5.5 Optimality Conditions
5.5.2 Formulation of the Optimality System
In this section, we characterise the minimisation of the control functional ˇJ(y, u) (5.4.13) by a Lagrangian multiplier formulation [28, 48, 103, 104] to derive the optimality system. This can be used to develop an efficient strategy to apply the reduced matrix Q. It also gives rise to two equivalent reformulations of the control problem.
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5.5. Optimality Conditions
Analogously to (5.2.13), where we have introduced the adjoint variable p ∈ Y as a Lagrangian multiplier, we employ here its wavelet discretisation p ∈ `2. This leads to the functional
L(y, u, p) := ˇJ(y, u) + pT(Ay − f − Eu) . (5.5.8) The first order Euler-Lagrange equations for δL(y, u, p) = 0 follow as
Ay = f + Eu , (5.5.9a)
ATp = −TTRs(Ty − y∗) , (5.5.9b)
ωR−1t u = ETp . (5.5.9c)
This system of equations is called optimality system. The three components, called the state, adjoint and design equations, are the wavelet representations of the variational formulations from (5.2.14). Note that for consistency with more general problems, we keep the notation AT although in our situation A is symmetric.
Remark 5.16. The order of differentiation and discretisation is much disputed in the finite element context, since the approaches differentiate-then-discretise and discretise-then-differentiate lead to differ-ent results for finite represdiffer-entations of functions. In the wavelet context however, where we deal with infinite-dimensional representations, this order is irrelevant. Expanding the system (5.2.14) in wavelet coordinates, which would correspond to differentiate-then-discretise, leads precisely to the equations from (5.5.9), which have been obtained via the approach discretise-then-differentiate.
Application of the Reduced Matrix
We now use the optimality system (5.5.9) to obtain some additional identities. These will prove helpful in the numerical realisation of the application of Q, which is the key ingredient for the numerical procedure proposed in Remark 5.15.
Theorem 5.17. Let u ∈ `2(IIU) be any given control vector. Inserting u into (5.5.9a) and solving for y(u), and subsequently inserting this solution y(u) into (5.5.9b) and solving for p(y) = p(u), the residual of (5.5.6) is
Qu − g = ωR−1t u − ETp(u) . (5.5.10)
Proof. For the proof, we proceed in the opposite direction, resolving first the second equation for p(y), and then the first equation for y(u). Using the definition of Z and g from (5.5.2) and (5.5.5), we obtain
ETp(u) = −ET A−TTTRs(Ty(u) − y∗)
= −ZTR1/2s T(A−1f + A−1Eu) − y∗
= −ZTZu + g .
(5.5.11)
Using the definition of Q (5.5.5), the residual attains the form
Qu − g = (ZTZ + ωR−1t )u − g = ωR−1t u − ETp(u) , (5.5.12) which is just the defect in the design equation (5.5.9c).
In the proof we have inverted the relations (5.5.9a) and (5.5.9b) to arrive at an expression of the residual of (5.5.6). Concentrating on the calculation of Qu alone, we could shift the vector g to the right hand side of (5.5.10). However, the result can be obtained slightly faster by the direct use of (5.5.5) and (5.5.2a).
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Chapter 5. A Linear-Quadratic Elliptic Optimal Control Problem in Wavelet Coordinates
Corollary 5.18. Let u ∈ `2(IIU) be any given control vector, and substitute f = 0 and y∗ = 0 into (5.5.9a) and (5.5.9b) to obtain their homogeneous forms
Ay0(u) = Eu , (5.5.13a)
ATp0(u) = −TTRsTy0(u) , (5.5.13b) hence defining y0(u) and p0(u). Then it holds that
Qu = ωR−1t u − ETp0(u) . (5.5.14)
Using this route, we save three additions and subtractions of vectors, and eliminate the dependence on the data. Equations (5.5.13) and (5.5.14) thus contain the final recipe for the computation of Qu.
Other Equivalent Formulations
To illustrate the connections between the variables y, p and u in more detail, we present two different but equivalent formulations of the optimality system. These give rise to other classes of numerical algorithms, which are applicable for selected special cases.
The first derivation consists in the elimination of the control. While in Section 5.5.1 we have eliminated y from the functional, we can also use (5.5.9c) to eliminate u from the optimality system. Then only two equations remain, namely
Ay = f + ω−1ERtETp , (5.5.15a)
ATp = −TTRs(Ty − y∗) . (5.5.15b)
We can rewrite these in system form according to
TTRsT AT
This so called saddle point formulation [25] shows that the problem is essentially symmetric with respect to the state y and the adjoint p. However, it is no longer positive definite. Moreover, when the norms used in the objective functional are not natural, the matrices T and E introduce a diagonal scaling which lets the lowest eigenvalues of the contributions on the block diagonal go to zero.
Alternatively, we can formulate the optimality system (5.5.9) as one large block-matrix equation,
The symmetric block system matrix again has the structure of a saddle point system. Defining U :=ωR−1t 0
When T has full rank (which is only possible for distributed observation), U is invertible, and we may form the Schur complement H,
H := FU−1FT = ω−1ERtET + A(TTRsT)−1AT. (5.5.20) 100
5.5. Optimality Conditions
Because of the uniformly bounded condition numbers of Rs, Rt and A, this is spectrally equivalent to
H ∼ ω−1ETE + (TTT)−1. (5.5.21)
While the right part is bounded from below because of (5.4.4), it is only bounded from above for natural norms on the observation space, that is s = 1, and unbounded for s < 1.
In both of these equivalent formulations of the optimality system, the system matrix is not positive definite, or it does in general not have a uniformly bounded condition number. Bounded condition numbers can only guaranteed for the second example in the special case of natural norms. In contrast, the matrix Q from (5.5.5) is symmetric positive definite and of uniformly bounded condition number for all combinations of parameters. In this sense, an iterative method based on the application of Q is general and robust.
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Chapter 5. A Linear-Quadratic Elliptic Optimal Control Problem in Wavelet Coordinates
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Chapter 6
A Fast Wavelet Algorithm for the Control Problem
6.1 Introduction
The purpose of this thesis is to develop and implement a fast wavelet solver for a linear-quadratic elliptic optimal control problem. In earlier chapters, we have addressed the practical importance of this class of problems and motivated a discretisation with wavelets based on two specific qualities. Firstly, the wavelet discretisation leads to well-conditioned systems of equations, which permits the fast iterative solution of the problem and allows for several millions of unknowns on a standard PC. Secondly, wavelets offer a unified framework for the numerical evaluation of Sobolev norms in the objective functional, which yields greater freedom in modelling.
After we have proved in the preceding chapters that the matrices A, Rsand Rtoccuring in the optimality system, and the reduced matrix Q are uniformly well-conditioned, we provide here a detailed specification of an algorithm for uniform discretisations, and a collection of systematic numerical results from one to three spatial dimensions. We confirm that all involved constants are effectively controlled, and that the algorithm is of optimal complexity in the sense that the numerical solution is computed with an effort proportional to the number of unknowns.
In addition, we examine various combinations of modelling parameters, namely the regularisation param-eter ω, and the smoothness indices s and t for the observation and control spaces, see (5.4.13). We study their effects on the shape and character of the control u and the state y, and also their interplay with each other, which gives rise to a diversity of results.
As discussed in detail in the previous chapter, the necessary and sufficient conditions for an optimal discrete control u are formulated as the following linear system of equations,
Qu = g , (6.1.1)
where Q is symmetric positive definite. As pointed out before, the wavelet discretisation ensures that it is also well-conditioned. Therefore we use an outer loop of conjugate gradient iterations on u, for which the central operation consists in the application of the system matrix Q. Considering (5.5.2) and (5.5.5), one such application of the matrix Q involves the solution of three elliptic systems of equations over the matrices A, AT and Rt, respectively, which are again well-conditioned in the wavelet setting. This is also accomplished by CG iterations, which constitute the inner layer of the algorithm. We have schematically lined out this structure in Algorithm 6.1.
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Chapter 6. A Fast Wavelet Algorithm for the Control Problem
Subroutine Apply-Q (u): Computes Qu.
(1) Perform CG method for u → y0 (5.5.13a).
(2) Perform CG method for y0→ p0 (5.5.13b).
(3) Perform CG method for u → R−1t u.
(4) Compute Qu := ωR−1t u − ETp0 (5.5.14).
Algorithm Control-Generic (f , y∗) → u: Solves Qu = g.
(i) Initialisation: Compute g according to (5.5.5).
(1) Perform CG method for (f , y∗) → G (5.5.2b).
(2) Perform CG method for G → g (5.5.2a), (5.5.5).
(ii) Outer CG method: Set k := 0, u0:= 0.
(1) Call Apply-Q (u0) to set d0:= −q0:= g − Qu0. (2) Call Apply-Q (dk) → hk, Compute αk:= qdTkTqk
khk. (4) Compute uk+1:= uk+ αkdk, qk+1:= qk+ αkhk. (5) Compute βk:=qTk+1qTqk+1
kqk , dk+1:= −qk+1+ βkdk. (6) If not ready, set k := k + 1, continue with (2).
(iii) Accept uk → u.
Algorithm 6.1: We provide an overview of the generic solver Control-Generic for the optimal control problem in wavelet coordinates. It consists of a CG method for the reduced matrix Q. Each application of Q is performed in the subroutine Apply-Q and needs three inner CG methods, two to invert A and AT and one to invert Rt. We have deliberately kept the infinite-dimensional formulation of matrices and vectors over `2, since it is most general and the structure for arbitrary finite subspaces is analogous.
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