NEUMANN–NEUMANN DOMAIN DECOMPOSITION …

SIAM J. SCI. COMPUT. c© 2006 Society for Industrial and Applied MathematicsVol. 28, No. 3, pp. 1001–1028

NEUMANN–NEUMANN DOMAIN DECOMPOSITIONPRECONDITIONERS FOR LINEAR-QUADRATIC ELLIPTIC

OPTIMAL CONTROL PROBLEMS∗

MATTHIAS HEINKENSCHLOSS† AND HOANG NGUYEN‡

Abstract. We present a class of domain decomposition (DD) preconditioners for the solutionof elliptic linear-quadratic optimal control problems. Our DD preconditioners are extensions ofNeumann–Neumann DD preconditioners, which have been successfully applied to the solution ofsingle PDEs. The DD preconditioners are based on a decomposition of the optimality conditions forthe elliptic linear-quadratic optimal control problem into smaller subdomain optimality conditionswith Dirichlet boundary conditions for the states and the adjoints on the subdomain interfaces.These subdomain optimality conditions are coupled through Neumann interface conditions for thestates and the adjoints. This decomposition leads to a Schur complement system in which theunknowns are the state and adjoint variables on the subdomain interfaces. The Schur complementoperator is the sum of subdomain Schur complement operators, the application of which is shownto correspond to the solution of subdomain elliptic linear-quadratic optimal control problems, whichare essentially smaller copies of the original optimal control problem. We show that, under suitableconditions, the application of the inverse of the subdomain Schur complement operators requires thesolution of a subdomain elliptic linear-quadratic optimal control problem with Neumann interfaceconditions for the state. The subdomain Schur complement operators are analyzed in the variationalsetting of the problem as well as the algebraic setting obtained after a finite element discretizationof the problem. Definiteness properties of the algebraic form of the (subdomain) Schur complementoperator(s) are studied. Numerical tests show that the dependence of these preconditioners on meshsize and subdomain size is comparable to its counterpart applied to elliptic equations only. These testsalso show that the preconditioners are insensitive to the size of the control regularization parameter.

Key words. optimal control, preconditioners, domain decomposition, Neumann–Neumannmethods

AMS subject classifications. 49M05, 65N55, 65F10, 90C06, 90C20

DOI. 10.1137/040612774

1. Introduction. This paper is concerned with a class of nonoverlapping domaindecomposition (DD) preconditioners for linear-quadratic elliptic optimal control prob-lems. The solution of linear-quadratic elliptic optimal control problems arises in manyapplications, either directly or as subproblems in Newton or sequential quadratic pro-gramming methods for the solution of nonlinear elliptic optimal control problems.After a finite element discretization, convex linear-quadratic elliptic optimal controlproblems lead to large-scale symmetric indefinite linear systems. The solution of theselarge systems is a very time consuming step and must be done iteratively, typicallywith a preconditioned Krylov subspace method. Developing good preconditioners forthese linear systems is an important part of improving the overall performance of thesolution method.

∗Received by the editors August 3, 2004; accepted for publication (in revised form) February 13,2006; published electronically June 23, 2006. This work was supported in part by NSF grant ACI-0121360 and by the Los Alamos National Laboratory Computer Science Institute (LACSI) throughLANL contract 03891-99-23 as part of the prime contract (W-7405-ENG-36) between the Departmentof Energy and the Regents of the University of California.

http://www.siam.org/journals/sisc/28-3/61277.html†Department of Computational and Applied Mathematics, MS-134, Rice University, 6100 Main

Street, Houston, TX 77005-1892 ([email protected]).‡Department of Biostatistics and Applied Mathematics, University of Texas, MD Anderson Cancer

Center, 1515 Holcombe Blvd., Unit 447, Houston, TX 77030-4009 ([email protected]).

1001

1002 MATTHIAS HEINKENSCHLOSS AND HOANG NGUYEN

To illustrate our ideas, we consider the example problem

minimize1

2

∫Ωo

(y(x) − y(x))2dx +α

2

∫∂Ω

u2(x)dx,(1.1a)

subject to − Δy(x) + σy(x) = f(x) in Ω,(1.1b)

∂

∂ny(x) = u(x) on ∂Ω,(1.1c)

where y, f are given functions and α > 0 and σ ≥ 0 are given parameters. Theproblem (1.1) has to be solved for y and u. Detailed model problem assumptions willbe introduced in section 3. While (1.1) is used to illustrate the DD preconditionersintroduced in this paper, their formulation and many of their properties carry over tomore general linear-quadratic elliptic optimal control problems.

Our DD preconditioners for linear-quadratic elliptic optimal control problemssuch as (1.1) are based on a decomposition of the system of necessary and suffi-cient optimality conditions. We decompose Ω into nonoverlapping subdomains. Wethen introduce state and adjoint variables on the subdomain interfaces and formu-late subdomain optimality conditions with Dirichlet boundary conditions for statesand adjoints on the subdomain interfaces. These subdomain optimality conditions arecoupled by requiring continuity of normal derivatives of subdomain state and adjoints,defined as solutions of the local optimality conditions, on the subdomain interfaces.This leads to subdomain Schur complement operators that map Dirichlet interfacedata into Neumann interface data. The original optimality conditions are equivalentto an optimality Schur complement system in which the operator is given by the sumof subdomain Schur complements. The properties of the subdomain Schur comple-ments allow the construction of preconditioners for the optimality condition Schurcomplement system that are extensions of the Neumann–Neumann methods, whichhave been successfully applied to single elliptic PDEs [13, 14, 24, 26, 33, 34, 36, 38].

We will show that the application of subdomain Schur complements correspondsto the solution of subdomain optimal control problems that are essentially smallercopies of the original optimal control problem but impose Dirichlet boundary condi-tions for the state on the subdomain interface and include an objective function termthat implies Dirichlet boundary conditions for the adjoint on the subdomain interface.Under suitable conditions, these subdomain Schur complements are invertible. Theapplication of the inverses of local Schur complement operators corresponds to the so-lution of subdomain optimal control problems that are also essentially smaller copiesof the original optimal control problem but impose Neumann boundary conditions onthe subdomain interfaces.

The paper is organized as follows. In the next section, we give a brief overview ofother DD approaches to solve linear-quadratic optimal control problems and describehow our approach differs from existing ones.

In section 3 we study the model problem (1.1) with emphasis on the case σ = 0.In this case, the state equation (1.1b), (1.1c) has a solution if and only if the controlsatisfies a certain compatibility condition; the solution of (1.1b), (1.1c), if it exists,is not unique. In section 3 we will show that the optimal control problem (1.1)has a unique solution. However, the issues outlined above regarding the existenceand uniqueness of the solution of the state equation (1.1b), (1.1c) later may lead tosingularities in the subproblems arising in our DD methods. This is one reason forchoosing the model problem (1.1).

NEUMANN–NEUMANN METHODS FOR CONTROL PROBLEMS 1003

Our DD approach is introduced in section 4. This section introduces the Schurcomplement formulation of the optimality system and investigates properties of thesubdomain Schur complement. The presentation in section 4 is based on the varia-tional form of the optimal control problem (1.1).

The algebraic view of our DD approach is presented in section 5. This viewemphasizes the formal similarity between the Neumann–Neumann methods for singleelliptic PDEs and our approach. In section 5 we will also investigate the inertia ofthe Schur complements arising in our DD preconditioners. We will see that the Schurcomplements arising in the optimal control context are highly indefinite; the numberof positive and negative eigenvalues is essentially the same and is equal to the numberof discrete state variables on the subdomain interfaces.

Numerical results for our (balancing) Neumann–Neumann preconditioners ap-plied to the model problem (1.1) are given in section 6. The results indicate that thenumerical performance of our (balancing) Neumann–Neumann DD methods appliedto linear-quadratic elliptic optimal control problems is similar to that of their coun-terparts applied to single elliptic PDEs. We also observe that the performance of ourpreconditioners applied to the model problem (1.1) is largely independent of the sizeof the regularization parameter.

This work significantly expands the material in [20] in which an additional nu-merical example may be found.

2. Comparison with other domain decomposition approaches. DD-basediterative methods for the solution of linear-quadratic elliptic optimal control problemssuch as (1.1) can be split into three categories, depending on how the DD is integratedwith the optimization. In all cases DD introduces parallelism into the optimization butaffects the (outer) iteration differently. We give an outline and refer to [32, sect. 1.2]for more details.

1. The methods in the first category apply preconditioned Krylov subspace meth-ods to the system of discretized optimality conditions (also known as the KKT system)for (1.1). This system has a 3 × 3 block structure, and the preconditioners for thesystem matrix (also known as the KKT matrix) are obtained from approximate blockfactorizations. DD methods are used for the approximate solution of systems involvingthe stiffness matrix and its transpose. DD is used at the PDE level. These methodsrequire a good preconditioner for the so-called reduced Hessian, which can be difficultto obtain. The methods in [9, 10] belong to this category.

2. The methods in the second category use DD to reformulate the optimizationproblem (1.1) or its discretization [11, 28]. For example, a reformulation may elim-inate state variables in the interior of subdomains by viewing them as functions ofthe controls in that subdomain as well as state variables on the subdomain interfaces.The optimization variables visible to the outer iteration then consist of the controlvariables and the state variables restricted to the subdomain interfaces. The optimal-ity conditions for the reformulated problem can be viewed as a Schur complement ofthe KKT matrix corresponding to the original problem (1.1). This Schur complementinvolves the state and adjoint variables on the subdomain interfaces as well as allcontrol variables. In contrast, the Schur complement formulation introduced in thispaper involves only the state and adjoint variables on the subdomain interfaces. Theapplication of the Schur complement arising in the DD methods in category 2 to avector requires the solution of subdomain PDEs. Typically, the Schur complementsarising in the methods in category 2 are better conditioned than the original KKTmatrix, but additional preconditioners are required, especially when α or the mesh


size h is small (see [32, sect. 1.2] for an example). The approach in [12] for a nonlinearparameter identification problem also falls into this category.

3. In [5, 6, 7] DD methods for linear-quadratic elliptic optimal control problemsare introduced that are based on a decomposition of the optimality conditions. Theresulting iterative method requires the solution of KKT-type systems restricted tosmaller domains, and the variables in the iteration are related to states and adjointson the subdomain interfaces. In [5, 6, 7] Robin-Robin-type transmission conditionsare used to connect the subdomain problems. Convergence of the method is provedin [6, 7], but no convergence rates are given. No results regarding the dependenceof the convergence on mesh size, subdomain size, and regularization parameter α aregiven. The KKT-type systems restricted to smaller domains are related to subdomainoptimal control problems, but these optimal control problems have a structure differ-ent from that of the original ones. In particular, the state space for the subdomainoptimal control problems in [6, p. 2411] is twice as large as that of the original optimalcontrol problems restricted to the same subdomain.

DD methods for a class of nonlinear parameter identification problems are inves-tigated in [25, 37]. The augmented Lagrangian method is used for the solution of thenonlinear optimization problem, and DD approaches for unconstrained minimizationare used to solve the subproblems within the augmented Lagrangian method.

The methods proposed in this paper, like those in [5, 6, 7], are based on a decom-position of the optimality conditions but use different transmission conditions. Oneconsequence is that the DD methods proposed in this paper require the solution ofKKT systems restricted to smaller subdomains, which are related to smaller copiesof the original linear-quadratic elliptic optimal control problem. Hence existing codecan be reused for subproblem solves. The DD methods proposed in this paper areextensions of the Neumann–Neumann methods, which have been successfully appliedto single elliptic PDEs [13, 14, 24, 26, 33, 34, 36, 38]. The numerical performance ofour (balancing) Neumann–Neumann DD methods applied to the model problem (1.1)indicates that their performance is similar to that of their counterparts applied to sin-gle elliptic PDEs. We also observe that their performance on the model problem (1.1)is largely independent of the size of the regularization parameter. This is importantsince the condition numbers of the KKT matrix corresponding to (1.1) and of thereduced Hessian grow linearly with α−1 (see [32, Chap. 1] for an example). The sub-domain problems that need to be solved in our (balancing) Neumann–Neumann DDmethods are essentially smaller subdomain copies of (1.1), and state and adjoint vari-ables restricted to subdomain interfaces have to be communicated between adjacentsubdomains. Hence our methods, like those in [5, 6, 7], involve more computationsper communication than those in categories 1 and 2 sketched above.

The DD preconditioners proposed in this paper complement existing precondition-ers for KKT systems, such as those in [1, 3, 4, 8, 9, 10, 21, 15, 16, 18, 19, 23, 29, 31],which can be used as subproblem solvers within our DD preconditioners. Such acombination introduces parallelism. For some applications, such a combination maymake existing preconditioners that rely on sparse matrix factorizations feasible, sincethey now are applied to smaller subproblems.

3. The example problem. In this section, we define the setting for the modelproblem (1.1), establish existence and uniqueness of its solution, and derive necessaryand sufficient optimality conditions. If σ > 0, the state equation (1.1b) has a uniquesolution y for given control u. In this case existence and uniqueness of solutions aswell as derivation of necessary and sufficient optimality conditions are standard (see,


e.g., [27]). The case σ = 0 is more interesting and is the focus of this section. In thecase σ = 0, the state equation (1.1b) is solvable if and only if the control satisfies acertain compatibility condition; if (1.1b) is solvable, the solution is not unique.

Let Ω ⊂ Rd, d = 2, 3, be an open, bounded set with Lipschitz boundary, let

Ωo ⊂ Ω be measurable with meas(Ωo) > 0, let y ∈ L2(Ω0), f ∈ L2(Ω) be givenfunctions, and let α > 0, σ ≥ 0 be given parameters. We define the (bi)linear forms

a : H1(Ω) ×H1(Ω) → R, a(y, ψ) =

∫Ω

∇y(x)∇ψ(x) + σy(x)ψ(x)dx,

b : L2(∂Ω) ×H1(Ω) → R, b(u, ψ) = −∫∂Ω

u(x)ψ(x)dx,

m : H1(Ω) ×H1(Ω) → R, m(y, ψ) =

∫Ωo

y(x)ψ(x)dx,

q : L2(∂Ω) × L2(∂Ω) → R, q(u, μ) = α

∫∂Ω

u(x)μ(x)dx,

f : H1(Ω) → R, f(ψ) =

∫Ω

f(x)ψ(x)dx.

We are interested in the solution y ∈ H1(Ω), u ∈ L2(∂Ω) of the optimal controlproblem

minimize 12m(y − y, y − y) + 1

2q(u, u),(3.1a)

subject to a(y, φ) + b(u, φ) = f(φ) ∀φ ∈ H1(Ω).(3.1b)

Theorem 3.1. Let σ ≥ 0. The optimal control problem (3.1) has a uniquesolution (u∗, y∗) ∈ L2(∂Ω) × H1(Ω), which, together with the adjoint variable p∗ ∈H1(Ω), is characterized by the necessary and sufficient optimality conditions

a(ψ, p) + m(y, ψ) = m(y, ψ) ∀ψ ∈ H1(Ω),(3.2a)

q(u, μ) + b(μ, p) = 0 ∀μ ∈ L2(∂Ω),(3.2b)

a(y, φ) + b(u, φ) = f(φ) ∀φ ∈ H1(Ω).(3.2c)

The necessary and sufficient optimality conditions (3.2) have a unique solution y∗ ∈H1(Ω), u∗ ∈ L2(∂Ω), p∗ ∈ H1(Ω).

Proof. For σ > 0 the state equation (3.1b) has a unique solution y for anyu∗ ∈ L2(∂Ω) and f ∈ (H1(Ω))′. In this case, the assertion of the theorem is wellknown and a proof can be found, e.g., in [27] or by a modification of the argumentsbelow.

Let σ = 0. We recall that for given l ∈ (H1(Ω))′ the equation

a(y, φ) = l(φ)

has a solution if and only if l(1) = 0. If l(1) = 0 and if y0 is a solution, then the setof solutions is given by y0 + γ, γ ∈ R.


(i) Existence and uniqueness of the optimal solution. We define

Y =

{y ∈ H1(Ω) :

∫Ω

y(x)dx = 0

}.

By the second Poincare inequality (see, e.g., [35, Cor. 6.100], [39, Thm. 7.7]),

‖y‖Y = ‖∇y‖L2(Ω)

defines a norm on Y which on Y is equivalent to ‖·‖H1(Ω). Moreover, Y equipped with‖ · ‖Y is a Hilbert space. Standard arguments can be used to show that if u ∈ L2(∂Ω)satisfies f(1) = b(u, 1), then (3.1b) has a unique solution y ∈ Y and there exists κ > 0,independent of u and f such that

‖y‖Y ≤ κ(‖u‖L2(∂Ω) + ‖f‖(H1(Ω))′).

The problem (3.1), which is an optimization problem in u ∈ L2(∂Ω), y ∈ H1(Ω), canbe written as an optimization problem in u ∈ L2(∂Ω), y0 ∈ Y , γ ∈ R given by

minimize 12m(y0 + γ − y, y0 + γ − y) + 1

2q(u, u),(3.3a)

subject to a(y0, ψ) + b(u, ψ) = f(ψ) ∀ψ ∈ H1(Ω).(3.3b)

Using the affine linear map u → y0(u) that maps u ∈ L2(∂Ω) with b(u, 1) = f(1) tothe solution y0(u) ∈ Y of (3.3b), we can write (3.3) as

minimize 12m(y0(u) + γ − y, y0(u) + γ − y) + 1

2q(u, u),(3.4a)

subject to b(u, 1) = f(1).(3.4b)

For fixed u satisfying (3.4b), the unique solution of

minγ

12m(y0(u) + γ − y, y0(u) + γ − y)

is given by

γ(u) = −∫

Ωo

y0(u;x) − y(x)dx

/∫Ωo

dx.(3.5)

Note that ∫Ωo

y0(u;x) + γ(u) − y(x)dx = 0.(3.6)

Using the definition (3.5) of γ(u), (3.4) can be written as the following optimizationproblem in u only:

Minimize 12m(y0(u) + γ(u) − y, y0(u) + γ(u) − y) + 1

2q(u, u),(3.7a)

subject to b(u, 1) = f(1).(3.7b)

Since u → y0(u) is an affine linear map, u → y0(u) + γ(u) is affine linear and (3.7a)is strictly convex. The existence and uniqueness of the optimal control u∗ followsby standard arguments (see, e.g., [22]). The unique corresponding state is given byy∗ = y0(u∗) + γ(u∗).


(ii) Optimality conditions. The optimality conditions follow from an applica-tion of the Lagrange multiplier theorem (see, e.g., [22]). First, we show that thelinear constraints are surjective. Let f ∈ (H1(Ω))′ be arbitrary. If we define u =−‖f‖(H1(Ω))′/

∫∂Ω

dx, then the state equation (3.2c) has a solution y ∈ H1(Ω). Thisproves the surjectivity of the constraints.

Since the constraints are surjective, there exists p ∈ H1(Ω) such that the partialFrechet derivatives of

L(y, u, p) = 12m(y − y, y − y) + 1

2q(u, u) + a(y, p) + b(y, p) − f(p)

with respect to y, u, and p are equal to zero. This gives (3.2). Since the optimal controlproblem (3.1) is convex, the necessary optimality conditions are also sufficient.

(iii) Uniqueness. In part (i) we have already shown the uniqueness of the optimalstate and control. The uniqueness of the associated adjoint p∗ remains to be shown.We write the optimal state, control, and adjoint as

y∗ = y0 + γ, u∗ = u0 + uf , p∗ = p0 + η,

where y0, p0 ∈ Y , u0 ∈{u ∈ L2(∂Ω) : b(u, 1) = 0

}, uf = −‖f‖(H1(Ω))′/

∫∂Ω

dx,and γ, η ∈ R. The function y0 is the unique solution (in Y ) of the state equa-tion (3.2c), and γ is given by (3.5). Since the optimal state y∗ = y0 + γ satis-fies (3.6), the adjoint equation (3.2a) has a solution p∗ = p0 + η, where p0 ∈ Y isunique. Using the representation of u∗ and p∗ as well as the decomposition L2(∂Ω) ={u ∈ L2(∂Ω) : b(u, 1) = 0

}⊕span{1} (by definition of b,

{u ∈ L2(∂Ω) : b(u, 1) = 0

}is the orthogonal complement of span{1} in L2(∂Ω)), we can write condition (3.2b)equivalently as

q(u0 + uf , 1) + b(1, p0 + η) = 0,

q(u0 + uf , μ) + b(μ, p0 + η) = 0, μ ∈{u ∈ L2(∂Ω) : b(u, 1) = 0

}.

The first condition implies

η = −(q(u0 + uf , 1) + b(1, p0))/b(1, 1),

and the second condition characterizes u0 ∈{u ∈ L2(∂Ω) : b(u, 1) = 0

}.

Note that in the case σ = 0, adjoint equation (3.2a) alone does not have a uniquesolution, but because the optimal state y∗ = y0(u∗)+γ(u∗) satisfies (3.6), the adjointequation (3.2a) is solvable. The entire optimality system (3.2) specifies optimal state,control, and adjoint uniquely.

4. Domain decomposition Schur complement formulation of the exam-ple problem.

4.1. Domain decomposition of the example problem. We discretize (3.1)using conforming linear finite elements. Let {Tl} be a triangulation of Ω. We divide Ωinto nonoverlapping subdomains Ωi, i = 1, . . . , s, such that each Tl belongs to exactlyone Ωi.

We define

Γi = ∂Ωi \ ∂Ω

and

Γ =

s⋃i=1

Γi.


The unit outward normal of Ωi is denoted by ni. By TrΓiwe denote the trace on Γi.

We split the subdomains into no-control subdomains

N = {i : int(∂Ωi ∩ ∂Ω) = ∅}

and control subdomains

C = {i : int(∂Ωi ∩ ∂Ω) �= ∅} .

Here int denotes the relative interior.The state y is approximated using piecewise linear functions. We define the finite

dimensional spaces

V h ={v ∈ H1(Ω) : v|Tl

∈ P 1(Tl) ∀l},(4.1)

V hi =

{v ∈ H1(Ωi) : v|Tl

∈ P 1(Tl) ∀Tl ⊂ Ωi

}, i = 1, . . . , s,

V hi,0 =

{v ∈ V h

i : v = 0 on Γi

}, i = 1, . . . , s.

We can identify vi ∈ V hi,0 with a function in V h if we extend vi by zero onto Ω.

Furthermore, we introduce the “trace” space V hi,Γi

of functions in V hi that are zero

at vertices in Ωi \ Γi and the “trace” space V hΓ of functions in V h that are zero at

vertices in Ω \ Γ. Hence, each function vh ∈ V h can be written as

V h =

s⊕i=1

V hi,0 ⊕ V h

Γ .(4.2)

For our discretization of the control, we use piecewise linear functions on ∂Ω.However, our discretization of the control is somewhat nonstandard. Typically, onewould choose the space of discretized controls to be

Uh ={u ∈ C0(∂Ω) : u is linear on ∂Ω ∩ ∂Tl ∀Tl ⊂ Ω

}.(4.3)

A DD formulation based on such a discretization would introduce “interface controls”(dotted hat function in the left-hand plot in Figure 4.1) defined on a “band” of widthO(h) around ∂Ω ∩ ∂Ωi ∩ ∂Ωj , i �= j. See the left-hand plot in Figure 4.1. Since theevaluation of u ∈ L2(∂Ω) on ∂Ω ∩ ∂Ωi ∩ ∂Ωj does not make sense, we avoid interfacecontrols.

We discretize the control u by a function which is continuous on each ∂Ωi, i =1, . . . , s, and linear on each ∂Ω∩∂Ωi∩Tl. The discretized control is not assumed to becontinuous at ∂Ω∩∂Ωi∩∂Ωj , i �= j. In particular, for each point xk ∈ ∂Ω∩∂Ωi∩∂Ωj ,i �= j, there are two discrete controls uki , ukj belonging to subdomains Ωi and Ωj ,respectively (see the right-hand plot in Figure 4.1). Hence, our control discretizationdepends on the partition {Ωi}di=1 of the domain Ω. We define the discrete controlspaces

Uhi =

{u ∈ C0(∂Ω ∩ ∂Ωi) : u is linear on ∂Ω ∩ ∂Ωi ∩ ∂Tl ∀Tl ⊂ Ωi

}.(4.4)

We identify Uhi with a subspace of L2(∂Ω) by extending functions ui ∈ Uh

i by zeroonto ∂Ω. We define

Uh =

s⋃i=1

Uhi ⊂ L2(∂Ω).


��

....................

∂Ω ∩ ∂Ωi ∂Ω ∩ ∂Ωjxk

��

∂Ω ∩ ∂Ωi ∂Ω ∩ ∂Ωjxk

Fig. 4.1. Sketch of the control discretization for the case Ω ⊂ R2.

In particular,

Uh =

s⊕i=1

Uhi .(4.5)

Our discretization of (3.2) is given by

a(ψ, p) + m(y, ψ) = m(y, ψ) ∀ψ ∈ V h,(4.6a)

q(u, μ) + b(μ, p) = 0 ∀μ ∈ Uh,(4.6b)

a(y, φ) + b(u, φ) = f(φ) ∀φ ∈ V h.(4.6c)

The system (4.6) is also the system of necessary and sufficient optimality conditionsfor the discretization of (3.1) using piecewise linear finite elements for the state andcontrols, as described above.

We decompose the optimality system (4.6) using (4.2), (4.5). For this purpose,we introduce the following (bi)linear forms. Let

ai : V hi × V h

i → R, ai(y, ψ) =

∫Ωi

∇y(x)∇ψ(x) + σy(x)ψ(x)dx,

mi : V hi × V h

i → R, mi(y, ψ) =

∫Ωi∩Ωo

y(x)ψ(x)dx,

fi : V hi → R, fi(ψ) =

∫Ωi

f(x)ψ(x)dx,

and, if Ωi is a control subdomain,

bi : Uhi × V h

i → R, bi(u, ψ) = −∫∂Ωi∩∂Ω

u(x)ψ(x)dx,

qi : Uhi × Uh

i → R, qi(u, μ) = α

∫∂Ωi∩∂Ω

u(x)μ(x)dx.

The parameters α > 0 and σ ≥ 0 as well as f ∈ L2(Ω) are given as in section 3.Now we write y = yΓ +

∑si=1 yi, p = pΓ +

∑si=1 pi, and u =

∑si=1 ui, where

yΓ, pΓ ∈ V hΓ , yi, pi ∈ V h

i,0, i = 1, . . . , s, and ui ∈ Uhi , i = 1, . . . , s (cf. (4.2) and (4.5)),

and insert these representations into (4.6). Using an analogous decomposition of thetest functions, this leads to the systems

ai(ψ, pi) + mi(yi, ψ) = mi(y, ψ) ∀ψ ∈ V hi,0,(4.7a)

bi(μ, pi) + qi(ui, μ) = 0 ∀μ ∈ Uhi ,(4.7b)

ai(yi, ψ) + bi(ui, ψ) = fi(ψ) ∀ψ ∈ V hi,0,(4.7c)

yi = yΓ, pi = pΓ on Γi(4.7d)


for i ∈ C,

ai(ψ, pi) + mi(yi, ψ) = mi(y, ψ) ∀ψ ∈ V hi,0,(4.8a)

ai(yi, ψ) = fi(ψ) ∀ψ ∈ V hi,0,(4.8b)

yi = yΓ, pi = pΓ on Γi(4.8c)

for i ∈ N , ands∑

i=1

ai(yi, qΓ) + bi(ui, qΓ) + ai(vΓ, pi) + mi(yi, vΓ)

=

s∑i=1

fi(qΓ) + mi(yi, vΓ) ∀ vΓ, qΓ ∈ V hΓ .(4.9)

The next theorem states that solving the optimality conditions (4.6) is equivalentto finding (yΓ, pΓ) ∈ V h

Γ × V hΓ such that the solutions (yi, ui, pi) ∈ V h

i × Uhi × V h

i ,i ∈ C, and (yi, pi) ∈ V h

i × V hi , i ∈ N , of (4.7) and (4.8), respectively, satisfy the

interface conditions (4.9).Theorem 4.1. If (y, u, p) ∈ V h×Uh×V h solves (4.6), then yi = y|Ωi

, pi = p|Ωi,

i = 1, . . . , s, ui = u|Ωi, i ∈ C, solve (4.7), (4.9).

If (yΓ, pΓ) ∈ V hΓ × V h

Γ is such that the solutions (yi, ui, pi) ∈ V hi × Uh

i × V hi ,

i ∈ C, and (yi, pi) ∈ V hi × V h

i , i ∈ N , of (4.7) and (4.8), respectively, satisfy theinterface conditions (4.9), then (y, u, p) ∈ V h×Uh×V h given by yi = y|Ωi

, pi = p|Ωi ,i = 1, . . . , s, ui = u|Ωi , i ∈ C, solves (4.6).

The proof of this theorem is analogous to the proof of [34, Lem. 1.2.1] and isomitted.

Our next goal is to view the solution of (4.7), (4.8) as an affine linear function of(yΓ, pΓ) and then consider (4.9) as a linear equation in (yΓ, pΓ). This Schur comple-ment system will be solved using a preconditioned Krylov subspace method. To derivethis Schur complement system, however, we first have to investigate the solvabilityof (4.7) and (4.8).

4.2. The subdomain optimality systems. We investigate the solvability ofthe systems (4.7) and (4.8). These are the weak forms of

−Δyi(x) + σyi(x) = f(x) in Ωi,(4.10a)

∂

∂nyi(x) = ui(x) on ∂Ω ∩ ∂Ωi,(4.10b)

yi(x) = yΓ(x) on Γi,(4.10c)

−Δpi(x) + σpi(x) = −(yi(x) − y(x))|Ωoin Ωi,(4.10d)

∂

∂npi(x) = 0 on ∂Ω ∩ ∂Ωi,(4.10e)

pi(x) = pΓ(x) on Γi,(4.10f)

αui(x) − pi(x) = 0 on ∂Ω ∩ ∂Ωi(4.10g)

and

−Δyi(x) + σyi(x) = f(x) in Ωi,(4.11a)

yi(x) = yΓ(x) on Γi,(4.11b)

−Δpi(x) + σpi(x) = −(yi(x) − y(x))|Ωo in Ωi,(4.11c)

pi(x) = pΓ(x) on Γi,(4.11d)

respectively.


We will show that (4.7), (4.8) have unique solutions, even if σ = 0 (see Theo-rems 4.3 and 4.4 below). This is due to the presence of Dirichlet boundary conditions,which ensure that the state equations (4.7c) and (4.8b) have a unique solution for eachui ∈ Uh

i and that the adjoint equations (4.7a) and (4.8a) have a unique solution foreach yi ∈ V h

i .Lemma 4.2. Let σ ≥ 0 and li ∈ (V h

i,0)∗. There exists a unique solution of

ai(yi, ψ) = li(ψ) ∀ψ ∈ V hi,0.

Furthermore, there exists κi > 0 independent of li such that ‖yi‖V hi≤ κi‖li‖(V h

i )∗ .

Proof. By the Poincare inequality there exists ci > 0 such that ci‖yi‖2V hi

≤ai(yi, yi) for all yi ∈ V h

i,0. The lemma now follows from a standard application of theLax–Milgram theorem.

The system (4.8) is a system of elliptic differential equations in triangular formthat can be easily solved by first determining yi ∈ V h

i from the second equationin (4.8) and then pi ∈ V h

i from the first equation in (4.8). The unique solvability ofthe two elliptic PDEs follows from the definition of V h

i,Γiand from Lemma 4.2.

Theorem 4.3. The system (4.8) has a unique solution (yi, ui, pi) ∈ V hi × Uh

i ×V hi .

Next we will show that (4.7) are the optimality conditions for a convex linear-quadratic optimal control problem restricted to the subdomain Ωi. Let Ωi be a controlsubdomain. Formally the system (4.7) (or its corresponding strong form (4.10)) maybe viewed as the necessary and sufficient optimality conditions for

minimize1

2

∫Ωo∩Ωi

(yi(x) − y(x))2dx +α

2

∫∂Ωi

u2i (x)dx +

∫Γi

∂

∂niyi(x)pΓ(x)dx,

(4.12a)

subject to − Δyi(x) + σyi(x) = f(x) in Ωi,

(4.12b)

∂

∂nyi(x) = ui(x) on ∂Ωi ∩ ∂Ω,(4.12c)

yi(x) = yΓ(x) on Γi.(4.12d)

The local optimal control problem (4.12) is an optimization problem in the variablesui, yi; all other quantities in (4.12) are given.

To rigorously establish a relationship between (4.7) and (4.12) we have to makeprecise the meaning of the state equation (4.12b)–(4.12d) and of the objective func-tion (4.12a). The state equation (4.12b)–(4.12d) is understood in the weak sense (4.7c),(4.7d). Concerning the objective function (4.12a) we note that equations (4.12b),(4.12c) imply∫

Γi

∂

∂nyi(x)ψ(x)dx

=

∫Ωi

∇yi(x)∇ψ(x) + σyi(x)ψ(x)dx−∫∂Ω∩∂Ωi

u(x)ψ(x)dx−∫

Ωi

f(x)ψ(x)dx,

= ai(yi, ψ) + bi(ui, ψ) − fi(ψ).

Thus, in (4.12a) we set∫Γi

∂

∂nyi(x)pΓ(x)dx = ai(yi,RipΓ) + bi(ui,RipΓ) − fi(RipΓ),


where Ri : V hi,Γi

→ V hi , is a continuous linear extension operator satisfying TrΓi

(RivΓ)

= vΓ for all vΓ ∈ V hi,Γi

. To summarize, the problem (4.12) is understood as follows:

Minimize1

2mi(yi − y, yi − y) +

α

2qi(ui, ui)

+ ai(yi,RipΓ) + bi(ui,RipΓ) − fi(RipΓ),(4.13a)

subject to ai(yi, ψ) + bi(ui, ψ) = fi(ψ) ∀ψ ∈ V hi,0,(4.13b)

yi = yΓ on Γi.(4.13c)

Theorem 4.4. The local optimal control problem (4.13) has a unique solution.The system (4.7) is the necessary and sufficient optimality conditions for the localoptimal control problem (4.13).

Proof. (i) By definition of V hi,Γi

and Lemma 4.2, the state equations (4.13b), (4.13c)

have a unique solution yi(ui) ∈ V hi for every ui ∈ Uh

i . Moreover, ui → yi(ui) is a con-tinuous affine linear map. Thus, the presence of α

2 qi(ui, ui) in the objective ensuresthat (4.13) is strictly convex and has a unique solution.

(ii) Since (4.13) is a convex linear-quadratic optimal control problem, yi ∈ V hi , ui ∈

Uhi is a solution of (4.13) if and only if (4.13b), (4.13c) are satisfied and the Frechet

derivative of (4.13a) applied to all vectors in the null-space of the constraints (4.13b),(4.13c) is zero, i.e., if

mi(yi − y, zi) + qi(ui, vi) + ai(zi,RipΓ) + bi(vi,RipΓ) = 0(4.14a)

for all vi ∈ Uhi and all zi ∈ V h

i,0 satisfying the homogeneous linear state equation

ai(zi, ψ) + bi(Vhi , ψ) = 0 ∀ψ ∈ V h

i,0.(4.14b)

Let yi ∈ V hi , ui ∈ Uh

i be a solution of (4.13). Let pi solve (4.7a) and pi = pΓ onΓi, and define p0

i = pi − RipΓ ∈ V hi,0. Let vi ∈ Uh

i be arbitrary and let zi ∈ V hi,0 be

the corresponding solution of (4.14b). Using (4.14b) with ψ = p0i , (4.7a) with φ = zi,

and (4.14a) implies that

0 = mi(yi − y, zi) + qi(ui, vi) + ai(zi,RipΓ) + bi(vi,RipΓ)

+ ai(zi, p0i ) + bi(vi, p

0i )

= mi(yi − y, zi) + qi(ui, vi) + ai(zi, pi) + bi(vi, pi)

= qi(ui, vi) + bi(vi, pi)

for all V hi ∈ Uh

i . Hence, if yi ∈ V hi , ui ∈ Uh

i is a solution of (4.13), then (4.7) issatisfied.

On the other hand, let yi, pi ∈ V hi , Uh

i ∈ Uhi solve (4.7). Define p0

i = pi −RipΓ ∈V hi,0. Let vi ∈ Uh

i , zi ∈ V hi,0 satisfy the homogeneous linear state equation (4.14b). If

we use (4.7a) with φ = zi, (4.7b) with μ = vi, and (4.14b) with ψ = p0i , we obtain

0 = mi(yi − y, zi) + qi(ui, vi) + ai(zi, pi) + bi(vi, pi) − ai(zi, p0i ) + bi(vi, p

0i )

= mi(yi − y, zi) + qi(ui, vi) + ai(zi,RipΓ) + bi(vi,RipΓ).

Thus, (4.14a) is satisfied and, therefore, yi ∈ V hi , ui ∈ Uh

i solve (4.13).


4.3. Schur complement formulation. Now we formulate the Schur comple-ment system; i.e, we view the solution of (4.7), (4.8) as an affine linear function of(yΓ, pΓ) and then consider (4.9) as a linear equation in (yΓ, pΓ).

For i ∈ C we define the operator

Hhi ∈ L(V h

Γ × V hΓ , V h

i × Uhi × V h

i )(4.15a)

with

Hhi (yΓ, pΓ) =

⎛⎜⎝ (Hhi )y(yΓ, pΓ)

(Hhi )u(yΓ, pΓ)

(Hhi )p(yΓ, pΓ)

⎞⎟⎠ =

⎛⎜⎝ y0i

u0i

p0i

⎞⎟⎠,(4.15b)

where (y0i , u

0i , p

0i ) is the solution of (4.7) with f = 0 and y = 0. For i ∈ N we define

the operator

Hhi ∈ L(V h

Γ × V hΓ , V h

i × V hi )(4.16a)

with

Hhi (yΓ, pΓ) =

((Hh

i )y(yΓ, pΓ)

(Hhi )p(yΓ, pΓ)

)=

(y0i

p0i

),(4.16b)

where (y0i , p

0i ) is the solution of (4.8) with f = 0 and y = 0. The linear opera-

tors (4.15), (4.16) may be viewed as generalizations of the harmonic extension usedin DD methods for linear elliptic PDEs to elliptic linear-quadratic optimal controlproblems.

Theorem 4.5. There exist constants 0 < c < C independent of h such that

c‖(yΓ, pΓ)‖(V hi,Γi

)2 ≤ ‖Hhi (yΓ, pΓ)‖V h

i ×Uhi ×V h

i≤ C‖(yΓ, pΓ)‖(V h

i,Γi)2(4.17)

for all yΓ, pΓ ∈ (V hi,Γi

)2.

Proof. We consider the case that Ωi is a control subdomain, i.e., that (y0i , u

0i , p

0i )

in (4.15b) is the solution of (4.7) with f = 0 and y = 0. The other case can be provedsimilarly.

(i) Let yΓ, pΓ ∈ (V hi,Γi

)2 be arbitrary and let (y0i , u

0i , p

0i ) be the corresponding

solution of (4.7) with f = 0 and y = 0. By the trace theorem, there exist constantsc1, c2 > 0, independent of yΓ, pΓ ∈ (V h

i,Γi)2, such that

‖yΓ‖V hi,Γi

≤ c1‖y0i ‖V h

i, ‖pΓ‖V h

i,Γi

≤ c2‖p0i ‖V h

i,

where (y0i , u

0i , p

0i ) is the solution of (4.7) with f = 0 and y = 0. This implies the

left-hand inequality in (4.17).(ii) Let Ri : V h

i,Γi→ V h

i be a continuous linear extension operator satisfying

TrΓi(RivΓ) = vΓ for all vΓ ∈ V h

i,Γi.

The y, p component of the solution (y0i , u

0i , p

0i ) of (4.7) with f = 0 and y = 0 can

be written as y0i = RiyΓ + yi, p

0i = RipΓ + pi, where yi, pi ∈ V h

i,0, and (yi, u0i , pi) ∈

V hi,0 × Uh

i × V hi,0 solve

ai(ψ, pi) + mi(yi, ψ) = −ai(ψ,RipΓ) −mi(RiyΓ, ψ) ∀ψ ∈ V hi,0,(4.18a)

bi(μ, pi) + qi(u0i , μ) = −bi(μ,RipΓ) ∀μ ∈ Uh

i ,(4.18b)

ai(yi, φ) + bi(u0i , φ) = −ai(RiyΓ, φ) ∀φ ∈ V h

i,0.(4.18c)


The properties of the bilinear forms ai, bi, equation (4.18c), and the continuity of Ri

imply the existence of c3, c4 > 0 independent of yi, u0i , yΓ such that

‖yi‖V hi≤ c3

(‖u0

i ‖Uhi

+ ‖RiyΓ‖V hi

)≤ c4

(‖u0

i ‖Uhi

+ ‖yΓ‖V hi,Γi

).(4.19)

Similarly, properties of the bilinear forms ai, mi, equation (4.18a), the continuity ofRi, and (4.19) imply the existence of c5, c6 > 0 independent of yi, u

0i , yΓ, pi, pΓ such

that

‖pi‖V hi≤ c5

(‖yi‖V h

i+ ‖RiyΓ‖V h

i+ ‖RipΓ‖V h

i

)≤ c6

(‖u0

i ‖Uhi

+ ‖yΓ‖V hi,Γi

+ ‖pΓ‖V hi,Γi

).(4.20)

If we set ψ = yi, μ = u0i , φ = −pi in (4.18) and add the equations, we obtain

mi(yi, yi) + qi(u0i , u

0i ) = −ai(yi,RipΓ) −mi(RiyΓ, yi) − bi(u

0,RipΓ) + ai(RiyΓ, pi).

The properties of the bilinear forms ai, bi, qi, mi, the continuity of Ri, and the inequal-ities (4.19), (4.20) guarantee the existence of c7, c8 > 0 independent of yi, u

0i , yΓ, pΓ

such that

α‖u0i ‖2

Uhi≤ mi(yi, yi) + qi(u

0i , u

0i )

≤ c7

(‖yi‖V h

i‖pΓ‖V h

i,Γi

+ ‖yi‖V hi‖yΓ‖V h

i,Γi

+ ‖u0i ‖Uh

i‖pΓ‖V h

i,Γi

+ ‖pi‖V hi‖yΓ‖V h

i,Γi

)≤ c8

(‖yΓ‖V h

i,Γi

+ ‖pΓ‖V hi,Γi

)‖u0

i ‖Uhi

+ c8

(‖yΓ‖V h

i,Γi

+ ‖pΓ‖V hi,Γi

)2

≤ α

2‖u0

i ‖2Uh

i+

c282α

(‖yΓ‖V h

i,Γi

+ ‖pΓ‖V hi,Γi

)2

+ c8

(‖yΓ‖V h

i,Γi

+ ‖pΓ‖V hi,Γi

)2

.

Hence,

‖u0i ‖Uh

i≤ c9

(‖yΓ‖V h

i,Γi

+ ‖pΓ‖V hi,Γi

),(4.21)

where c9 =√

2c8/α + c28/α2. The right-hand inequality in (4.17) now follows from

(4.19), (4.20), (4.21).

Let 〈〈·, ·〉〉 denote the duality pairing between (V hΓ )2 and ((V h

Γ )∗)2. We define thesubdomain Schur complement operator

Shi ∈ L((V h

Γ )2, ((V hΓ )∗)2),(4.22a)

i = 1, . . . , s, with

〈〈Shi (yΓ, pΓ), (vΓ, qΓ)〉〉

= ai((Hhi )y(yΓ, pΓ), (Hh

i )p(vΓ, qΓ)) + bi((Hhi )u(yΓ, pΓ), (Hh

i )p(vΓ, qΓ))

+ ai((Hhi )y(vΓ, qΓ), (Hh

i )p(yΓ, pΓ)) + mi((Hhi )y(yΓ, pΓ), (Hh

i )y(vΓ, qΓ))(4.22b)


for i ∈ C, and



i )p(vΓ, qΓ))



i )y(vΓ, qΓ))(4.22c)

for i ∈ N . The definition of Hhi implies that for i ∈ C,




i )p(vΓ, qΓ))

+ bi((Hhi )u(vΓ, qΓ), (Hh

i )p(yΓ, pΓ)) + qi((Hhi )u(yΓ, pΓ), (Hh

i )u(vΓ, qΓ))



i )y(vΓ, qΓ)).(4.22d)

We define ri ∈ ((V hΓ )∗)2, i = 1, . . . , s, as follows. If i ∈ C, then

〈〈ri, (vΓ, qΓ)〉〉 = fi((Hhi )p(vΓ, qΓ)) + mi(yi, (Hh

i )y(vΓ, qΓ))

− ai(yi, (Hhi )p(vΓ, qΓ)) − bi(ui, (Hh

i )p(vΓ, qΓ))

− ai((Hhi )y(vΓ, qΓ), pi) −mi(yi, (Hh

i )y(vΓ, qΓ)),(4.23)

where (yi, ui, pi) is the solution of (4.7) with yΓ = 0 and pΓ = 0. If i ∈ N , then

〈〈ri, (vΓ, qΓ)〉〉 = fi((Hhi )p(vΓ, qΓ)) + mi(yi, (Hh

i )y(vΓ, qΓ)) − ai(yi, (Hhi )p(vΓ, qΓ))

− ai((Hhi )y(vΓ, qΓ), pi) −mi(yi, (Hh

i )y(vΓ, qΓ)),(4.24)

where (yi, pi) is the solution of (4.8) with yΓ = 0 and pΓ = 0.Theorem 4.1 implies that the system (4.6) of optimality conditions is equivalent

to the Schur complement system

s∑i=1

Shi (yΓ, pΓ) =

s∑i=1

ri in ((V hΓ )∗)2.(4.25)

From (4.22c) and (4.22d) it is easy to see that the subdomain Schur complementoperator is symmetric; i.e.,

〈〈Shi (yΓ, pΓ), (vΓ, qΓ)〉〉 = 〈〈Sh

i (vΓ, qΓ), (yΓ, pΓ)〉〉 ∀yΓ, vΓ, pΓ, qΓ ∈ V hi,Γi

.

However, unlike subdomain Schur complement operators arising in DD methods forelliptic PDEs, the subdomain Schur complement operator Sh

i is not positive definite.The inertia will be investigated in Theorem 5.5 and we will show that the Schurcomplement operator essentially has the same number of positive and negative eigen-values.

Remark 4.6. If Ωi is a control subdomain, Theorem 4.4 states that the evalu-ation of Si(yΓ, pΓ) requires the solution of a linear-quadratic elliptic optimal controlproblem (4.13). If Ωi is a no-control subdomain, the evaluation of Si(yΓ, pΓ) requiresthe solution of a triangular system of elliptic PDEs (4.8).

We will solve (4.25) using a preconditioned Krylov subspace method. Our pre-conditioner is constructed using the inverses of the subdomain Schur complementoperators Sh

i (yΓ, pΓ), i = 1, . . . , s. Their existence and computation will be studiednext.


4.4. The inverse subdomain Schur complement. The next result estab-lishes the invertibility of the subdomain Schur complement operator Sh

i in the casethat σ > 0.

Theorem 4.7. Let σ > 0 and let ri = (ryi , rpi ) ∈ ((V h

i,Γi)∗)2.

(i) If Ωi is a control subdomain, then the unique solution (yΓ, pΓ) ∈ (V hi,Γi

)2 of

Shi (yΓ, pΓ) = ri in ((V h

i,Γi)∗)2(4.26)

is given by

(yΓ, pΓ) = (TrΓiyi,TrΓipi),

where (yi, ui, pi) ∈ V hi × Uh

i × V hi is the unique solution of

ai(ψ, pi) + mi(yi, ψ) = 〈ryi ,TrΓiψ〉 ∀ψ ∈ V h

i ,(4.27a)


ai(yi, ψ) + bi(ui, ψ) = 〈rpi ,TrΓiψ〉 ∀ψ ∈ V h

i ,(4.27c)

and where 〈·, ·〉 denotes the duality pairing between V hi,Γi

and (V hi,Γi

)∗.

(ii) If Ωi is a no-control subdomain, then the unique solution (yΓ, pΓ) ∈ (V hi,Γi

)2

of (4.26) is given by

(yΓ, pΓ) = (TrΓiyi,TrΓi

pi),

where (yi, ui, pi) ∈ V hi × Uh

i × V hi is the unique solution of

ai(ψ, pi) + mi(yi, ψ) = 〈ryi ,TrΓiψ〉 ∀ψ ∈ V hi ,(4.28a)

ai(yi, ψ) = 〈rpi ,TrΓiψ〉 ∀ψ ∈ V hi .(4.28b)

Proof. (i) By definition (4.22) of Shi , the equality (4.26) can be written as

ai((Hhi )y(yΓ, pΓ), (Hh


i )p(vΓ, qΓ))



i )y(vΓ, qΓ))

= 〈ryi , vΓ〉 + 〈rpi , qΓ〉(4.29)

for all vΓ, qΓ ∈ V hi,Γi

. Using the definition (4.15) of Hhi (yΓ, pΓ) together with (4.29),

we see that (4.26) is equivalent to

ai(yi, (Hhi )p(vΓ, qΓ))

+ bi(ui, (Hhi )p(vΓ, qΓ))

+ ai((Hhi )y(vΓ, qΓ), pi)

+mi(yi, (Hhi )y(vΓ, qΓ)) = 〈ryi , V h

Γ 〉 + 〈rpi , qΓ〉 ∀vΓ, qΓ ∈ V hi,Γi

,(4.30a)

ai(ψ0, pi) + mi(yi, ψ

0) = 0 ∀ψ0 ∈ V hi,0,(4.30b)

bi(μ, pi) + qi(ui, μ) = 0 ∀μ ∈ Uhi ,(4.30c)

ai(yi, φ0) + bi(ui, φ

0) = 0 ∀φ0 ∈ V hi,0,(4.30d)

yi = yΓ, pi = pΓ on Γ.(4.30e)


If we set ψ = ψ0 + (Hhi )p(vΓ, qΓ) ∈ V h

i and φ = φ0 + (Hhi )y(vΓ, qΓ) ∈ V h

i , then (4.30)is equivalent to

ai(ψ, pi) + mi(yi, ψ) = 〈ryi ,TrΓiψ〉 ∀ψ ∈ V h

i ,(4.31a)


ai(yi, φ) + bi(ui, φ) = 〈rpi ,TrΓiφ〉 ∀φ ∈ V h

i ,(4.31c)

yi = yΓ, pi = pΓ on Γ.(4.31d)

The assertion follows if we prove that (4.27) has a unique solution (yi, ui, pi) ∈V hi × Uh

i × V hi . Let (y1

i , u1i , p

1i ), (y

2i , u

2i , p

2i ) ∈ V h

i × Uhi × V h

i be solutions of (4.27).Then (eyi , e

ui , e

pi ) = (y1

i − y2i , u

1i − u2

i , p1i − p2

i ) ∈ V hi × Uh

i × V hi satisfies

ai(ψ, epi ) + mi(e

yi , ψ) = 0 ∀ψ ∈ V h

i ,(4.32a)

bi(μ, epi ) + qi(e

ui , μ) = 0 ∀μ ∈ Uh

i ,(4.32b)

ai(eyi , φ) + bi(e

ui , φ) = 0 ∀φ ∈ V h

i .(4.32c)

If we set ψ = eyi , μ = eui , and φ = −epi in (4.32) and add the resulting equations, weobtain

0 = mi(eyi , e

yi ) + qi(e

ui , e

ui ) ≥ α‖eui ‖2

Uhi.

Hence eui = 0. Equation (4.32c) implies eyi = 0 and, finally, (4.32a) implies epi = 0.(ii) The second statement can be proved analogously.Remark 4.8. Equations (4.27) can be interpreted as the weak form of

−Δyi(x) + σyi(x) = 0 in Ωi,(4.33a)

∂

∂nyi(x) = Uh

i (x) on ∂Ω ∩ ∂Ωi,(4.33b)

∂

∂nyi(x) = ryi (x) on Γ,(4.33c)

−Δpi(x) + σpi(x) = −yi(x) in Ωi,(4.33d)

∂

∂npi(x) = 0 on ∂Ω ∩ ∂Ωi,(4.33e)

∂

∂npi(x) = rpi (x) on Γ,(4.33f)

αui(x) − pi(x) = 0 on ∂Ω ∩ ∂Ωi.(4.33g)

Remark 4.9. If σ = 0, then the subdomain Schur complement operator Si mayno longer be invertible. In this case we define

ai : V hi × V h

i → R, ai(y, ψ) =

∫Ωi

∇y(x)∇ψ(x) + εy(x)ψ(x)dx,

where ε > 0 and we introduce the perturbed subdomain Schur complement operatorSi, which is defined as in (4.22) but with ai replaced by ai. Theorem 4.7 implies that Si

is continuously invertible. A situation somewhat comparable to the case σ = 0 arisesin so-called Neumann–Neumann DD methods for the Laplace equation. Following,e.g., [36, p. 175], we use ai with ε = 1/H2

i , where Hi is the diameter of Ωi.


5. Algebraic formulation. In this section we present the matrix view of theDD Schur complement formulation introduced in the previous section and introducethe Neumann–Neumann preconditioners for the Schur complement. Furthermore, westudy some properties of the Schur complement matrix arising in the optimal controlcontext and discuss the numerical solution of the preconditioned Schur complementsystem.

5.1. Matrix representation of the discretized optimal control problem.A finite element discretization of the optimal control problem (3.1) leads to a large-scale linear-quadratic problem of the form

minimize1

2yTMy + cTy + yTNu +

α

2uTQu + dTu,(5.1a)

subject to Ay + Bu + b = 0,(5.1b)

with A ∈ Rm×m, B ∈ R

m×n, N ∈ Rm×n, M ∈ R

m×m, Q ∈ Rn×n, b ∈ R

m, c ∈ Rm,

d ∈ Rn, y ∈ R

m, u ∈ Rn, and α > 0.

For our model problem the matrices and vectors A, . . . are given as follows. Let{xj}nv

j=1 be the set of vertices of {Tl} and let {φj}nvj=1 be the piecewise linear nodal

basis for V h defined in (4.1). Furthermore, let {μij}

nibv

j=1 be the piecewise linear nodal

basis for Uhi defined in (4.4), where ni

bv is the number of vertices on ∂Ω ∩ ∂Ωi. Weidentify μi

j with a function in L2(∂Ω) by extending μij by zero outside ∂Ωi. We set

μ1 = μ11, . . . , μni

bv= μ1

n1bv, μn1

bv+1 = μ21, . . . , μn1

bv+n2bv

= μ2n2bv, . . . .

Then m = nv, n =∑s

i=1 nibv, and

Ajk = a(φk, φj), Mjk = m(φk, φj), cj = −m(y, φj), bj = f(φj)

for j, k = 1, . . . ,m, and

Bjk = b(μk, φj), Qjk = q(μk, μj)

for j, k = 1, . . . , n, and N = 0, d = 0.Lemma 5.1. Let σ ≥ 0. For our model problem the condition

range(A | B) = Rm(5.2)

is satisfied, M,Q are symmetric, and the Hessian in (5.1a) is positive definite on thenull-space of the constraints (5.1b); i.e.,(

zv

)T (M NNT αQ

)(zv

)> 0 ∀ z ∈ R

m,v ∈ Rn with Az + Bv = 0.(5.3)

Proof. If σ > 0, then A is symmetric positive definite and (5.2) follows immedi-ately. It holds that

null(A | B) =

{(−A−1Bv

v

): v ∈ R

n

}and (5.3) is equivalent to

vT (αQ + BTA−TMA−1B)v > 0 ∀v ∈ Rn,(5.4)


where we have used the fact that in the model problem N=0. Condition (5.4) follows,since in our model problem M is symmetric positive semidefinite and Q is symmetricpositive definite.

If σ= 0, then A is symmetric positive semidefinite and null(A) = span{e}, wheree ∈ R

m is the vector of all ones. If (zT ,vT )T ∈ null(A | B), then 0 = eT (Az +Bv) = eTBv. Hence, v ∈ span{BTe}⊥. Moreover, for each v ∈ span{BTe}⊥, theequation Az = −Bv has a solution and the minimum norm solution of this equationis characterized by eT z = 0. This implies that

null(A | B) = span

{(e0

)}+

{(−A†Bv

v

): v ∈ span{BTe}⊥

},

where A† denotes the pseudoinverse of A, and dim(null(A | B)) = n. Consequently,dim(range(A | B)) = m, which implies (5.2). Condition (5.3) follows, since in ourmodel problem M is symmetric positive semidefinite with eTMe =

∫Ωo

1dx > 0, Q issymmetric positive definite, and N = 0.

The following result is known, and it is stated here for completeness. A proof canbe found in [17].

Theorem 5.2. (i) Let (5.2) and (5.3) be satisfied. The necessary and sufficientoptimality conditions for (5.1) are given by⎛⎝ M N AT

NT αQ BT

A B 0

⎞⎠⎛⎝yup

⎞⎠ = −

⎛⎝ cdb

⎞⎠.(5.5)

(ii) If (5.2) and (5.3) are satisfied, then the system matrix in (5.5) has m + npositive and m negative eigenvalues.

The equation (5.5) is the matrix representation of the discretized optimality con-ditions (4.6).

5.2. Domain decomposition Schur complement formulation. We can usethe decomposition of Ω to decompose the matrices A, etc. Let the local (bi)linearforms ai, . . . be defined as in section 4.

Let Ryi , i = 1, . . . , s, be the restriction operator which maps from the vector of

coefficient unknowns on the artificial boundary, yΓ, to only those associated with theboundary of Ωi, and define

Ri =

(Ry

i

Rpi

), Rp

i = Ryi .(5.6)

For i = 1, . . . , s, we define the submatrices

(AiII)jk = ai(φk, φj), xj , xk ∈ Ωi,

(AiIΓ)jk = ai(φk, φj), xj ∈ Ωi, xk ∈ ∂Ωi \ ∂Ω,

(AiΓΓ)jk = ai(φk, φj), xj , xk ∈ ∂Ωi \ ∂Ω,

AiΓI = (Ai

IΓ)T , and we define AΓΓ =∑s

i=1(Ryi )

TAiΓΓR

yi . After a suitable reordering

of rows and columns, the stiffness matrix can be written as

A =

⎛⎜⎜⎜⎝A1

II A1IΓR

y1

. . ....

AsII As

IΓRys

(Ry1)

TA1ΓI · · · (Ry

s)TAs

ΓI AΓΓ

⎞⎟⎟⎟⎠.


Similar decompositions can be introduced for M and c, as well as y,p.For i = 1, . . . , s, we define the submatrices

(BiII)jk = bi(μk, φj), xj ∈ Ωi, xk ∈ ∂Ωi ∩ ∂Ω,

(BiΓI)jk = bi(μk, φj), xj ∈ ∂Ωi \ ∂Ω, xk ∈ ∂Ωi ∩ ∂Ω.

After a suitable reordering of rows and columns, the matrix B can be written as

B =

⎛⎜⎜⎜⎝B1

II

. . .

BsII

(Ry1)

TB1ΓI · · · (Ry

s)TBs

ΓI

⎞⎟⎟⎟⎠.

Note that in our particular control discretization, all basis functions μik for the dis-

cretized control uh have support in only one subdomain boundary ∂Ωi (see the right-hand plot in Figure 4.1). Consequently, there is no Bi

ΓΓ. The matrix Q and thevector u can be decomposed analogously. Note that there is no uΓ.

We can now insert the DD structure of the matrices A,M,B,Q into (5.5). Aftera symmetric permutation, (5.5) can be written as⎛⎜⎜⎜⎝

K1II (K1

ΓI)T R1

. . ....

KsII (Ks

ΓI)T Rs

RT1 K1

ΓI · · · RTs Ks

ΓI KΓΓ

⎞⎟⎟⎟⎠⎛⎜⎜⎜⎝

x1I...

xdI

xΓ

⎞⎟⎟⎟⎠ =

⎛⎜⎜⎜⎝g1I...

gdI

gΓ

⎞⎟⎟⎟⎠,(5.7)

where

KiΓΓ =

(Mi

ΓΓ (AiΓΓ)T

AiΓΓ

), i = 1, . . . , s, KΓΓ =

s∑i=1

RTi Ki

ΓΓRi,

KiII =

⎛⎝ MiII Ni

II (AiII)

T

(NiII)

T αQiII (Bi

II)T

AiII Bi

II

⎞⎠, KiΓI =

(Mi

ΓI NiΓI (Ai

IΓ)T

AiΓI Bi

ΓI

), i ∈ C,

and

KiII =

(Mi

II (AiII)

T

AiII

), Ki

ΓI =

(Mi

ΓI (AiIΓ)T

AiΓI

), i ∈ N .

Furthermore,

xΓ =

(yΓ

pΓ

), gΓ =

(cΓ

bΓ

), xi

I =

⎛⎝ yiI

uiI

piI

⎞⎠, giI =

⎛⎝ ciIdiI

biI

⎞⎠, i ∈ C,

and

xiI =

(yiI

piI

), gi

I =

(ciIbiI

), i ∈ N .


Frequently, we use the compact notation(KII KT

ΓI

KΓI KΓΓ

)(xI

xΓ

)=

(gI

gΓ

)(5.8)

or even Kx = g instead of (5.7).The matrix representation of the operators Hh

i defined in (4.15) and (4.16) isgiven by

Hi =

(−(Ki

II)−1Ki

IΓ

I

).

The matrix representation of the local Schur complement operators Shi defined in (4.22)

is given by

Si = KiΓΓ − Ki

ΓI(KiII)

−1(KiΓI)

T .(5.9)

The matrix

Ki =

(Ki

II (KiΓI)

T

KiΓI Ki

ΓΓ

)plays an important role in the computation of the inverse of Si (assuming it exists).In fact, if Ki

II is invertible,

Ki =

(I 0

KiΓI(K

iII)

−1 I

)(Ki

II 00 Si

)(I (Ki

II)−1(Ki

ΓI)T

0 I

),(5.10)

and Si is invertible if and only if Ki is invertible. In this case,

S−1i v = (0 I) (Ki)−1

(0I

)v

(see, e.g., [36, p. 113]).The following theorem is concerned with the invertibility of the submatrices Ki

II ,which is important for the computation of Si, and with the invertibility of the sub-matrices Ki, which is important for the computation of (Si)

−1.Theorem 5.3. Consider the following model problem.(i) If σ ≥ 0, then the following hold:

a. The subdomain matrices AiII ∈ R

mIi×mI

i are invertible and MiII ∈ R

mIi×mI

i

are symmetric positive semidefinite, i = 1, . . . , s.b. For i ∈ C, (

ziIIviII

)T (Mi

II NiII

(NiII)

T αQiII

)(ziIIviII

)> 0(5.11)

for all ziII ∈ RmI

i ,viII ∈ R

nIi with Ai

IIziII + Bi

IIviII = 0.

c. The matrices KiII , i = 1, . . . , s, are invertible.

(ii) If σ > 0, then the following hold:a. The subdomain matrices Ai ∈ R

mi×mi are invertible and Mi ∈ Rmi×mi

are symmetric positive semidefinite, i = 1, . . . , s.


b. For i ∈ C, (zi

vi

)T (Mi Ni

(Ni)T αQi

)(zi

vi

)> 0(5.12)

for all zi ∈ Rmi ,vi ∈ R

ni with Aizi + Bivi = 0.c. The matrices Ki, i = 1, . . . , s, are invertible.

(iii) If σ = 0, then null(Ai) = span{e}, where e ∈ Rmi is the vector of all

ones and Mi ∈ Rmi×mi are symmetric positive semidefinite, i = 1, . . . , s.

Furthermore, for i ∈ C, condition (5.12) is satisfied for all zi ∈ Rmi ,vi ∈ R

ni

with Aizi + Bivi = 0, provided that eTMie > 0.Proof. The matrices Ai

II ∈ Rmi×mi and Ai ∈ R

mi×mi are the stiffness matricescorresponding to the problems

−Δy + σy = f in Ωi, y = 0 on ∂Ωi,

and

−Δy + σy = f in Ωi, ∂y/∂n = 0 on ∂Ωi,

respectively. This implies (a) that AiII is symmetric positive definite, (b) that Ai is

symmetric positive definite if σ > 0, and (c) that null(Ai) = span{e} if σ = 0.The proof of assertions (i)–(iii) is identical to the proof of Theorem 5.2.Remark 5.4. Consider the model problem with σ = 0.For i ∈ N , the matrix Ki is singular. After a symmetric permutation, Ki is given

by

Ki =

(Mi Ai

Ai 0

),

where

Ai =

(Ai

II (AiΓI)

T

AiΓI Ai

ΓΓ

), Mi =

(Mi

II (MiΓI)

T

MiΓI Mi

ΓΓ

).

The vector (0T , eT )T ∈ R2mi is in the null-space of Ki and, if Mie = 0, so is the

vector (eT ,0T ) ∈ R2mi .

If i ∈ C and Mie = 0, the matrix Ki is singular. In fact, after a symmetricpermutation, Ki is given by

Ki =

⎛⎝Mi 0 Ai

0 QiII (Bi)T

Ai Bi 0

⎞⎠,

where Ai,Mi are defined as before and

Bi =

(Bi

II

BiΓI

).

The vector (eT ,0T ,0T )T ∈ R2mi+ni is in the null-space of Ki.

By definition of Mi,

eTMie =∑

xj ,xk∈Ωi

∫Ωo

φj(x)φk(x)dx.

Hence Mie = 0 if and only if Ωi ∩ Ωo = ∅.


Theorem 5.3(i) guarantees that KII is invertible. Hence, we can form the Schurcomplement system

SxΓ = r,(5.13)

where

S = KΓΓ − KΓIK−1II KT

ΓI(5.14)

and

r = gΓ − KΓIK−1II gI .

The Schur complement matrix S can be written as a sum of subdomain Schur com-plement matrices,

S =s∑

i=1

RTi SiRi,(5.15)

where Si, i = 1, . . . , s, is defined in (5.9).Theorem 5.5. (i) Consider the model problem with σ ≥ 0. The Schur com-

plement matrix S has m−∑s

i=1 mIi positive and m−

∑si=1 m

Ii negative eigenvalues,

where mIi is the number of finite element nodes in Ωi \ Γ.

(ii) Consider the model problem with σ > 0. The subdomain Schur complementmatrix Si, i = 1, . . . , s, has mi−mI

i positive and mi−mIi negative eigenvalues, where

mi is the number of finite element nodes in Ωi and mIi is the number of finite element

nodes in Ωi \ Γ.(iii) Consider the model problem with σ = 0. If i ∈ N or if i ∈ C and eTMie = 0,

then the subdomain Schur complement matrix Si is singular. The eigenvalue zero ofSi has multiplicity one or two.

Proof. (i) Recall (5.14). It is easy to verify that(KII KT

ΓI

KΓI KΓΓ

)=

(KII 0KΓI I

)(K−1

II 00 S

)(KII KT

ΓI

0 I

).

The matrix K is a symmetric permutation of the system matrix in (5.5) and, hence,both matrices have the same eigenvalues. By Theorem 5.2, the system matrix in (5.5),and hence K, both have m + n positive and m negative eigenvalues.

Let mIi be the number of finite element nodes in Ωi and let nI

i be the number offinite element nodes in ∂Ωi \ ∂Ω. Applying the same arguments used to analyze theinertia of K to KII , one can show that the matrix KII has

∑si=1 m

Ii +nI

i positive and∑si=1 m

Ii negative eigenvalues. By Sylvester’s law of inertia, the number of positive

(negative) eigenvalues of K is equal to the number of positive (negative) eigenvaluesof K−1

II plus the number of positive (negative) eigenvalues of S. Since n =∑s

i=1 nIi ,

this implies that S has m−∑s

i=1 mIi positive and m−

∑si=1 m

Ii negative eigenvalues.

(ii) The second assertion can be proved using arguments analogous to those ap-plied in (i). We omit the details.

(iii) By Lemma 5.4, KiII is invertible. In Remark 5.4 we have shown that Ki is

singular if i ∈ N or if i ∈ C and eTMie = 0. In these cases the zero eigenvalue of Ki

has mutiplicity one or two. Hence, the assertion follows from (5.10).


5.3. The Neumann–Neumann preconditioners. It is now relatively easy togeneralize the Neumann–Dirichlet and Neumann–Neumann preconditioners used inthe context of elliptic PDEs (see, e.g., [36, Chap. 4] for an overview) to the optimalcontrol context. We focus on Neumann–Neumann preconditioners.

Let Dyi be the diagonal matrix whose entries are computed as follows. If xk ∈ Γi,

then (Dyi )

−1kk is the number of subdomains that share node xk. Note that

∑i D

yi = I.

Furthermore, let Dpi = Dy

i and

Di =

(Dy

i

Dpi

).

By Theorem 5.3(i), Si, i = 1, . . . , s, is well defined. If σ > 0, then Theorem 5.5(ii)guarantees the existence of S−1

i , i = 1, . . . , s. The one-level Neumann–Neumannpreconditioner is given by

P =∑i

DiRTi S−1

i RiDi,(5.16)

where S−1i is equal to S−1

i if Si is invertible and otherwise S−1i is the inverse of the

Schur complement matrix Si obtained by adding a positive multiple of∫Ωi

ψk, ψj to the

bilinear form ai (cf. Remark 4.9). (With this modified bilinear form Theorem 5.5(ii)

guarantees the invertibility of Si.)It is well known that the performance of one-level Neumann–Neumann precondi-

tioners for elliptic PDEs deteriorates quickly as the number of subdomains increases.The same is observed for the Neumann–Neumann preconditioner (5.16) in the optimalcontrol context (see section 6). To avoid this, we include a coarse grid. More precisely,we adapt the balancing Neumann–Neumann preconditioner proposed in [30] to theoptimal control context. Following the description in [36, sect. 4.3.3], the balancingNeumann–Neumann preconditioner for the optimal control problem is given by

P =(I − RT

0 S−10 R0S

)(s∑

i=1

DiRTi S−1

i RiDi

)(I − SRT

0 S−10 R0

)+ RT

0 S−10 R0,

(5.17)

where S−1i is defined as in (5.16), and where S0 = R0SRT

0 and R0 is defined as in

(5.6) with Ry0 being the restriction operator which returns for each subdomain the

weighted sum of the values of all the nodes on the boundary of that subdomain. Theweight corresponding to an interface node is one divided by the number of subdomainsin which the node is contained.

6. Numerical results. We consider the model problem (1.1) with Ω = Ωo =

(−1, 1)2, σ = 0, f(x) = (2π2 + 1) sin(πx1) sin(πx2), and y(x) = sin(πx1) sin(πx2).Additional numerical results for the model problem (1.1) with σ = 1 can be found

in [20], and results using a problem with distributed control are given in [32, Chap. 4].The domain Ω = (−1, 1)2 was partitioned into equal-sized square subdomains in a

checkerboard pattern. The side length of each subdomain was denoted by H. Regularmeshes consisting of triangular elements of various widths (denoted as h) were gen-erated. The preconditioned KKT system was solved using symmetric QMR (sQMR)with the Neumann–Neumann and the balancing Neumann–Neumann precondition-ers. In all cases, the stopping criteria was 10−8 for the �2-norm of the preconditionedresidual.


Table 6.1

Number of iterations for the sQMR method with Neumann–Neumann and balancing Neumann–Neumann preconditioners for varying mesh sizes h, subdomain sizes H, and regularization parame-ters α.

Neumann–Neumann, α = 1.0sQMR iterations

H \ h 14

18

116

132

164

1128

12

12 15 19 24 26 2814

53 69 94 107 11918

170 226 287 345116

509 679 798132

1578 2233

Bal. Neumann–Neumann, α = 1.0sQMR iterations

H \ h 14

18

116

132

164

1128

12

5 6 8 10 11 1214

5 9 12 14 1518

5 10 13 15116

5 9 13132

4 9

Neumann–Neumann, α = 10−4

sQMR iterations

H \ h 14

18

116

132

164

1128

12

13 14 16 19 21 2314

55 54 63 67 7118

129 143 165 187116

432 439 465132

1544 1588

Bal. Neumann–Neumann, α = 10−4

sQMR iterations

H \ h 14

18

116

132

164

1128

12

6 9 9 11 11 1314

8 11 13 16 1918

7 10 13 17116

6 10 14132

5 11

Neumann–Neumann, α = 10−8

sQMR iterations

H \ h 14

18

116

132

164

1128

12

15 16 16 21 21 2314

58 61 63 74 7618

217 191 202 214116

583 536 584132

1255 1249

Bal. Neumann–Neumann, α = 10−8

sQMR iterations

H \ h 14

18

116

132

164

1128

12

9 10 13 15 17 2014

11 17 21 26 3018

13 18 24 30116

12 19 24132

11 16

The subdomain Schur complements are well defined, but for σ = 0 subdomainSchur complements corresponding to no-control subdomains are not invertible (see

Theorem 5.5 and Remark 5.4 and recall that Ω = Ωo). In this case we construct Si,

where Si is computed as Si but with the bilinear form ai replaced by

ai : Vi × Vi → R, ai(y, ψ) =

∫Ωi

∇y(x)∇ψ(x) + H−2y(x)ψ(x)dx;

cf. Remark 4.9. The subproblem solves involving KiII and Ki (the latter constructed

with bilinear form ai) were performed using a sparse LU decomposition.

Table 6.1 summarizes the sQMR iteration counts for both preconditioners. Thesenumerical results show that both the Neumann–Neumann and the balancing Neumann–Neumann preconditioners grow worse slowly when the mesh size h is reduced whileother parameters are kept constant. This behavior is consistent with the PDE versionof these preconditioners. As expected, the Neumann–Neumann preconditioner grows


worse quickly as the number of subdomains increases while the balancing Neumann–Neumann preconditioner remains effective. A notable result is that both precondi-tioners depend only weakly on the regularization parameter α. As α is reduced from1 to 10−8, the iteration count for the balancing Neumann–Neumann preconditionergrows by only a factor of two or less in most cases. The one-level Neumman–Neumannpreconditioner appeared to be even less sensitive to small α.

7. Conclusion. We have introduced Neumann–Neumann preconditioners forlinear-quadratic elliptic optimal control problems, which are obtained by introducingauxiliary states and adjoints on the subdomain interfaces and formulating subdomainoptimality conditions with Dirichlet boundary conditions for states and adjoints onthe subdomain interfaces. These subdomain optimality conditions are coupled byrequiring continuity of normal derivatives of subdomain states and adjoints, definedas solutions of the local optimality conditions, on the subdomain interfaces. Thisleads to subdomain Schur complement operators that map Dirichlet interface datainto Neumann interface data. The original optimality conditions are equivalent to aSchur complement system in which the operator is given by the sum of subdomainSchur complements.

The application of subdomain Schur complements has been shown to correspondto the solution of subdomain optimal control problems with Dirichlet boundary con-ditions on the subdomain interfaces that are otherwise smaller copies of the originaloptimal control problem. Conditions are given that establish the invertibility of sub-domain Schur complements. The application of the inverses of local Schur complementoperators has been shown to correspond to the solution of subdomain optimal controlproblems with Neumann boundary conditions on the subdomain interfaces that areotherwise smaller copies of the original optimal control problem. For a finite elementdiscretization, the inertia of the local Schur complement matrices as well as that ofthe sum of the local matrices has been analyzed.

Numerical results for our (balancing) Neumann–Neumann preconditioners ap-plied to a model problem indicate that the numerical performance of our (balancing)Neumann–Neumann DD methods applied to linear-quadratic elliptic optimal con-trol problems is similar to that of their counterparts applied to single elliptic PDEs.We also observe that the performance of our preconditioners applied to the modelproblem (1.1) is largely independent of the size of the regularization parameter. Aconvergence theory for Neumann–Neumann methods applied to elliptic optimal con-trol problems comparable to the convergence theory for these methods applied toindividual symmetric elliptic PDEs [13, 14] or to the Stokes equation [33] is stillmissing.

While we have used an elliptic model optimal control problem with Neumannboundary control to illustrate the preconditioners, they can be applied to many otherelliptic model optimal control problems. For example, it is straightforward to admitstate equations with nonconstant coefficient functions (although the scaling matricesDi may have to be adjusted if there are large jumps in coefficients across subdo-mains [13, 14]). It is also easy to extend the methods to distributed controls (see[32, Chap. 4]). Recently, we have extended the approach in this paper to advectiondominated optimal control problems [2]. This requires a careful adjustment of theinterface conditions to account for advection terms.

Acknowledgment. The authors thank the anonymous referees for their carefulreading and suggestions, which have led to an improvement in the presentation of thepaper.


REFERENCES

[1] U. M. Ascher and E. Haber, A multigrid method for distributed parameter estimation prob-lems, in Tenth Copper Mountain Conference on Multigrid Methods (Copper Mountain,CO, 2001), Electron. Trans. Numer. Anal., 15 (2003), pp. 1–17.

[2] R. Bartlett, M. Heinkenschloss, D. Ridzal, and B. van Bloemen Waanders, Domaindecomposition methods for advection dominated linear-quadratic elliptic optimal controlproblems, Comput. Methods Appl. Mech. Engrg., to appear.

[3] A. Battermann and M. Heinkenschloss, Preconditioners for Karush–Kuhn–Tucker matri-ces arising in the optimal control of distributed systems, in Control and Estimation ofDistributed Parameter Systems, W. Desch, F. Kappel, and K. Kunisch, eds., Internat. Ser.Numer. Math. 126, Birkhauser-Verlag, Basel, 1998, pp. 15–32.

[4] A. Battermann and E. W. Sachs, Block preconditioners for KKT systems in PDE-governedoptimal control problems, in Fast Solution of Discretized Optimization Problems (Berlin,2000), K.-H. Hoffmann, R. H. W. Hoppe, and V. Schulz, eds., Internat. Ser. Numer.Math. 138, Birkhauser-Verlag, Basel, 2001, pp. 1–18.

[5] J.-D. Benamou, A Domain Decomposition Method with Coupled Transmission Conditionsfor the Optimal Control of Systems Governed by Elliptic Partial Differential Equations,Technical report 2246, INRIA, Rocquencourt, France, 1994.

[6] J.-D. Benamou, A domain decomposition method with coupled transmission conditions forthe optimal control of systems governed by elliptic partial differential equations, SIAM J.Numer. Anal., 33 (1996), pp. 2401–2416.

[7] J.-D. Benamou, A domain decomposition method for control problems, in Proceedings of the9th International Conference on Domain Decomposition, P. Bjørstad, M. Espedal, andD. Keyes, eds., Domain Decomposition Press, Bergen, Norway, 1998, pp. 266–273.

[8] M. Benzi, G. H. Golub, and J. Liesen, Numerical solution of saddle point problems, ActaNumer., 14 (2005), pp. 1–137.

[9] G. Biros and O. Ghattas, Parallel preconditioners for KKT systems arising in optimal con-trol of viscous incompressible flows, in Proceedings of Parallel CFD ’99, Williamsburg,VA, 1999, North–Holland, Amsterdam, London, New York, 1999; available online fromhttp://www.cs.cmu.edu/∼oghattas.

[10] G. Biros and O. Ghattas, Parallel Lagrange–Newton–Krylov–Schur methods for PDE-constrained optimization. Part I: The Krylov–Schur solver, SIAM J. Sci. Comput., 27(2005), pp. 687–713.

[11] A. Bounaim, A Lagrangian approach to a DDM for an optimal control problem, in Proceedingsof the 9th International Conference on Domain Decomposition, P. Bjørstad, M. Espedal,and D. Keyes, eds., Domain Decomposition Press, Bergen, Norway, 1998, pp. 283–289.

[12] J. E. Dennis and R. M. Lewis, A Comparison of Nonlinear Programming Approaches toan Elliptic Inverse Problem and a New Domain Decomposition Approach, Technical re-port TR94–33, Department of Computational and Applied Mathematics, Rice University,Houston, TX, 1994.

[13] M. Dryja, B. F. Smith, and O. B. Widlund, Schwarz analysis of iterative substructuringalgorithms for elliptic problems in three dimensions, SIAM J. Numer. Anal., 31 (1994),pp. 1662–1694.

[14] M. Dryja and O. B. Widlund, Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element methods, Comm. Pure Appl. Math., 48 (1995), pp. 121–155.

[15] P. E. Gill, W. Murray, D. B. Ponceleon, and M. A. Saunders, Preconditioners for indefi-nite systems arising in optimization, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 292–311.

[16] G. H. Golub and C. Greif, On solving block-structured indefinite linear systems, SIAM J.Sci. Comput., 24 (2003), pp. 2076–2092.

[17] N. I. M. Gould, On practical conditions for the existence and uniqueness of solutions to thegeneral equality quadratic programming problem, Math. Programming, 32 (1985), pp. 90–99.

[18] N. I. M. Gould, M. E. Hribar, and J. Nocedal, On the solution of equality constrainedquadratic programming problems arising in optimization, SIAM J. Sci. Comput., 23 (2001),pp. 1376–1395.

[19] E. Haber and U. Ascher, Preconditioned all-at-once methods for large, sparse parameterestimation problems, Inverse Problems, 17 (2001), pp. 1847–1864.

[20] M. Heinkenschloss and H. Nguyen, Balancing Neumann-Neumann methods for elliptic op-timal control problems, in Domain Decomposition Methods in Science and Engineering,R. Kornhuber, R. H. W. Hoppe, J. Periaux, O. Pironneau, O. B. Widlund, and J. Xu, eds.,Lect. Notes Comput. Sci. Eng. 40, Springer-Verlag, Heidelberg, 2004, pp. 589–596.


[21] R. H. W. Hoppe, S. I. Petrova, and V. Schulz, Primal-dual Newton-type interior-pointmethod for topology optimization, J. Optim. Theory Appl., 114 (2002), pp. 545–571.

[22] J. Jahn, Introduction to the Theory of Nonlinear Optimization, 2nd ed., Springer-Verlag,Berlin, Heidelberg, New York, 1996.

[23] C. Keller, N. I. M. Gould, and A. J. Wathen, Constraint preconditioning for indefinitelinear systems, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1300–1317.

[24] A. Klawonn and O. B. Widlund, FETI and Neumann-Neumann iterative substructuringmethods: Connections and new results, Comm. Pure Appl. Math., 54 (2001), pp. 57–90.

[25] K. Kunisch and X.-C. Tai, Nonoverlapping domain decomposition methods for inverse prob-lems, in Proceedings of the 9th International Conference on Domain Decomposition Meth-ods, Domain Decomposition Press, Bergen, Norway, 1998, pp. 517–524.

[26] P. Le Tallec, Domain decomposition methods in computational mechanics, Comput. Mech.Adv., 1 (1994), pp. 121–220.

[27] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, Heidelberg, New York, 1971.

[28] J.-L. Lions and O. Pironneau, Sur le controle parallele des systemes distribues, C.R. Acad.Sci. Paris Ser. I Math., 327 (1998), pp. 993–998.

[29] L. Luksan and J. Vlcek, Indefinitely preconditioned inexact Newton method for large sparseequality constrained non-linear programming problems, Numer. Linear Algebra Appl., 5(1998), pp. 219–247.

[30] J. Mandel, Balancing domain decomposition, Comm. Numer. Methods Engrg., 9 (1993),pp. 233–241.

[31] M. F. Murphy, G. H. Golub, and A. J. Wathen, A note on preconditioning for indefinitelinear systems, SIAM J. Sci. Comput., 21 (2000), pp. 1969–1972.

[32] H. Q. Nguyen, Domain Decomposition Methods for Linear-Quadratic Elliptic Optimal Con-trol Problems, Ph.D. thesis, Department of Computational and Applied Mathematics,Rice University, Houston, TX, 2004; available online from http://www.caam.rice.edu/caam/trs/2004/TR04-16.pdf.

[33] L. F. Pavarino and O. B. Widlund, Balancing Neumann-Neumann methods for incompress-ible Stokes equations, Comm. Pure Appl. Math., 55 (2002), pp. 302–335.

[34] A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equa-tions, Oxford University Press, Oxford, UK, 1999.

[35] M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, TextsAppl. Math. 13, Springer-Verlag, Berlin, Heidelberg, New York, 1993.

[36] B. Smith, P. Bjørstad, and W. Gropp, Domain Decomposition. Parallel Multilevel Meth-ods for Elliptic Partial Differential Equations, Cambridge University Press, Cambridge,London, New York, 1996.

[37] X.-C. Tai, J. Frøyen, M. S. Espedal, and T. F. Chan, Overlapping domain decomposi-tion and multigrid methods for inverse problems, in Domain Decomposition Methods 10(Boulder, CO, 1997), Contemp. Math. 218, AMS, Providence, RI, 1998, pp. 523–529.

[38] A. Toselli and O. Widlund, Domain decomposition methods—Algorithms and theory,Springer Ser. Comput. Math. 34, Springer-Verlag, Berlin, 2005.

[39] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, UK, 1987.

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