Optimal control problems with PDE constraints

Transcription

1 Optimal control problems with PDE constraints Maya Neytcheva CIM, October 2017

2 General framework

3 Unconstrained optimization problems min f (q) q x R n (real vector) and f : R n R is a smooth function. Definition: A point q is a global minimizer of f if f (q ) f (q) for all q R n. Definition: A point q is a (strict) local minimizer of f if there exist a neighbourhood E of q, such that f (q ) f (q) for all q E. <

4 Taylor s theorem: Let f : R n R be continuously differentiable and p R n. Then, f (q + p) = f (q) + f (q + εp) T p for ε (0, 1). If f is twice continuously differentiable, then and 1 f (q + p) = f (q) f (q + εp)p 2 f (q + εp) T pf (q + p) = f (q) + f (q) T p pt 2 f (q + εp)p.

5 Necessary and sufficient conditions for optimality Theorem:[First order necessary condition] If q is a local minimiser and f is continuously differentiable in an open neighbourhood of q, then f (q ) = 0. Theorem:[Second order necessary condition] If q is a local minimiser and 2 f is continuous in an open neighbourhood of q, then f (q ) = 0 and 2 f (q ) is positive definite.

6 Theorem:[Second order sufficient condition] Let (i) 2 f be continuous in an open neighbourhood of q, (ii) f (q ) = 0 and (iii) 2 f (q ) be positive definite. Then q is a strict local minimiser of f. Theorem: When f is convex, any local minimizer is a global minimizer. If in addition f is differentiable, then any stationary point q is a global minimiser. Stationary point: any point where the derivative is zero.

7 Recall: Line search We are after a direction vector d, along which the gradient descends. Descend direction d: d T f (q) < 0. Then seek a constant α to construct q = q + αd, such that α solves min αf (q + αd). Iteratively (put a subscript k).

8 Recall: Depending on the choice of the search direction d k, the following methods are mostly used: Steepest descent: d k = f (q k ) (linear convergence) Newton s method: d k = 2 f 1 f (q k ) (quadratic convergence) Quasi-Newton method: d k = B 1 k f (q k) (super-linear convergence) Here B k is an approximation of the Hessian.

9 Example of a unconstrained problem from 1788 I An excellent article: Martin Gander, Felix Kwox, Gerhard Wanner Constrained optimization: From Lagrangian Mechanics to Optimal Control and PDE constraints R. Hoppe (ed.) Optimization with PDE constraints, Lecture Notes in Computational Science and Engineering, 101, Springer 2014.

10 Example of a unconstrained problem from 1788 II Example: Bernoulli s regle, discussed by Lagrange in 1788: A point mass, attached to three forces. The forces P, Q, R cause (infinitely) small displacements dp, dq, dr.

11 Example of a unconstrained problem from 1788 III As a condition of equilibrium we get Pdp + Qdq + Rdr = 0. The terms are called energies by Bernoulli and moments by Lagrange. Important: this is an unconstrained problem and the displacements are independent on each other!

12 Example of a unconstrained problem from 1788 IV Wrt a Cartesian coordinate system, introduce: p = (x x 1 ) 2 + (y y 1 ) 2 + (z z 1 ) 2 q = (x x 2 ) 2 + (y y 2 ) 2 + (z z 2 ) 2 r = (x x 3 ) 2 + (y y 3 ) 2 + (z z 3 ) 2 dp = 1 p ((x x 1)dx + (y y 1 )dy + (z z 1 )dz) dq = 1 q ((x x 2)dx + (y y 2 )dy + (z z 2 )dz) dr = 1 r ((x x 3)dx + (y y 3 )dy + (z z 3 )dz) The equilibrium condition becomes Xdx + Yxy + Zdz = 0 with X = P x x 1 p + Q x x 2 q + R x x 3 r, analogously for Y and Z. As the point of mass is free to take any position in space, then dx, dy, dz are independent. Thus, the condition of equilibrium becomes X = 0, Y = 0, Z = 0.

13 Constrained optimization problems I min f (q, y) q,y subject to c(q, y) = 0, where q R m, y R n, f : R m+n R, c : R m+n R. In general the constraint is a nonlinear equation. Many sources state at this point the following: The Lagrangian functional for the above problem is defined as L(q, y, λ) = f (q, y) + λ T c(q, y), where λ R m is the vector of Lagrange multipliers or the adjoint variables.

14 Constrained optimization problems II Next to it comes the statement: the necessary optimality condition leads to the following system of equations to be solved: L f y y c λt y = 0 L q f q c λt q = 0 L c(q, y) = 0 λ What is the Lagrange multiplier and why the Lagrangian is constructed?!

15 Example of a constrained problem from 1788 I The discovery of the Lagrange Multiplier method Constraint: the point mass should not leave the surface! The displacements are NOT independent of each other.

16 Example of a constrained problem from 1788 II The displacements are restricted to the tangent space of L = 0, i.e., dl = L x L L dx + dy + y z dz = 0 Thus, the vectors [ L x, L y, L z ] and [dx, dy, dz] are orthogonal. However, the balance equation must also be satisfied: Xdx + Ydy + Zdz = 0. Therefore, the vectors [X, Y, Z] and [ L x, L y, L z ] are parallel, thus, there exist a constant λ, such that X + λ L x = 0, Y + λ L y = 0, Z + λ L z = 0. Thus, Xdx + Ydy + Zdz + λdl = 0.

17 Example of a constrained problem from 1788 III From this point on, adding another constraint becomes elementary: Pdp + Qdq + Rdr + + λdl + µdm + = 0 This is the method of Multipliers, Lagrange OBS: The meaning of λ is that it tires the point to the surface (ensures that the constraint is satisfied).

18 How all this is related to minimum and maximum? Lagrange noted: if a function F (x, y, z) is such that for some independent dx, dy, dz we have F F F dx + dy + x y z dz = 0 then we must have F x = F y = F z = 0. Then, F must have min or max at (x, y, z) (in the unconstrained case). Analogously, the conditions in the constrained case mean that we are minimizing/maximizing the function F (x, y, z) + λl(x, y, z) which is the Lagrangian functional.

19 Variational problems I Optimization problems, where the unknown is not only a scalar value but a function b min J, J = y a F (x, y(x), y x )dx. F is a given function, y = y(x) is unknown.

20 Variational problems II An example of constrained variational problem Bernoulli gave to his brother (1697) (oldest known such): Given two points A and B, find a curve of a given length L, such that the area ADEFGA is maximized. In addition, for any distance y, g(y) is given.

21 Variational problems III In math terms, F A g(y(x))dx max subject to F A 1 + p 2 dx = L. The corresponding Lagrangian for the case A = 0, F = 1 L = 1 0 [ g(y) + λ( ] 1 + p 2 L) dx.

22 Solving opt.control problems with Lagrange multipliers I Example: Consider J = b a k(x, y(x), u(x))dx min s.t. dy dx = f (x, y, u), y(a) = y a, y(b) = y b. Now we need to determine two functions, y (state) and u (control). As x varies, there are infinite number of constraints that control the behaviour of y. Introduce λ(x).

23 Solving opt.control problems with Lagrange multipliers II Multiply the constraint dy dx f (x, y, u) = 0 by λ and insert in the integral: L(x, y, p, u, λ) = b a [ ] k(x, y(x), u(x)) + (p T + f T )λ(x) dx p is the derivative of y. The necessary optimality conditions read: L = 0 : y f (x, y, u) = 0 λ L y L y p = 0 : λ = k y f T y λ L u = 0 : k u f T u λ = 0 (1) Unknowns: y, u, λ - functions!

24 Lagrange multipliers in finite-dimensional problems I Consider a somewhat simpler setting min x f (x) subject to g(x) = 0, where f : R n R (objective) g : R m R, (constraint), x R n, m < n. We want to eliminate the constraints. To this end, partition x = (y, u, y R m, u R n m. Since g(x) = 0, applying the implicit function theorem we can eliminate u and express y = y(u). Then we have to minimize a problem of reduced dimension, min f (y(u), u). x

25 Lagrange multipliers in finite-dimensional problems II The necessary conditions for a local minimum: f u = f y y u + f u = (Y u T y f + u f ) T = 0. where Y u : R n m R m (n m) is the Jacobian of the implicit function y(u). Thus, the necessary optimality condition is a small system of equations of n m unknowns of the vector u only. Very limited cases when we can construct y(u).

26 Lagrange multipliers in finite-dimensional problems III Implicit function theorem

27 Lagrange multipliers in finite-dimensional problems IV In practice we work with the complete optimality system: Yu T y f + u f = 0 n m eq. Yu T Gy T + Gu T = 0 m(n m) eq. g = 0 m eq. Here Gu T : R n R m (n m) is the Jacobian of g wrt u and Gy T : R n R m (n m) is the Jacobian of g wrt u. In total, n + m(n m) equations!

28 Apply Lagrange multipliers: I Miraculous effect! Assume that G y is invertible and introduce λ = Gy T y f. Multiply the matrix equation from the right: Yu T Gy T λ + Gu T λ = Yu T Gy T Gy T y f + Gu T λ = Yu T y f + Gu T λ = 0 Add to Y T y y f + u f = 0 and we obtain an equivalent system of

29 Apply Lagrange multipliers: II first order necessary conditions that is much simpler: In total; n + m equations! y f + Gu T λ = 0 n m eq. u f + Gu T λ = 0 m eq. g = 0 m eq. The key observation: The simpler necessary optimality system is obtained from the Lagrangian functional L(y, u, λ) = f (y, u) + g(y, u) T λ!

30 Apply Lagrange multipliers: III OBS; New difficulty - the control cannot be arbitrarily large, i.e., u does not vary freely but only in some closed set U.

31 Pontryagin s maximum principle In the paper by Gander et al. a connection is made between Lagrange multiplies framework and Pontryagin s maximum principle. (Interesting, but out of the scope of this course.)

32 An example of a variational problem (to get the flavour) Given a system with a state variable y = y(t) R and a control variable u = u(t) R, described by y = u, y(0) = y 0, subject to box constraints u(t) 1 for each t. Find u such that y(1) = 1/2 and the following cost functional is minimized J = y 2 dt.

33 PDE-constrained optimal control problems

34 PDE-constrained optimal control problems Given a functional J to minimize, deal with a constraint, that is a PDE. The ultimate problem to solve Control of various processes (oil reservoir simulations, name it..) Shape and topology optimization (airfoil) Inverse problems (parameter estimation with all flavours - uncertainty, noisy data,...)

35 OPT-PDE The model problem: min J (y, u) subject to L(y, u) = 0, y,u where L is a differential operator. Many questions arise: How to define the cost functional to ensure well-posed problem? Should we first optimize, then discretize or vice versa? How to handle additional constraints? How to solve the arising linear or nonlinear algebraic systems?

36 OPT-PDE: variations min J (y, u; m) subject to L(y, u; m) = 0, y,u where m is a vector of parameters. min J (Q(y), u) subject to L(y, u) = 0, y,u Q(y) is some function of y, for instance, it takes only y on the boundary of the domain.

37 OPT-PDE: variations How to handle time-dependent problems? We obtain very large algebraic problems, in particular for large time intervals. Non-stationary heat equation Eddy current simulations Non-stationary Navier-Stokes equations

38 OPT-PDE: Discretize Optimize? The problems mentioned so far are continuous, where the variable functions live in some Hilbert space. The important message is that in general DO and OD do not commute. O-D: we need to discretize the derivatives of the Lagrangian that may not be true gradients of any objective function. Thus, we may get wrong descent directions. D-O: We may need to differentiate computational facilitators, for instance, meshes. Some aspects, such as mesh-independent convergence are only tractable via the continuous framework.

39 OPT-PDE: Discretize Optimize? Discrete PDEs are inherently large scale. Discrete OPT-PDEs (the KKT system) are even larger! Discretization and solution of OPT-PDEs should not be considered as independent, but rather we should see how to interwine those in order to obtain efficient solution algorithms.

40 Optimization and regularization The regularization can be seen as reformulation that is needed to ensure well-posed formulation of the OPT-PDE problem. In other words, to guarantee unique and stable solution, we add some regularization to the cost functional. J = 1 2 y y d β 2 u 2 2 J = 1 2 Q(y) d β 2 R(m) Q(y) part of the domain d given data R(m) how well data should be fitted Tikhonov regularization; comment on the value of β.

41 Regularization, cont. What norms to use depends on the problem at hand. L 1 -regularization and sparse controls Moreau-Yosida regularization Lavrentiev regularization Important: What is the effect of the type of the regularization on the (condition of the) so-arising algebraic systems to solve.