v(t)y(t, ) dt K = Y Y = R = YY = T T y(t) y(t), x dt L(X) y(t, ), y(t, ) X dt L(L (, T ), L (, T ))... Hilbert-Schmidt Proposition. Let y L (, t; X). Then K is compact. Except for possibly the eigenvalues of K and R coincide. They are positive with identical multiplicities and ψ is EVE of R iff Y ψ = y(t, ), ψ( ) x = ϕ is eigenvalue of K.

9 error formula: n l d y j y j, ψ j X ψ i X = j= i= i=l+ l n choice of l: I(l) = λ k/ λ k α <. k= k= λ i i

10 Convergence rate for state (*) y t ν y + (y )y = f in (, T ] Ω, div y = y(t) = on (, T ] Ω, y(, ) = y in Ω POD-Galerkin scheme y l = l i= x i(t)ψ i ( ) { Eẋ = A x + N(x) x() = x, A full grid = {t i } m i=, t i = i t, for snapshots and backwards Euler V = span {y(t i ), ( t) (y(t i ) y(t i ))} m i=

11 Convergence rate for state (*) y t ν y + (y )y = f in (, T ] Ω, div y = y(t) = on (, T ] Ω, y(, ) = y in Ω POD-Galerkin scheme y l = l i= x i(t)ψ i ( ) { Eẋ = A x + N(x) x() = x, A full grid = {t i } m i=, t i = i t, for snapshots and backwards Euler V = span {y(t i ), ( t) (y(t i ) y(t i ))} m i=

12 fully discrete Galerkin-POD approximation Theorem Let y be solution to (*), {Y } m k= solution to backwards Euler (POD) scheme. There exists C independent of l and t (dep. on T ) such that m m Y i y(t i ) L (Ω) C i= for t t. i=l+ ( y o, ψ i H + λ i ) +C( t) y W, (,T ;H (Ω)), Crank Nicolson: ( t) 4, grid can be decoupled, without finite differences in snapshot set: i=l+ ( t) λ i

13 semi-continuous Galerkin-POD approximation (E l ) { d dt (y l(t), v) H + a(y l, v) = f (t), v V,V, t (, T ] (y l (), v) H = (y, v) H, for all v span{ψ i } l i=. ỹ l (t) = (y(t), ψ k ) V ψ k. k=l+ ỹ l (t) as l for t [, T ]. Theorem Choose X = V, set ρ l := dỹ l dt L (,T ;H) + ỹ l () H. Then for the solution y l of (E l ) Henri, Yvon y y l L (,T ;V ) ρ l + i=l+ λ i l.

14 a-posteriori estimate for open loop optimal control for linear quadratic optimal control (P) J(y, u) = ( T y z + β u ) dt y t = Ay(t) + Bu(t), y() = y u u l L (,T ;U) β J l (u l ) where J l (u l ) = βu l p l (u l ) Tröltzsch Volkwein

15 Galerkin POD procedure for optimal control y l = l x i (t)ψ i, u l = i= l β i (t) ψ i i= Eẋ = f (x, β) = A x + N(x) + β ( ) x() = x, A full low order optimization problem (P POD ) min J(x, β) = solve HJB equation for discretized system e µt L(x(t), β(t)dt subject to ( ). µv (x) + min β U [ f (x, β) V (x) + L(x, β)] =, β closed loop (t) = β (V (x (t))) utilize POD-HJB control in infinite dimensional system

16 Reynolds number, Re Time horizon, T No. of basis functions 6 No. of controls Bounds on controls [.5,.5] Grid system Discount rate, µ Observe region, Ω o [, ] [, ] Table: Parameter Settings

17 Streamline, Re= Streamline, Re=

18 unmodelled dynamics update POD basis where to take snapshots? ( in discrete POD case) how many basis elements?

19 OS-POD optimality-system POD min J(y, u) = T Ω y z dxdt + β T u dt (P) y t = Ay + N (y) in (, T ] Ω y(t, ) = Bu = m k= u i(t)b i (x) on (, T ) Ω y(, ) = y on Ω

20 OS-POD min J l (x, ψ, u) over (x, ψ, u) L (R l ) X l L (, T ; R m ) subject to E(ψ) ẋ(t) + A(ψ)x(t) + N(x(t), ψ) = B(ψ)u(t), t (, T ], (POSPOD l ) E(ψ) x() = x (ψ), d dt y(t) + Ay(t) + N (y(t)) = Bu(t), t (, T ], y() = y, R(y)ψ i = λ i ψ i, i =,..., l, ψ i, ψ j = δ ij i, j =,..., l. where R(y)ψ i = T y(t), ψ i X y(t)dt.

21 OS-POD min J l (x, ψ, u) over (x, ψ, u) L (R l ) X l L (, T ; R m ) subject to E(ψ) ẋ(t) + A(ψ)x(t) + N(x(t), ψ) = B(ψ)u(t), t (, T ], (POSPOD l ) E(ψ) x() = x (ψ), d dt y(t) + Ay(t) + N (y(t)) = Bu(t), t (, T ], y() = y, R(y)ψ i = λ i ψ i, i =,..., l, ψ i, ψ j = δ ij i, j =,..., l. where R(y)ψ i = T y(t), ψ i X y(t)dt.

22 L(z, ξ) = J l (x, ψ, u)+ E(ψ)ẋ + A(ψ)x + N(x, ψ) B(ψ)u, q L (,T,R n ) (y t + Ay + N (y) Bu, p L (,T ;L (Ω)) + l i= (R λ ii)ψ i, µ i X + l i= ( ψ i )η i primal variables adjoint variables z = (x, ψ, u, y), ξ = (q, p, µ, η).

23 The following optimality system holds: { E(ψ) q(t) + (A(ψ) + N T x (x, ψ)) q(t) = J l x(x, ψ, u), (ADJ ) q(t ) = ṗ(t) + Ap(t) + N (y(t)) p(t) (ADJ ) = l i= y(t), µ i X I ψ i + y(t), ψ i X I µ i p(t ) = { ηi = G i(x, ψ, u, q), ψ i X,X, I = X X and µ i = (R λ i I) [ η i ψ i + I G i (x, ψ, u, q)], for i =,..., βu(t) = B T (ψ) q(t) + B p(t).

24 ( T ( l ) l G i (x, ψ, u, q) = x i x j ψ j z + q i ẋ j ψ j + ẋ j j= j= ( T l l + q i x j Aψ j + x i q j A ψ j q i + j= j= ) l q j ψ j dt j= m k= l (x j () ψ j q i () + x i ()ψ j q j ()) y q i () j= ( l ) + N x k ψ k q i + x i k= analogous result for K formulation. l j= N ( l k= b k u k ) dt x k ψ k ) q j ψ j.

25 Algorithm implemented. initial (POD) basis {ψ i } l i=. solve (PPOD l ) inexactly for u 3. compute G i, y(u ), p(u ) 4. use optimality condition in gradient step to update u + 5. compute y(u + ) and update POD basis ψ +

26 Test examples min J(y, u, u ) = subject to T y z dxdt + β T y t νy xx + yy x = f in (, T ] (, ) (u + u )dt νy x (, ) + σ y(, ) = u in (, T ] νy x (, ) + σ y(, ) = u in (, T ] y(, ) = y in (, )

27 Desired state z=z(t,x) Finite element solution y=y(t,x) t axis..4 t axis.6.8 Figure: Desired state (left) and FE solution to the uncontrolled Burgers equation, i.e., u = v = (right).

28 Desired state z=z(t,x) POD optimal state y * =y * (t,x) t axis..4 t axis.6.8 Figure: Desired state :: Optimal POD state for the OS-POD strategy.

29 .5 POD basis ψ 3 POD basis ψ.5 POD basis ψ 3 4 POD basis ψ POD basis ψ 4 POD basis ψ POD basis ψ 3 4 POD basis ψ POD basis functions: uncontrolled:: final OS-POD update of the controls.

30 POD basis ψ 4 POD basis ψ.5.5 POD basis ψ 4 POD basis ψ POD basis ψ 3 4 POD basis ψ 4 POD basis ψ 3 4 POD basis ψ POD basis functions: final OS-POD update of the controls::pod basis functions associated with the optimal FE-SQP controls.

31 n = n = 4 with FE controls λ /tr (K h ) λ /tr (K h ) λ 3 /tr (K h ) λ 4 /tr (K h ) Table: Decay of the first four eigenvalues λ i for the OS-POD strategy for n =, n = 4 and for the first four eigenvalues associated with the snapshots that are computed by using the optimal FE controls. For n = we have the decay of the first eigenvalues associated with the uncontrolled solution. J(y, u, v) Uncontrolled solution.34 OS-POD.383 (.3856) FE-SQP.3765 Table: Values of the cost functional for the different approaches (in brackets: value of the cost using POD solution in original system).

32 5 Optimal POD controls u(t) v(t) 5 Optimal FE controls u(t) v(t) t axis t axis Optimal control for OS-POD strategy (left) and optimal FE-SQP controls (right).

33 partial steps Generate snapshots POD computation Compute ROM SQP solve P l Compute µ i s FE dual solve Backtracking in step (4) CPU time.6 seconds.9 seconds.3 seconds 4.6 seconds.7 seconds.7 seconds.3 seconds Table: averaged CPU times for one iteration of OS-POD.

34 use these (optimal open loop) POD basis elements to solve closed loop problems, where now, only unresolved dynamics result from noise... and other systems theoretical tasks. Arian, Borggard, Gunzburger, Hinze, Ly, Ravindran, Rowley, Sachs, Sirovich, Tran, Willcox,...

35 Optimal Snapshot Location where to place additional snapshots? ẏ(t) = Ay(t) + f (t) for t (, T ] y() = y, fixed time grid t j = j t, j m, with t = T /m, snapshots y j = y(t j ) V, j m, new time instances t k [, T ], k k, corresponding snapshots ȳ k = y( t k ), k k, number of POD basis functions l {,..., m}.

36 R( t)ψ = m y j, ψ H y j + j= k k= ȳ k, ψ H ȳ k. The POD basis functions {ψ i } l i= are eigenvectors of R( t), i.e. POD-Galerkin approximation R( t)ψ i = λ i ψ i, ẋ(t) = A l x(t) + f(t) for t (, T ], x() = x

37 subject to min J(x, t, ψ, λ) = T y(t) l x i (t) ψ i i= H dt. ẋ A l x f(t) = in L (, T ; R l ), ( R( t) λ ) ψ. ( R( t) λ l ) ψl x() x = in R l, = in Hl, ψ H. ψ l H = in R l

38 ( ẏ(t) = f (t) for t (, T ] and y() = ) f (t) = ( ) for t [, T /4) and f (t) = ( ) for t [T /4, T ] y l (t) = Exact solution constant on (, T /4) POD: Snapshots: t = and t, l = { (ψ ( t)) ψ ( t) for t [, T /4), ( (t T /4)(ψ ( t)) + (ψ ( t)) ) ψ ( t) for t [T /4, T ].

39 Value of the cost for varying tbar tbar axis Value of the gradient of the cost for varying tbar tbar axis