Multigrid and stochastic sparse-grids techniques for PDE control problems with random coefficients

Transcription

1 Multigrid and stochastic sparse-grids techniques for PDE control problems with random coefficients Università degli Studi del Sannio Dipartimento e Facoltà di Ingegneria, Benevento, Italia

2 Random fields A random field is a generalization of a stochastic process such that the underlying parameter can be a multidimensional vector space or a manifold. Values in a random field are spatially correlated. c(y, u, δ, σ) = 0, with δ(ω), σ(ω) where ω O.

3 Random parameter fields The triple (O, A, P) denotes a probability space where O is the space of elementary events, A is the sigma-algebra of subsets of O, and P the probability measure on O. We assume that δ = δ(t, ω), σ = σ(x, ω) > 0 where t (0, T ), x Ω, and ω O. Examples of finite-dimensional random fields and δ(t, ω) = δ 0 + W (ω) sin(2πt/t ) σ(x 1, x 2, ω) = σ 0 + exp{ [Y 1 (ω) cos(πx 2 ) + Y 3 (ω) sin(πx 2 )] e 1/8 where W, Y j [ 1, 1], j = 1, 2, 3, 4. +[Y 2 (ω) cos(πx 1 ) + Y 4 (ω) sin(πx 1 )] e 1/8 }

4 Random field modeling The values of the stochastic functions δ(t, ω) and σ(x, ω) are timely and spatially correlated in a way characterized by a covariance structure. The mean and the covariance of σ(x, ω) are nown respectively as σ 0 (x) = E(σ)(x) := σ(x, ω)dp(ω) and C σ (x, x ) = O O (σ(x, ω) σ 0 (x))(σ(x, ω) σ 0 (x ))dp(ω). C σ defines the ernel of a compact and self-adjoint operator.

5 Karhunen Loève random field representation The Karhunen Loève (KL) expansion of the random field σ(x, ω) is based on a spectral decomposition of the covariance ernel of the stochastic process. Ω C σ (x, x ) z j (x )dx = λ j z j (x), x Ω Define the following uncorrelated random variables Y j (ω) = 1 (σ(x, ω) σ 0 (x)) z j (x)dx, j = 1, 2,... λj Ω with zero mean and unit variance. The truncated KL expansion of σ(x, ω) is given by N σ N (x, ω) = σ 0 (x) + λj z j (x) Y j (ω) We assume the eigenvalues decay sufficiently fast. j=1

6 Optimal control of PDE with random data A model of the stochastic dynamical system A description of the control mechanism A criterion that models the purpose of the control (deterministic) and the cost of its action We have a constrained minimization problem minimize J(y, u) under the constraint c(y, u, ω) = 0 ω O

7 Optimality system For an event ω O, we have a deterministic control problem min u Uad J(y, u) := h(y) + ν g(u) s.t. c(y, u, ω) = 0 J : Y U R The existence of cy 1 enables a distinction between y, the state variable, and u U ad U, the optimization variable in the admissible set. So we have the mapping u J(y(u), u) in the form u IFT y(u) J(y(u), u) =: Ĵ(u) The solution of this optimization problem is characterized by the following optimality system c(y, u, ω) = 0 c y (y, u, ω) p = h (y) (ν g (u) + cu p, v u) 0 for all v U ad We have Ĵ(u) = ν g (u) + c u p(u).

8 Steady models with random inputs An explosive reaction-diffusion problem { y + δ e y = f in Ω y = g on Ω, δ is a reaction coefficient that is considered subject to randomness and is described by a spatially varying random field. Two realizations of δ(x 1, x 2, ω):

9 A singular optimal control problem We discuss the following min u L2 (Ω) J(y, u) y + δ(ω) e y = f + u in Ω, ω O y = 0 on Σ = Ω Control required to attain a desired target configuration y d (x) A cost functional of tracing type can be considered J(y, u) = 1 2 y y d 2 L 2 (Ω) + ν 2 u 2 L 2 (Ω)

10 Optimality system for the steady model Corresponding to ω O, the optimal solution is characterized by y + δ(ω) e y = u in Ω, p + δ(ω) e y p + (y y d ) = 0 in Ω, νu p = 0 in Ω, y = 0, p = 0 on Ω. Solving an optimal control problem with random coefficients means to determine the -dim space of controls for all ω O. This space represents the solution of the stochastic control problem.

11 Time-dependent models with random inputs t y + G δ (y) + σ y = u in Q = Ω (0, T ) y = y 0 in Ω {t = 0} y = 0 on Σ = Ω (0, T ) The nonlinear term G δ (y) models the reaction inetics for the state y and δ is the reaction parameter. Here σ is the diffusion coefficient. Two realizations of σ(x 1, x 2, ω):

12 Reaction-diffusion process controlled through source terms We discuss the following min u L2 (Q) J(y, u) t y + G δ(ω) (y) + σ(ω) y = u in Q = Ω (0, T ), ω O y = y 0 in Ω {t = 0} y = 0 on Σ = Ω (0, T ) Control required to trac a desired trajectory y d (x, t) reach a desired terminal state y T (x) A cost functional of tracing type can be considered J(y, u) = α 2 y y d 2 L 2 (Q) + β 2 y(, T ) y T 2 L 2 (Ω) + ν 2 u 2 L 2 (Q)

13 Optimality system Corresponding to the event ω O, the optimal solution is characterized by t y + G δ(ω) (y) + σ(ω) y = u in Q t p + G δ(ω) (y)p + σ(ω) p + α(y y d) = 0 in Q νu p = 0 in Q y = 0, p = 0 on Σ With initial condition y(x, 0) = y 0 (x) for the state variable (evolving forward in time). And the terminal condition for the adjoint variable (evolving bacward in time) p(x, T ) = β(y(x, T ) y T (x)). By solving optimal control problems with random coefficients, we explore the -dim space of controls for all configurations of the parameters space. The space of controls represents the solution of the stochastic control problem.

14 The issue of a robust control An important tas is to define a unique control for the governing random PDE model that provides good tracing features for all configurations of the random coefficients. Consider the following problem min u U E(Ĵ)(u) := min u U O Ĵ ω (u) dp(ω) where Ĵω(u) = Ĵ(y ω(u), u). A modeling issue... E(J(σ, δ)) = J(E(σ), E(δ)) J 2 σ 2 s2 σ J 2 δ 2 s2 δ J 2 σ δ s σs δ + O(s 4 ) with standard deviations (s σ, s δ ). J(y(u E(σ), E(δ) ), u E(σ), E(δ) ) J(y(E(u)), E(u)).

15 Collocation approach in the probability space In this framewor, the PDE model is approximated by FEM, FVM or FDM discretization and the stochastic domain is approximated using multidimensional interpolating functions. Stochastic optimal solutions are sought in a tensor-product space P(Γ) [Y U] where P represents the span of tensor-product interpolation polynomials in the N + M-dimensional stochastic space We have the random fields σ(x, ω) = σ(x, Y 1 (ω),..., Y N (ω)) and δ(t, ω) = δ(t, W 1 (ω),..., W M (ω)). ρ denotes the joint probability density of (Y 1,..., Y N, W 1,..., W M ) with support Γ = N j=1 ΓY j M j=1 ΓW j. N = N + M.

16 Lagrange interpolation Consider a function f on the one-dim space Γ j = [ 1, 1] with m j nodal values (z 1,..., z mj ). Lagrange interpolation operator m j m j I (f )(z) = f (z i ) L j i (z), Lj i (z) = i=1 =1, i z z z i z. Assume p mj (f ) is the best approximation polynomial of degree m j 1 f p mj (f ) f I (f ) (1 + Λ) f p mj (f ), mj Λ = max z Γj i=1 Lj i (z) is the Lebesgue constant depending on the distribution of the nodes. The Chebyshev Gauss Lobatto (CGL) nodes are given by ( ) (i 1)π z i = cos, i = 1,..., m j, m j 1 m j = 2 j (m 1 = 1). Here j > 1 is the order index for the grid with nodes z j = (z j 1, zj 2,..., zj m j ).

17 Multivariate interpolation In the multidimensional case the determination of the best approximation polynomial is an unsolved problem. For f : R N R the practice is to consider tensor-product interpolation polynomials on a full tensor-product grid. m 1 I(f )(z) =... i 1=1 m N i N =1 ( ) f (z j1 i 1,..., z j N in ) L j1 i 1... L j N in (z), Thus the mean value of a function f : R N R is given by m 1 E(f ) =... i 1=1 m N i N =1 and the variance is Var(f ) = E([f E(f )] 2 ). ( ) f (z j1 i 1,..., z j N in ) L j1 i 1... L j N in (z) ρ(z) dz, Γ

18 Smolya sparse-grids The full tensor-product interpolation formulae require m 1... m N function evaluations and this effort grows exponentially with the number of dimensions: Curse of dimensionality. A Smolya scheme provides multivariate interpolation as linear combination of tensor-product formulas. A full tensor-product N -dimensional grid of order J with j 1 = j 2 = j N = J in each dimension is given by ẑ J = z j1 z j2 z j N The sparse grid z J of order J is composed of a strict subset of full grids. For example, for N = 2 and J = 4 we have {z 1 z 4 } {z 2 z 3 } {z 3 z 2 } {z 4 z 1 }

19 Sparse-grid interpolation and integration The sparse grids are the union of all possible reduced full grids with orders that sum up to J z J = j =J +N 1 z j1 z j2 z j N with vector index notation j = (j 1,.., j N ), where j = j j N. On the sparse-grids we have an interpolation formula and we have a sparse-grid quadrature.

20 Approximation error in the probabilistic and physical spaces The optimality systems are discretized by finite differences and bacward Euler scheme. The discretization error of the deterministic solution y h y c h 2, p h p c h 2, and u h u c h 2. The stochastic solution is affected by an error having three components: 1) The error due to truncation of the KL expansion, e N ; 2) The discretization error (above), e d ; 3) The Smolya sparse grid interpolation error due to the collocation, e S c χ g/(1+log(2n )). (χ is the total number of sparse-grids points.) The total error is e total = e N + e d + e S.

21 CS-Multigrid - Sparse-grids solution process For i = 1,..., χ 1. Define coefficient functions δ i, σ i. 2. Call CSMG to solve the ith optimality system to Ĵ(u) < tol. 3. Collect results for statistics. End Statistics: compute E( y ), Var ( y ), Sew ( y ), and E(u).

22 Collective Smoothing Multi-Grid -V(m 1, m 2 )-Cycle Set B 1 (w (0) 1 ) A 1 1 (e.g., iterating with S 1 starting with w (0) 1 ). For = 2,..., L define B in terms of B 1 as follows. 1. Set the starting approximation w (0). 2. Pre-smoothing. Define w (l) w (l) for l = 1,..., m 1, by = S (w (l 1), f ) 3. Coarse grid correction. Set w (m1+1) = w (m1) + I 1 (w 1 Î 1 w 1 = B 1 (Î 1 w (m 1 ) h ) 4. Post-smoothing. Define w (l) w (l) I 1 5. Set B (w (0) ) f = w (m1+m2+1). w (m1) ) where (f A (w (m 1 ) )) + A 1 (Î 1 w (m 1 ) i ). for l = m 1 + 2,, m 1 + m 2 + 1, by = S (w (l 1), f )

23 Multigrid components: Smoothing Coupling between state and control variables. For time dependent problems preserve opposite orientation of state and adjoint equations. t = 0 y 0 y t = T p p 0 Pointwise smoother: Collective Gauss-Seidel Newton Iteration (CGSN), Time-Splitted Collective Gauss-Seidel Iteration (TS-CGS) Blocwise smoother: Time-Line Collective Gauss-Seidel Iteration (TL-CGS)

24 Mean MG Convergence factors -2-4 Log nu gamma Log nu gamma Convergence factors of TL-CGS scheme (left) and TS-CGS scheme E(η) TL-CGS TS-CGS γ \ ν The mean E(η) for TL-CGS and TS-CGS multigrid schemes Università (ν 1 = degli ν 2 Studi = 1); del Sannio, Italy

25 Elliptic case: experimental setting We consider a random reaction coefficient given by δ(x 1, x 2, ω) = δ 0 + exp{ [Y 1 (ω) cos(πx 2 ) + Y 3 (ω) sin(πx 2 )] e 1/8 +[Y 2 (ω) cos(πx 1 ) + Y 4 (ω) sin(πx 1 )] e 1/8 } where δ 0 = 2 and Y j [ 1, 1], j = 1, 2, 3, 4. This field is characterized by a squared exponential covariance typical of Gaussian processes The desired target configuration is given by y d (x 1, x 2 ) = 1 sin(π x 1 ) sin(π x 2 ). This target is not attainable by any control. Multigrid The and mean stochastic field reaction sparse-grids coefficient techniques E(δ) for PDE control problems with random coefficients

26 Numerical results for the elliptic case Table: Results with the CSMG sparse-grids scheme; y = y y d. J = 2, χ = 41 ν N x N y CPU(s) E( y ) Var ( y ) Sew ( y ) E(ρ obs ) J = 3, χ = 137 ν N x N y CPU(s) E( y ) Var ( y ) Sew ( y ) E(ρ obs )

27 Parabolic case: experimental setting We consider the case of a nonmonotone nonlinear reaction term given by G δ (y) = δ e y that is used to model explosive combustion phenomena. The desired target trajectory is given by y d (x, t) = (1 + t) (x 1 x 2 1 ) (x 2 x 2 2 ) cos(4π t). This is an oscillating function whose amplitude increases linearly with time. We tae y T (x) = y d (x, T ). We use one pre- and one post-smoothing steps (ν 1 = ν 2 = 1) and h = 1/4 is the coarsest space mesh size. We tae Ω = (0, 1) (0, 1) and T = 1. The stopping criteria is r y + r p /ν <

28 TS-CGS MG convergence and tracing properties Results for α = 1, β = 0, with TS-CGS MG scheme. y = y y d. J = 2, χ = 61 ν N x N y N t CPU(s) E( y ) Var ( y ) Sew ( y ) E(ρ obs ) J = 3, χ = 241 ν N x N y N t CPU(s) E( y ) Var ( y ) Sew ( y ) E(ρ obs )

29 TL-CGS MG convergence and tracing properties Results for α = 1, β = 0 and J = 2, χ = 61, with TL-CGS MG scheme. y = y y d. ν N x N y N t CPU(s) E( y ) Var ( y ) Sew ( y ) E(ρ obs )

30 Searching for a robust control We consider the function E(u) defined as the mean function of the optimal controls corresponding to each point of the coefficient configuration space. Another possible way to define a unique control function is to consider the control computed using average coefficient values. The question is then how good are y E(u) and y E(σ), E(δ) for tracing the desired target for different optimization settings.

31 Elliptic case: results with robust control candidates Table: Results with a given control (left) and moments of tracing ability for controls corresponding to different realization of the reaction coefficient; N x N y = , χ = 137. ν E( y y d ) Var ( y y d ) E( y ) Var ( y ) y E(u) y ue(δ) y E(u) y ue(δ) y E(u) y ue(δ)

32 Parabolic case: results with robust control candidates α = 1, β = 0, and ν = 10 6 ; N x N y N t = , χ = 241. E( y y d ) Var ( y y d ) Sew ( y y d ) y E(u) y ue(σ), E(δ) α = 0, β = 1, and ν = 10 6 ; N x N y N t = , χ = 241. E( y y d ) Var ( y y d ) Sew ( y y d ) y E(u) y ue(σ), E(δ) The difference in the tracing property of the two strategies is minimal thus motivating forthcoming investigation efforts.

33 A model reduction approach Define a robust control as û = argmin u L 2 (Q) O J(y ω (u), u) dp(ω) To approximately solve this problem we consider argmin u L2 (Q) J(y ω (u), u) dp(ω) argmin u L2 (Q) O χ w j J(y ωj (u), u) Further, we construct a POD basis of the optimal controls. Let u j (x, t), j = 1,..., χ represent the controls corresponding to different configurations, the POD basis functions are obtained from T A j = u j (x, t)u (x, t)dxdt, A = V SV. Ω 0 The th POD basis function is φ (x, t) = 1 s m < χ and s = S. j=1 m V j u j (x, t), where j=1

34 Minimization in the POD space We use the representation u(x, t) = m c φ (x, t) =1 and find min J(u) := χ j=1 w j J(y ωj (u), u). The gradient w.r.t the POD coefficients is given by c J(u) = Then we can use c (l+1) where u (l) = m χ w j (νc p ωj (u)) = νc j=1 =1 c(l) χ w j p ωj (u) j=1 = c (l) α c J(u (l) ), = 1,..., m φ (x, t).