shown by Bock et al. [3], Hazra et al. [6] that for ...... Adsorption, 4:113â147, 1998. [11] A. Potschka, Hans Georg Bock, S. Engell, A. Küpper, and Johannes P.
A NEWTON-PICARD APPROACH FOR EFFICIENT NUMERICAL SOLUTION OF TIME-PERIODIC PARABOLIC PDE CONSTRAINED OPTIMIZATION PROBLEMS † , AND H.G. BOCK∗† ¨ A. POTSCHKA∗† , M.S. MOMMER† , J.P. SCHLODER
Abstract. We investigate an iterative method for the solution of time-periodic parabolic PDE constrained optimization problems. It is an inexact Sequential Quadratic Programming (iSQP) method based on the Newton-Picard approach. We present and analyze a linear quadratic model problem and prove optimal mesh-independent convergence rates. Additionally, we propose a two-grid variant of the Newton-Picard method. Numerical results for the classical and the two-grid variants of the Newton-Picard iSQP method as a solver and as a preconditioner for GMRES are presented. Key words. optimal control, periodic boundary condition, parabolic PDE, Newton-Picard, simultaneous optimization, inexact SQP, Krylov method preconditioner AMS subject classifications. 35K20, 65F08, 65F10, 65K05, 65N55
1. Introduction. Let Ω ⊂ Rd be a bounded open domain with Lipschitz boundary ∂Ω and let Σ := (0, 1) × ∂Ω. We look for controls u ∈ L2 (Σ) and corresponding states y ∈ W (0, 1) = {y ∈ L2 (0, 1; H 1 (Ω)) : yt ∈ L2 (0, 1; H 1 (Ω)∗ )} which solve the time-periodic PDE constrained optimization problem Z ZZ γ 1 2 (y(1; .) − yˆ) + u2 (1.1a) minimize J(y(1; .), u) := 2 Ω 2 u∈L2 (Σ),y∈W (0,1) Σ subject to
yt = D∆y,
in (0, 1) × Ω,
(1.1b)
yν + βy = αu,
in (0, 1) × ∂Ω,
(1.1c)
y(0; .) = y(1; .),
in Ω,
(1.1d)
with α, β ∈ L∞ (∂Ω) non-zero and positive a.e., and D, γ > 0. The work of Barbu [2] considers a similar problem with a more general formulation of the underlying linear parabolic PDE in a Hilbert-space valued ODE setting. However, existence and uniqueness results do not carry over to problem (1.1) because of the different choice of objective function which excludes the case of point evaluations of the state y at time t = 1. An analysis for periodic optimal control of the Boussinesq equation can be found in Trenchea [15]. Besides analytical results, we are interested in efficient numerical methods for problem (1.1) and their analysis. Most of the work on numerical methods for these problems can be found in the more application oriented literature which we shall describe in the following. For instance, many processes in chemical engineering are operated in a periodic manner, e.g., separation processes like Simulated Moving Bed (SMB) or Pressure Swing Adsorption (PSA), or reaction processes like the Reverse Flow Reactor. The models consist of partial differential equations of parabolic type. For the efficient simulation of periodic steady states it is important to exploit the inherent dynamics of dissipative processes, which are characterized by relatively low-dimensional dynamics even for problems with a high number of variables after ∗ This work was supported by the German Research Foundation (DFG) within the priority program SPP1253 under grant BO864/12-1 and by the German Federal Ministry of Education and Research (BMBF) under grant 03BONCHD. † Interdisciplinary Center for Scientific Computing, Im Neuenheimer Feld 368, 69123 Heidelberg, GERMANY. Corresponding authors: (potschka|mario.mommer)@iwr.uni-heidelberg.de
1
discretization. One typical example is the Newton-Picard method proposed by Lust et al. [9]. Numerical optimization of periodic processes is currently an active field of research. General purpose structure-exploiting direct optimal control techniques have been employed to optimize the SMB process [8, 14] and the PSA process [10]. In all these works, however, the inherent system dynamics were not exploited in the numerical computations. In van Noorden et al. [17], the Newton-Picard method is used to optimize periodic processes exploiting the inherent dynamics, however, the employed projected gradient descent method is used in a sequential mode, i.e., after each update of the control and design parameters the periodic steady state must be calculated to high accuracy, which makes the overall algorithm computationally expensive. It was shown by Bock et al. [3], Hazra et al. [6] that for similar problems in shape optimization for fluid dynamics a simultaneous method can save up to 70 percent of CPU time. The similarity between the problems is that an efficient iterative method (a solver) is available for the constraint of the optimization problem given control and design parameters. The savings can be gained from performing only one (or few) steps of the solver for each step of the optimization algorithm. One purpose of this article is to investigate a simultaneous optimization algorithm on problem (1.1). The model problem is an extension of the parabolic optimal control problem presented, e.g., in the textbook of Tr¨oltzsch [16]. The algorithm is based on the Newton-Picard method and thus exploits the inherent system dynamics. The resulting method belongs to the class of inexact Sequential Quadratic Programming (iSQP) methods (see, e.g., Griewank and Walther [5]). Preliminary numerical results of the Newton-Picard iSQP method for the non-linear SMB process are available in Potschka et al. [11]. We also present a variant of the Newton-Picard method, which relies on a two-grid approach. Multigrid approaches for parabolic optimal control problems have been studied by, e.g., Borz`ı [4]. Recently, Agarwal et al. [1] presented a reduced order model approach for the optimization of PSA, where a surrogate model based on Proper Orthogonal Decomposition (POD) is used in the optimization. We persue a different approach, where the reduced order model (generated by the NewtonPicard method) is only used for the generation of derivatives while still using the residuals of the full model. The article is organized as follows: In Section 2.1 we establish the existence and uniqueness of the optimal solution of the model problem (1.1). We discuss the optimality conditions for the model problem in a Hilbert space setting in Section 2.2 and investigate the regularity of the Lagrange multipliers in Section 2.3. In Section 3 we review the Newton-Picard method in a general setting of Newton-type methods in Banach space. We continue with a presentation of the Newton-Picard iSQP method for the model problem in Section 4. In Section 5 we discuss the discretization of the optimal control problem. In Section 6 we present the Newton-Picard iSQP method for the discretized problem in the case of a classical Newton-Picard projective approximation and of a coarse-grid approach for the constraint Jacobians. The importance of the choice of the scalar product for the projection is highlighted, a question that has so far not been addressed. We establish an optimal, mesh-independent convergence result for the Newton-Picard iSQP method with classical projective approximation and discuss its property of bounded retardation. We also outline the fast solution of the subproblems in this section. In Section 7 we present numerical results for different sets of problem parameters for the Newton-Picard iSQP method as such and as a preconditioner for GMRES. In Section 8 we give an outlook on how the Newton-Picard 2
iSQP method can be employed for non-linear optimization problems. 2. On solutions of the model problem. 2.1. Existence and uniqueness of the optimal solution. To show existence and uniqueness of the solution of problem (1.1), we use the approach persued in Tr¨ oltzsch [16]: We first prove existence of a linear, continuous “solution operator” S that maps a given control u to a solution y of the constraint system (1.1b)–(1.1d). Then we invoke the following theorem that can be found in Tr¨oltzsch [16], Satz 2.14– 2.17. Theorem 2.1 (Existence and uniqueness). Let real Hilbert spaces {U, k.kU } and {H, k.kH }, yˆ ∈ H, and γ > 0 be given. Suppose further that S : U → H is a linear, continuous operator. Then, the Hilbert space optimization problem γ 1 minimize kSu − yˆk2H + kuk2U u∈U 2 2 has a unique solution. We shall now construct the solution operator S : L2 (Σ) → H 1 (Ω) in three steps. Lemma 2.2. Let a pair of initial state and control (y(0; .), u) ∈ L2 (Ω) × L2 (Σ) be given. Then, there exists a unique solution y ∈ W (0, 1) to equations (1.1b)–(1.1c) and there exists a linear, continuous operator S˜ : L2 (Ω) × L2 (Σ) → H 1 (Ω) which maps (y(0; .), u) to the final state y(1; .). Proof. According to, e.g., Wloka [18], equations (1.1b)–(1.1c) have a unique solution y ∈ W (0, 1) for given y(0; .) ∈ L2 (Ω) and u ∈ L2 (Σ) and the solution operator SW : (y(0; .), u) 7→ y is a linear and continuous operator. Furthermore, y can be modified on a set of measure zero such that y ∈ C(0, 1; H 1 (Ω)). Thus, the trace operator T : y 7→ y(1; .) exists and is linear and continuous. Therefore, also the operator S˜ := T SW is linear and continuous. ˜ 0) in H 1 (Ω). For ease of presenIn the second step we show contractivity of S(., tation, define the linear, continuous operators G1 : L2 (Ω) → L2 (Ω) and G2 : L2 (Σ) → L2 (Ω) via ˜ 0) G1 := S(.,
˜ .). and G2 := S(0,
(2.1)
Lemma 2.3 (Contractivity). There exists κ < 1 such that for all v ∈ H 1 (Ω) kG1 vkH 1 (Ω) ≤ κkvkH 1 (Ω) . Proof. We start by investigating the spatial derivative terms of (1.1b)–(1.1c) with homogeneous Robin boundary condition (i.e., u = 0). They can be expressed in variational form as a symmetric bilinear form a : H 1 (Ω) × H 1 (Ω) → R Z a(ϕ, v) := −D
ϕ∆v Z Z =D ∇ϕ · ∇v − D Ω
Ω
Z ϕ(ν · ∇v) = D
∂Ω
Z ∇ϕ · ∇v + D
Ω
We define the operator A : H 1 (Ω) → H 1 (Ω) via Z Z hϕ, AviH 1 (Ω) := a(ϕ, v) = D ∇ϕ · ∇v + D Ω
3
βϕv. ∂Ω
∂Ω
βϕv.
(2.2)
The operator A is self-adjoint, its spectrum is real, and A has a set of orthogonal eigenvectors {ϕi } which form a basis of H 1 (Ω) (see, e.g., Renardy and Rogers [12]). Furthermore, A is coercive (see Tr¨ oltzsch [16], equation 2.17), i.e., there exists a constant ω > 0 such that for all v ∈ H 1 (Ω) hv, AviH 1 (Ω) ≥ ωkvk2H 1 (Ω) . For v = ϕi we obtain the lower bound λi ≥ ω > 0 for the eigenvalue λi . Thus, the Lumer-Phillips theorem (cf. Renardy and Rogers [12], Theorem 11.22) yields that the operator A generates a strongly continuous semigroup exp(−tA) which satisfies the inequality kexp(−tA)kH 1 (Ω) ≤ exp(−ωt). Because y(t; .) = exp(−tA)y(0; .) is another representation of the unique solution of equations (1.1b)–(1.1c) with u = 0, we finally obtain ˜ 0)kH 1 (Ω) = kexp(−A)vkH 1 (Ω) ≤ exp(−ω)kvkH 1 (Ω) = κkvkH 1 (Ω) , kG1 vkH 1 (Ω) = kS(v, with κ := exp(−ω) < 1. Remark 2.4. G1 = exp(−A) = exp(−A∗ ) = exp(−A)∗ = G∗1 is self-adjoint. To finish the proof of existence and uniqueness of the optimal solution, we use a fixed point argument to show existence and continuity of the solution operator S. Lemma 2.5. Let u ∈ L2 (Σ). Then, there exists a unique y0 ∈ H 1 (Ω) such that ˜ 0 , u) solves the system of constraints (1.1b)–(1.1d). Furthermore, the operator y = S(y S : L2 (Σ) → L2 (Ω) defined by S(u) := y0 is linear and continuous. Proof. We seek for a solution y0 of one of the three equivalent equations G1 y0 + G2 u − y0 = 0
⇔
y0 = G1 y0 + G2 u
⇔
y0 = (I − G1 )−1 G2 u.
(2.3)
The middle equation, Lemma 2.3, and the Banach Fixed Point Theorem yield the existence and uniqueness of y0 ∈ H 1 (Ω). The eigenvectors ϕi of A defined in equation (2.2) are also eigenvectors of I − G1 . Thus, the eigenvalues of I − G1 can be bounded away from zero by 1 − κ in modulus. This yields kSukL2 (Ω) ≤ kSukH 1 (Ω) = k(I − G1 )−1 G2 ukH 1 (Ω) ≤
kG2 kH 1 (Ω) kukL2 (Σ) . 1−κ
(2.4)
Hence, S : L2 (Σ) → L2 (Ω) is a linear and continuous operator (even into H 1 (Ω)). Corollary 2.6. The optimization problem (1.1) has a unique optimal solution. Proof. Lemma 2.5 and Theorem 2.1. 2.2. Optimality conditions. We eliminate constraints (1.1b) and (1.1c) from problem (1.1) via S˜ from Lemma 2.2 and use the Lagrange-formalism to derive necessary optimality conditions. By convexity they are also sufficient. For notational convenience we shall from now on write y instead of y(0; .). Define the Lagrangian L : L2 (Ω) × L2 (Σ) × L2 (Ω) → R via L(y, u, λ) = J(y, u) + hλ, (G1 − I) y + G2 uiL2 (Ω) .
(2.5)
Regarding the chosen spaces for the Lagrangian, we are now looking for states in the larger space L2 (Ω) instead of H 1 (Ω). This does not introduce additional solutions to 4
the original problem because y ∈ H 1 (Ω) is ensured via the periodicity constraint (2.1). The optimality condition is then L0 (y, u, λ)(δy, δu, δλ) = 0,
∀δy ∈ L2 (Ω), δu ∈ L2 (Σ), δλ ∈ L2 (Ω).
A simple calculation yields L0 (y, u, λ)(δy, δu, δλ) = hy − yˆ, δyi + γhu, δui + hλ, (G1 − I) δy + G2 δui + hδλ, (G1 − I) y + G2 ui. Written in block-operator form, we obtain the Karush-Kuhn-Tucker system I 0 G∗1 − I y yˆ 0 γI G∗2 u = 0 . G1 − I G2 0 λ 0
(2.6)
The block-operator on the left-hand side of equation (2.6) shall be denoted by K : L2 (Ω) × L2 (Σ) × L2 (Ω) → L2 (Ω) × L2 (Σ) × L2 (Ω). 2.3. Regularity of the Lagrange multiplier. We are going to show that the Lagrange multiplier of the periodicity constraint inherits higher regularity from the target yˆ. ¯ ∈ L2 (Ω) be the optimal Theorem 2.7. Let the subset E ⊂ Ω be open and λ ¯ ∈ H 1 (E). Lagrange multiplier of problem (1.1). If yˆ E ∈ H 1 (E), then λ E Proof. Let the optimal state be denoted by y¯ ∈ H 1 (Ω). From the first line of ¯ the fixed point equation KKT system (2.6), we obtain for λ ¯ = G∗ λ ¯ λ y − y¯). 1 − (ˆ We construct a sequence λk+1 = G∗1 λk − (ˆ y − y¯), which converges in L2 (Ω) due to Remark 2.4 and Lemma 2.3. Let now E ⊂ Ω be open and such that yˆ E ∈ H 1 (Ω). Then, the sequence (λk E ) converges in H 1 (E). ¯ ∈ H 1 (E). Thus, λ E 3. The Newton-Picard method for general fixed point problems. Let V be a Banach space and Φ : V → V be a possibly non-linear continuously Fr´echet differentiable mapping with derivative Φ0 ∈ L(V, V ). We are interested in finding a solution v ∈ V of the fixed point equation v = Φ(v).
(3.1)
If v¯ ∈ V is a solution of equation (3.1), and if furthermore Φ0 (¯ v ) has a spectral radius of κ < 1, then there exists a neighborhood U of v¯ such that for each v0 ∈ U the sequence (vk ) defined by vk+1 := Φ(vk )
(3.2)
converges to v¯ with an asymptotic linear convergence rate of κ. A different way of solving equation (3.1) is to use Newton’s method for finding a root of the problem Φ(v) − v = 0. 5
(3.3)
This leads to the iteration vk+1 = vk − (Φ0 (vk ) − I)
−1
(Φ(vk ) − vk ) ,
(3.4)
which is also known to converge to v¯ in a neighborhood U of v¯ but with quadratic convergence rate and without the need for contractive Φ0 (¯ v ). From a computational point of view, the evaluation of the derivative Φ0 (vk ) and the inversion of Φ0 (vk ) − I may be prohibitively expensive. The cost for evaluation and inversion can be reduced by approximating (Φ0 (vk ) − I)−1 ≈ Jk suitably, resulting in the iteration vk+1 = vk − Jk (Φ(vk ) − vk ) .
(3.5)
This leads to the class of so-called Newton-type methods, which do not exhibit quadratic local convergence any more. By approximating Φ0 (vk ) with zero, we see that the fixed point recurrence (3.2) can also be interpreted as a Newton-type method with Jk = −I. For fixed u, the periodicity equation (2.3) is of the described form with Φ0 (vk ) = G1 . The operator G1 maps the initial state values to the end values of the solution of the dynamic system. Thus, the iterate vk of the fixed point iteration (3.2) corresponds to a continuous integration of the system dynamics over k periods. This iteration is called Picard method. It was proposed by Lust et al. [9] to use a low-rank projective approximation of Φ0 (vk ). It can be shown that this is equivalent with using Newton’s method on a (hopefully small) subspace of “slow” modes (in the sense of contraction of the fixed point iteration (3.2)) and performing pure Picard iteration on the anyway “fast” remaining modes, leading to an overall fast linear convergence with cheap evaluation of Jk . 4. Newton-Picard for optimal control problems: The Newton-Picard inexact Sequential Quadratic Programming method. In this section we investigate how the Newton-Picard method for the forward problem (i.e., solving for a periodic state for given controls) can be exploited in a so-called “simultaneous optimization” approach as opposed to a “sequential optimization” approach. In the sequential approach, one completely eliminates the states from the optimization problem, because for each control u the Newton-Picard method yields a corresponding state y(u) which satisfies the periodicity constraint (1.1d). The resulting unconstrained problem is solved by standard descent methods, e.g., gradient methods or variants of Newton’s method, yielding only feasible iterates. The main drawback of this approach is that a lot of effort (i.e., Newton-Picard iterations) is spent for resolving the system dynamics far away from the solution. The sequential approach does not have the property of bounded retardation, i.e., the numerical effort for solving the constrained optimization problem is not bounded by a small constant times the effort spent on solving one forward problem. The simultaneous approach tries to overcome this disadvantage by performing only a small number of steps (ideally only one) of the solver for the forward problem per optimization step. The intermediate iterates are not feasible with respect to periodicity any more, but each iterate is less expensive than in the sequential approach. The simultaneous approach has been used successfully in, e.g., one-shot methods for shape optimization in fluid dynamics [3, 6]. The simultaneous Newton-Picard optimization approach can be written as an inexact Sequential Quadratic Programming (iSQP) method with approximated con6
˜ 1 denote the approximation of G1 and regard the approxistraint derivatives. Let G mated KKT system ˜∗ − I I 0 G 1 ˜ := 0 γI G∗2 . K ˜ 1 − I G2 G 0 ˜ 1 = 0 is used. The inexact SQP method generates In the case of pure Picard, G iterates via yk+1 yk yk yˆ ˜ −1 K uk − 0 . uk+1 = uk − K (4.1) λk+1 λk λk 0 ˜ − K. The contraction rate is determined Let ∆K = K ˜ ∗ − I −1 0 I 0 G 1 ˜ −1 ∆K = 0 γI G∗2 0 K ˜ 1 − I G2 ∆G1 G 0
by the spectral radius of 0 ∆G∗1 0 0 , 0 0
˜ 1 − G1 . where ∆G1 = G 5. Discretization of the optimal control problem. First, we discretize the controls in space with nu form functions ql whose amplitude can be controlled in time, i.e., u(t, x) =
nu X
ul (t)ql (x),
ul ∈ L2 (0, 1), ql ∈ L2 (∂Ω).
l=1
We continue with discretizing the initial state y using Finite Elements. Let ϕi denote the i-th Finite Element basis function and define the following Finite Element matrices and vectors: Z Z Z Sij = D ∇ϕi · ∇ϕj , Qij = D βϕi ϕj , Bil = D αϕi ql , ∂Ω Z Ω Z ∂Ω ˆi = Mij = ϕi ϕj , y yˆϕi . Ω
Ω
We can now discretize the PDE with the Method Of Lines: The matrix of the discretized spatial differential operator of equation (2.2) is L = S + Q, which leads to the ODE ˙ M y(t) = Ly(t) + Bu(t), Pny
(5.1)
where y(t) = i=1 y i (t)ϕi . Then, we discretize ul (t) using piecewise constant functions on the equidistant grid with grid size τ = 1/m, which leads to a full discretization of the operators G1 and G2 defined in equation (2.1). Let ψi , i = 1, . . . , nu m, denote a basis of the discrete control space and define the control mass matrix ZZ Nij = ψi ψj . Σ
7
We arrive at the following finite dimensional linear-quadratic optimization problem: minimize
1
y∈Rny ,u∈Rnu m 2
ˆ T y + γuT N u yT M y − y
subject to (G1 − I)y + G2 u = 0. Lemma 5.1. Problem (5.2) has a unique solution. Proof. The optimality conditions of (5.2) are ˆ y M 0 GT y 1 −I u = 0 . 0 γN GT 2 z 0 G1 − I G2 0
(5.2a) (5.2b)
(5.3)
The constraint Jacobians have full rank due to G1 − I being invertible. The Hessian blocks M and γN are positive definite. Thus, the symmetric indefinite linear system (5.3) is non-singular and, thus, has a unique solution. Lemma 5.2. The Finite Element approximation of the Lagrange multiplier λ is given by λ = M −1 z. Proof. The finite dimensional Lagrange multiplier z is a Riesz representation of an element from the dual space of Rny with respect to the Euclidian scalar product. To recover a Riesz representation of the Lagrange multiplier in an L2 sense from z, one has to find λ such that hλ,
ny X
v i ϕi iL2 (Ω) = z T v,
for all v ∈ Rny .
i=1
Thus, one can recover a Finite Element approximation λ of the Lagrange multiplier λ via λ = M −1 z. Remark 5.3. Fast convergence of the Finite Element approximation can only be ¯ ∈ H 1 (Ω). expected for λ 6. The Newton-Picard iSQP algorithm. 6.1. General considerations. Solving equation (5.3) by a direct method is prohibitively expensive for large values of ny because the matrix G1 is a large, dense ny -by-ny matrix. Clearly, iterative methods have to be employed. We observe that matrix-vector products are relatively economical to evaluate: The cost of an evaluation of G1 v is the cost of a numerical integration of the ODE (5.1) with initial value v and zero controls. The evaluation of GT 1 v can be computed from the solution ζ(0) of the adjoint end value problem ζ˙ = −LM −1 ζ,
ζ(1) = v,
which can be transformed into the original ODE (5.1) due to the symmetry of M and ˜ ˜ L with the transformation ζ˜ = M −1 ζ. This yields GT 1 v = M ζ(1) where ζ solves the initial value problem ˜˙ = Lζ, ˜ Mζ
˜ ζ(0) = M −1 v.
2 Define the inner product hv 1 , v 2 iM := v T 1 M v 2 , which is a discrete version of the L inner product. We observe:
8
Lemma 6.1. The matrix G1 is symmetric with respect to h., .iM and there exists an M -orthonormal matrix Z ∈ Rny ×ny consisting of eigenvectors of G1 with real eigenvalues µi , i = 1, . . . , ny , i.e., G1 Z = Z diag(µi ),
Z T M Z = I.
Proof. The M -symmetry of G1 follows from −1 ˜ GT v 1 v = M ζ(1) = M G1 M
⇔
˜ = M G1 v ˜, GT 1 Mv
and the Spectral Theorem completes the proof. 6.2. Discretized version of the Newton-Picard method for the model ˜ 1 , which problem. The G1 blocks of system (5.3) are substituted by approximations G ˜ have the property that products with G1 − I and its inverse are cheap to compute. The iteration is then k k ˜T − I ˆ ∆y y M 0 G M 0 GT y 1 1 −I ∆uk = 0 0 uk − 0 , γN GT γN GT 2 2 ˜ 1 − I G2 0 G1 − I G2 0 ∆z k zk G 0 (6.1a) k+1 k k y y ∆y uk+1 = uk − ∆uk . (6.1b) z k+1 zk ∆z k ˜ 1 : The first is based on the classical NewtonWe shall investigate two choices for G Picard projective approximation for the forward problem, the second is based on a two-grid idea. 6.2.1. Classical Newton-Picard projective approximation. The principle of the Newton-Picard approximation is based on observations about the spectrum of the monodromy matrix G1 (see Figure 7.1). The eigenvalues µi cluster around zero and there are only few eigenvalues that are close to the unit circle. The cluster is a direct consequence of the compactness of the infinite dimensional operator G1 . Let the range of the orthonormal matrix V ∈ Rny ×p be spanned by the p eigenvectors of G1 with largest eigenvalues µi such that E ∈ Rp×p .
G1 V = V E,
Now, the monodromy matrix is approximated with ˜ 1 = G1 Π, G where Π is a projector onto the dominant subspace of G1 . Lust et al. [9] proposed to use Π = V V T, which is an orthogonal projector in the Euclidean sense. This works well for the solution of the pure forward problem but inside the iSQP framework, this choice may lead to undesirable loss of contraction. We propose to use a projector that instead takes the scalar product of the infinite dimensional space into account. The projector 9
maps a vector w to the closest point V v of the dominant subspace in an L2 sense, by solving the minimization problem 1 1 1 minimize kw − vk2L2 (Ω) = v T V T M V v − v T V T M w + wT w, v 2 2 2 P P where w = wi ϕi and v = (V v)i ϕi . The projector is therefore given by Π = V V TM V
−1
V T M.
(6.2)
Thus, we approximate G1 in equation (6.1a) with ˜1 = V E V TM V G
−1
V T M.
˜ 1 − I is then given by The inverse of G ˜ 1 − I)−1 = V (E − I)−1 + I V T M V −1 V T M − I, (G which only needs the inversion of a small p-by-p system and of the projected mass matrix. In contrast to the first choice, this formulation also allows for the use of non-orthonormal but full-rank V . The dominant subspace basis V can be determined by computing the p-dimensional dominant eigenspace of M −1 L by solving the (generalized) eigenvalue problem ˜=0 M −1 LV˜ − V˜ E
⇔
˜ = 0. LV˜ − M V˜ E
The matrix exponential for calculating the fundamental system of ODE (5.1) yields ˜ = V˜ exp(E) ˜ =: V E. G1 V˜ = exp(M −1 L)V˜ = exp(V˜ E) Thus, V = V˜ and the dominant eigenvalues are simply µi = exp(˜ µi ), ˜ where µ ˜i are the eigenvalues of E. 6.2.2. Two-grid Newton-Picard. This variant is based on the observation that for parabolic problems the slow-decaying modes are the low-frequency modes and the fast-decaying modes are the high-frequency modes. Low-frequency modes can be ˜1 approximated well on coarse grids. Thus, we propose a method with two grids: G is calculated only on a coarse grid. The remaining calculations are performed on the fine grid. Let P and R denote the prolongation and restriction matrices between the two grids and let superscripts c and f denote coarse and fine grid, respectively. Then, Gf1 is approximated by ˜ f = P Gc R, G 1 1 i.e., we first project from the fine grid to the coarse grid, evaluate the exact Gc1 on the coarse grid, and prolongate the result back to the fine grid. We use nested grids, i.e., the Finite Element basis on the coarse grid can be represented exactly in the FE basis on the fine grid. Thus, the prolongation P is obtained by interpolation. We define the restriction R in an L2 sense, such that given 10
uf =
Pnfy
uc =
Pncy
i=1
ufi ϕfi on the fine grid we look for the projector R : uf 7→ uc such that with
i=1
uci ϕci it holds that hϕci , uc iL2 (Ω) = hϕci , uf iL2 (Ω)
for all i = 1, . . . , ncy ,
or, equivalently, M c uc = P T M f uf . It follows that R = (M c )
−1
P TM f.
Due to P being an exact injection, it follows that P T M f P = M c and thus RP = I. ˜ f − I is given by The inverse of G 1
˜ f1 − I G
−1
h i −1 = P (Gc1 − I) + I R − I,
which can be computed by only an inversion of a ncy -by-ncy matrix from the coarse grid. 6.3. Convergence of the classical Newton-Picard iSQP method. In this section we show that the Newton-Picard iSQP method for the model problem converges at least as fast as the classical Newton-Picard method for the pure forward problem and that the convergence rate is grid independent in the sense of the following definitions. Let X be a Banach space. A family of finite dimensional problems (Ph , w ¯h )h∈R+ with unique solutions w ¯h ⊂ X is said to be a consistent discretization of an infinite dimensional problem (P0 , w ¯0 ) with unique solution w ¯0 ∈ X if and only if lim kw ¯h − w ¯0 kX = 0.
h→0
Let an iterative method for a consistent discretization (Ph , w ¯h ) of (P0 , w ¯0 ) be given. Denote its iterates by (whk )k∈N . The iterative method is said to have a mesh independent linear convergence rate if there exists a constant κ < 1 independent of h, ¯ > 0, and k¯ ∈ N such that for all h < h ¯ and all k ≥ k¯ h h kwkh − w ¯ h kX ≤ κkwk−1 −w ¯ h kX .
We state the central theorem of this section and defer the proof for later. Theorem 6.2. Let µi , i = 1, . . . , ny , denote the eigenvalues of G1 ordered in descending modulus, let p ≤ ny , and let µp > µp+1 . Then, the Newton-Picard inexact SQP method for the model problem converges with a contraction rate of at most µp+1 . The main result of this section is now at hand: Corollary 6.3. The contraction rate of the Newton-Picard iSQP method is mesh independent. Proof. For finer discretizations the eigenvalue µp+1 converges towards an eigenvalue µ ¯p+1 of the infinite dimensional operator G1 . Theorem 6.2 then yields that for every ε > 0 the assumption of Definition 6.3 is satisfied with κ = µ ¯p+1 + ε. 11
˜ − K denote the difference between the approxiIn the remainder, let ∆K = K mated and the exact KKT matrix. For the proof of Theorem 6.2 we need the following lemma whose technical proof we defer until the end of this section. The lemma asserts the existence of a variable transformation which transforms the Hessian blocks to identity, and furthermore reveals the structure of the matrices on the subspaces of fast and slow modes. Lemma 6.4. Let p ≤ ny , and let EV = diag(µi , i = 1, . . . , p) and EW = diag(µi , i = p+1, . . . , ny ). Then, there exist matrices V ∈ Rny ×p and W ∈ Rny ×(ny −p) such that with Z = V W the following conditions hold: (i) Z is a basis of eigenvectors of G1 , i.e., G1 Z = V EV W EW . (ii) Z is M -orthonormal, i.e., Z T M Z = I. (iii) There exists a non-singular matrix T such that −1 ˜ −1 ∆KT = T −1 K ˜ −1 T −T T T ∆KT = T T KT ˜ T −1 K T T ∆KT = −1 I 0 0 0 −I 0 0 0 0 −EW 0 0 I 0 EV − I 0 0 0 0 0 T T 0 0 γN G2 M V G2 M W 0 0 0 0 0 . 0 0 EV − I V T M G2 0 0 0 0 0 0 −EW 0 0 0 0 −I 0 W T M G2 0 0 Proof of Theorem 6.2. The contraction rate is given by the spectral radius ˜ −1 ∆K). We can use the similarity transformation with T given by Lemma 6.4 σr ( K to obtain the eigenvalue problem −1 ˜ T T KT T T ∆KT v − σv = 0, which is equivalent to solving the generalized eigenvalue problem ˜ v = 0. −T T ∆KT v + σT T KT We assume that there is an eigenpair (v, σ) such that |σ| > µp+1 . Division by σ yields the system (1/σ)EW v 5 + v 1 − v 5 = 0,
(6.3a)
v 2 + (EV − I) v 4 = 0,
(6.3b)
γN v 3 +
GT 2M
(V v 4 + W v 5 ) = 0,
(6.3c)
T
(6.3d)
T
(6.3e)
(EV − I) v 2 + V M G2 v 3 = 0, (1/σ)EW v 1 − v 1 + W M G2 v 3 = 0,
where v was divided into five parts v 1 , . . . , v 5 corresponding to the blocks of the system. With the assumption on |σ| we obtain invertibility of I − (1/σ)EW and thus we can eliminate −1
v 5 = (I − (1/σ)EW ) v 4 = (I − EV )
−1
v2 ,
v 2 = (I − EV )
−1
T
v1 ,
(6.4b)
V M G2 v 3 , −1
v 1 = (I − (1/σ)EW ) 12
(6.4a)
T
V M G2 v 3 .
(6.4c) (6.4d)
Resubstituting these in equation (6.3c) yields
γN + GT 2 M V (I − EV )
−2
V T M G2 + GT 2 M W (I − (1/σ)EW )
−2
W T M G2 v 3 = 0.
The matrix on the left hand side is symmetric positive definite and thus it follows that v 3 = 0, which implies v = 0 via equations (6.4). Thus, (v, σ) cannot be an eigenpair. The only thing that remains to be proven is Lemma 6.4. Proof of Lemma 6.4. The existence of the matrices V and W , as well as conditions (i) and (ii) follow from Lemma 6.1. To show (iii), we choose W T =0 0
V 0 0
0 I 0
0 0 MV
0 0 . MW
Due to M -orthonormality (ii) of V , the Newton-Picard projector from equation (6.2) simplifies to Π = V V T M . Using V T M W = 0, V T M V = I, and GT 1 M V = M G1 V = M V EV we obtain W V 0 0 0 0 0 ΠT − I GT 1 T T 0 0 I 0 0 0 0 0 T ∆KT = T 0 0 0 MV MW G1 (Π − I) 0 0 T W 0 0 V T 0 0 0 0 0 −M G1 W 0 0 0 0 0 0 I 0 = 0 0 0 0 V T M −G1 W 0 0 0 0 0 W TM 0 0 0 0 −EW 0 0 0 0 0 0 0 0 0 0 = . 0 0 0 0 0 −EW 0 0 0 0 ˜ the form Similarly, we obtain for K M 0 M V V T GT W V 0 0 0 1 −I ˜ = TT 0 0 I 0 0 0 γN GT T T KT 2 T 0 0 0 MV MW G1 V V M − I G2 0 T W 0 0 V T 0 MV 0 M V (EV − I) −M W 0 MW 0 I 0 0 γN GT GT = 2 MV 2 MW 0 T 0 0 0 0 V M −W V (EV − I) G2 0 0 W TM I 0 0 0 −I 0 I 0 EV − I 0 T T . 0 0 γN G M V G M W = 2 2 0 EV − I V T M G2 0 0 −I 0 W T M G2 0 0 13
6.4. Numerical solution of the approximated KKT system. The solution of the step equation (6.1a) can be carried out by block elimination. To simplify notation we denote the residuals by r i . We want to solve ˜T − I M 0 G y r1 1 0 u r2 . γN GT = 2 ˜ 1 − I G2 z r3 G 0 Compute the nu m-by-nu m symmetric positive-definite matrix −1 −1 T ˜ ˜ B = γN + GT G − I M G − I G2 . 1 2 1 Then, −1 −1 T ˜ ˜ Bu = r 2 − GT G − I r − M G − I r3 , 1 1 2 1
(6.5)
which can be solved for u via Cholesky decomposition of B if nu m is small. An alternative is a suitably preconditioned Conjugate Gradient method, which is not investigated further in this article. Solving for y and z is then simple: −1 ˜1 − I y= G (r 3 − G2 u) , (6.6) −1 ˜T z= G (r 1 − M y) . (6.7) 1 −I ˜ 1 (in a suitable representation) are calculated, the step Note, that once G2 and G equation (6.1a) can be solved without further numerical integration of the system dynamics. 6.5. Bounded retardation. Theorem 6.2 states that the Newton-Picard inexact SQP method has a contraction rate of at least as fast as the classical NewtonPicard method for the pure forward problem. For the optimization problem, one needs the evaluation of one adjoint per iteration. Thus, the Newton-Picard inexact SQP method has a retardation factor of 2 for the model problem if the numerical effort for the computation of matrix G2 and for the solution of the necessary eigenvalue problems is not taken into account. In the general context, this is of course an unreasonable assumption. One has to add the effort for a solve of equation (6.5) and two additional evaluations of GT 2 and G2 for the residual of equation (6.5) and in equation (6.6). 7. Numerical results. 7.1. General parameters and methods. The calculations were performed on Ω = [−1, 1]2 . We varied the diffusion coefficient in D ∈ {0.1, 0.01, 0.001} which results in problems with almost only fast modes for D = 0.1 and problems with more slow modes in the case of D = 0.001. The functions α and β were chosen identically to be a multiple of the characteristic function of the subset Γ = Γ1 ∪ Γ2 := {1} × [−1, 1] ∪ [−1, 1] × {1} ⊂ ∂Ω, with α = β = 100χΓ . Throughout, we used the two boundary control functions q1 (x) = χΓ1 (x),
q2 (x) = χΓ2 (x). 14
1 0.8
µi
0.6 0.4 0.2 0
0
50
100 i
150
200
Fig. 7.1. The eigenvalues µi of the spectrum of the monodromy matrix G1 decay exponentially fast. Only few eigenvalues are greater than 0.5. Shown are the first 200 eigenvalues calculated with D = 0.01 and β = 100χ(Γ) on a grid of 8-by-8 elements of order K = 5. KKT system size GMRES iterations
442 2083 3562 7642 223 294 327 342 Table 7.1 Number of unpreconditioned GMRES iterations to a residual tolerance of 10−6 for D = 0.01, β = 100χ(Γ), γ = 0.00005.
In other words, the two controls act each uniformly on one edge Γi of the domain. With γ = 0.00005, we chose the regularization parameter rather small. In order to see the difference in the regularity of the Lagrange multipliers between smooth and discontinuous target functions we used the two target functions yˆsmooth (x) = (2 + cos(x1 π/2) + cos((1 − x2 )π/2)) /4, yˆdiscont (x) = 1 + χ[0,1]×[−1,0] (x) /2. We discretized the controls in time on an equidistant grid of m = 100 intervals. For the discretization of the initial state y we use quadrilateral high-order nodal Finite Elements. The reference element nodes are the Cartesian product of the GaussLobatto nodes on the 1D reference element. Let ϕi denote the i-th nodal basis function. We used part of the code which comes with the book of Hesthaven and Warburton [7], and extended the code with continuous elements in addition to discontinuous elements. The evaluations of matrix-vector products with G1 and G2 were obtained from the NDF time-stepping scheme implemented in ode15s [13], which is part of the commercial software package MATLABr , with a relative integration tolerance of 10−11 . A typical spectrum of the monodromy matrix G1 can be seen in Figure 7.1. ˜ 1 are calculated directly from the fundamental system projected The approximations G on the slow modes or on the coarse grid. Table 7.1 shows slow, mesh-dependent convergence of unpreconditioned GMRES on the KKT system of the discretized model problem. 7.2. Solutions. Figure 7.2 shows the solution triples (y, u, λ) for the smooth and the discontinuous target function. One can observe the smoothness of the Lagrange multiplier in the regions where the target function is smooth. The jump in yˆdiscont leads to a jump in the Lagrange multiplier at the same location. 15
Fig. 7.2. Optimal controls u (top row), optimal states y (middle row), and corresponding Lagrange multipliers λ (bottom row) for the smooth target function yˆsmooth (left column) and the discontinuous target function yˆdiscont (right column), calculated for D = 0.01 on a grid of 32-by-32 elements of order 5. The displayed meshes are not the finite element meshes but evaluations of the Finite Element basis functions on a coarser equidistant mesh.
7.3. Euclidean vs. L2 projector. Figure 7.3 summarizes the spectral prop˜ −1 K. The erties of the discretized KKT-system and of the iteration matrices I − K spectrum of the iteration matrix can also be interpreted as the deviation of the preconditioned KKT matrix from the identity. The discretization with 4-by-4 elements of order 5 is moderately fine in order to shorten the computation of the spectra. One sees that the spectrum of K is spread between 10−6 and 101 without larger gaps, which is the reason for the poor performance of unpreconditioned GMRES on this 16
Fig. 7.3. Top row: Spectrum of KKT-Matrix K, with diffusion coefficient D = 0.01, discretized on a 4-by-4 grid of 5-th order elements (441 Dof ). Middle row: Unit circle and spectrum of iteration matrix for the classical Newton-Picard with p = 10 using Euclidean projector (left column) and L2 projector (right column). Bottom row: Like middle row with p = 50.
special problem. The figure also shows how the appropriate choice of the projector for the Newton-Picard approximation leads to fast convergence which is monotonic in p. Both the Euclidean and the L2 projector eliminate many large eigenvalues, but the Euclidean projector leaves out a few large eigenvalues. These eigenvalues can even lead to divergence, e.g., in the case p = 50. This behavior is further demonstrated in Figure 7.4. Numerically we observe that the Euclidean projector leads to a nonmonotone behavior of the contraction rate with respect to the subspace dimension, 17
Fig. 7.4. Contraction of the iSQP iteration versus psub for the Euclidean projector (left) and the L2 projector. Note that only the right plot is in logarithmic scale.
and also exhibits clear plateaus. The L2 projector leads to an exponential decay of the contraction rate with respect to the subspace dimension and is by far superior to the Euclidean projector. Thus, only the L2 projector will be considered further. 7.4. Mesh independence. Tables 7.2 and 7.3 show the number of iterations of the classical Newton-Picard iSQP method for diffusion coefficients of D = 0.1 and D = 0.01, respectively, with respect to the subspace dimension. The tables show the contraction rate of the Newton-Picard iSQP method and the number of iterations for pure Newton-Picard iSQP and for a GMRES method that was preconditioned by ˜ The stopping criterion for GMRES was the Newton-Picard KKT approximation K. −8 set to 10 for the residual. The Newton-Picard iSQP method was stopped when the increment was smaller than 10−8 , which slightly favors GMRES in the iterations counts because the residual norm was always below the increment norm. Tables 7.4, 7.6, and 7.5 show the number of iterations of the Newton-Picard iSQP method for diffusion coefficients of D = 0.1, D = 0.01, and D = 0.001, respectively, with respect to the coarse grid discretizations. The same termination criteria were used. We observe the following: The number of Newton-Picard iterations does not depend on the dimension of the discretized problem, i.e., the contraction rate is mesh independent. Using Newton-Picard as a preconditioner for GMRES results in an almost mesh independent method: We observe only a a slight increase of iterations with respect to the dimension of the discretized problem. To further compare the NewtonPicard iSQP method with Newton-Picard PCGMRES, we observe that for rather low subspace dimensions or rather coarse meshes, PCGMRES needs less iterations than Newton-Picard iSQP. On the other hand, if we use higher subspace dimensions or finer coarse grids, the Newton-Picard iSQP needs less iterations than PCGMRES. In that sense, the Newton-Picard iSQP method can be regarded as a solver rather than ˜ 1 = −I, a preconditioner. Interestingly, pure Picard preconditioning, i.e., p = 0 and G results in an extremely cheap preconditioner because no eigenvalue problems have to be solved, while the decrease in the number of iterations compared to unprecon18
Subspace GMRES Newton-Picard dimension iters iters contr 0 28 77 0.75 0 29 77 0.75 0 33 77 0.75 1 24 35 0.51 1 26 35 0.51 1 27 35 0.51 2 21 21 0.33 2 21 21 0.33 2 25 21 0.33 5 16 13 0.16 5 16 13 0.16 5 18 13 0.16 10 12 8 0.035 10 12 8 0.035 10 13 8 0.035 20 7 5 0.0015 20 8 5 0.0015 20 8 5 0.0015 50 7 3 8.4e-05 50 7 3 2.2e-05 50 7 3 2.6e-05 Table 7.2 Mesh independence of classical Newton-Picard iSQP method for different choices of the subspace dimension, D = 0.1. Elms 16 64 256 16 64 256 16 64 256 16 64 256 16 64 256 16 64 256 16 64 256
Ord 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
Dof 441 1681 6561 441 1681 6561 441 1681 6561 441 1681 6561 441 1681 6561 441 1681 6561 441 1681 6561
ditioned GMRES is large. In general, already low dimensions p for the dominant subspace approximation and rather low-dimensional coarse grids lead to fast convergence rates of the Newton-Picard iSQP method and Newton-Picard PCGMRES. The two-grid Newton-Picard iSQP method is more efficient than the classical NewtonPicard method because computation of full constraint Jacobians on an appropriate coarse grid is cheaper than solving eigenvalue problems on the fine grid. Especially for low diffusion coefficients the two-grid version can yield solutions much faster than the classical Newton-Picard version. 8. Outlook: Non-linear problems. The Newton-Picard inexact SQP method was demonstrated on a linear-quadratic model problem in this article but the method is well-suited also for non-linear problems. The biggest difference is that the projector Π changes due to changes of the dominant subspace of G1 (.), and that G2 (.) changes from iteration to iteration. The technique of using one to few subspace iterations per Newton-Picard iteration to update Π was proposed in Lust et al. [9]. For the two-grid version of Newton-Picard, this technical difficulty vanishes entirely. 9. Conclusions. We have investigated existence, uniqueness, and regularity of the solution of a time-periodic parabolic PDE constrained optimization problem and have analyzed the numerical performance of different versions of the Newton-Picard inexact SQP method applied to the model problem. We have shown that the choice of the discrete L2 scalar product instead of the Euclidean scalar product in the projector of the Newton-Picard method is crucial to obtain fast convergence. We have proved mesh-independet contraction of the method with the L2 projector. We have also proposed a novel two-grid version of the Newton-Picard inexact SQP method and have shown that it is numerically more efficient than the classical Newton-Picard inexact SQP method. Additionally, we have shown numerically that Newton-Picard 19
Subspace GMRES Newton-Picard dimension iters iters contr 0 74 >1000 0.98 0 79 >1000 0.98 0 85 >1000 0.98 1 62 290 0.94 1 68 294 0.94 1 71 301 0.94 2 58 243 0.92 2 63 244 0.92 2 69 246 0.92 5 43 100 0.84 5 45 111 0.84 5 49 114 0.84 10 33 61 0.73 10 36 60 0.73 10 38 60 0.73 20 23 29 0.52 20 24 29 0.52 20 26 30 0.52 50 16 14 0.22 50 17 14 0.22 50 17 13 0.22 100 10 8 0.038 100 11 8 0.041 100 13 8 0.041 Table 7.3 Mesh independence of classical Newton-Picard inexact SQP method for different choices of the subspace dimensions, D = 0.01. Elms 16 64 256 16 64 256 16 64 256 16 64 256 16 64 256 16 64 256 16 64 256 16 64 256
Ord 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
Coarse grid GMRES Newton-Picard Elms Ord Dof iters iters contr 4 2 25 15 10 0.099 4 2 25 16 10 0.099 4 2 25 16 10 0.099 16 2 81 10 6 0.012 16 2 81 10 6 0.012 16 2 81 12 6 0.012 4 3 49 10 7 0.018 4 3 49 11 7 0.018 4 3 49 13 7 0.018 16 3 169 8 5 0.0012 16 3 169 9 5 0.0012 16 3 169 9 5 0.0012 4 4 81 9 5 0.0038 4 4 81 9 5 0.0038 4 4 81 10 5 0.0038 16 4 289 7 4 0.00016 16 4 289 7 4 0.00015 16 4 289 7 4 0.00015 4 5 121 8 4 0.00094 4 5 121 8 4 0.00095 4 5 121 8 4 0.00096 16 5 441 7 4 5.5e-05 16 5 441 7 4 5.1e-05 Table 7.4 Mesh independence of the two-grid Newton-Picard inexact SQP method D = 0.1. Elms 16 64 256 16 64 256 16 64 256 16 64 256 16 64 256 16 64 256 16 64 256 64 256
Fine grid Ord 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
Dof 441 1681 6561 441 1681 6561 441 1681 6561 441 1681 6561 441 1681 6561 441 1681 6561 441 1681 6561 441 1681 6561
Dof 441 1681 6561 441 1681 6561 441 1681 6561 441 1681 6561 441 1681 6561 441 1681 6561 441 1681 6561 1681 6561
20
Coarse grid GMRES Newton-Picard Elms Ord Dof iters iters contr 4 5 121 38 81 0.79 4 5 121 42 89 0.80 4 5 121 42 89 0.80 4 5 121 48 89 0.80 16 5 441 22 25 0.46 16 5 441 25 25 0.46 16 5 441 29 26 0.46 4 3 49 66 197 0.91 16 3 169 40 55 0.70 64 3 625 23 18 0.31 Table 7.5 Mesh independence of the two-grid Newton-Picard inexact SQP method D = 0.001. Elms 16 64 256 1024 64 256 1024 1024 1024 1024
Fine grid Ord Dof 5 441 5 1681 5 6561 5 25921 5 1681 5 6561 5 25921 5 25921 5 25921 5 25921
Coarse grid GMRES Newton-Picard Elms Ord Dof iters iters contr 4 1 9 50 131 0.87 4 1 9 57 132 0.87 4 1 9 59 136 0.87 4 1 9 63 141 0.87 16 1 25 30 49 0.64 16 1 25 35 49 0.64 16 1 25 38 49 0.64 16 1 25 40 49 0.64 4 2 25 32 49 0.63 4 2 25 35 49 0.63 4 2 25 37 49 0.63 4 2 25 40 49 0.63 16 2 81 20 14 0.23 16 2 81 21 15 0.23 16 2 81 21 15 0.23 16 2 81 23 15 0.23 4 3 49 22 24 0.41 4 3 49 24 25 0.41 4 3 49 27 25 0.41 4 3 49 29 25 0.41 16 3 169 15 10 0.13 16 3 169 15 11 0.13 16 3 169 16 11 0.13 16 3 169 18 11 0.13 4 4 81 19 18 0.31 4 4 81 20 17 0.31 4 4 81 22 17 0.31 4 4 81 26 18 0.31 16 4 289 12 7 0.039 16 4 289 13 8 0.041 16 4 289 13 8 0.041 16 4 289 14 8 0.041 4 5 121 17 13 0.19 4 5 121 17 13 0.19 4 5 121 18 13 0.19 4 5 121 19 13 0.19 16 5 441 10 6 0.014 16 5 441 10 6 0.014 16 5 441 12 6 0.014 Table 7.6 Mesh independence of the two-grid Newton-Picard inexact SQP method D = 0.01. Elms 16 64 256 1024 16 64 256 1024 16 64 256 1024 16 64 256 1024 16 64 256 1024 16 64 256 1024 16 64 256 1024 16 64 256 1024 16 64 256 1024 64 256 1024
Fine grid Ord Dof 5 441 5 1681 5 6561 5 25921 5 441 5 1681 5 6561 5 25921 5 441 5 1681 5 6561 5 25921 5 441 5 1681 5 6561 5 25921 5 441 5 1681 5 6561 5 25921 5 441 5 1681 5 6561 5 25921 5 441 5 1681 5 6561 5 25921 5 441 5 1681 5 6561 5 25921 5 441 5 1681 5 6561 5 25921 5 1681 5 6561 5 25921
21
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