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A PRIORI ERROR ANALYSIS FOR DISCRETIZATION OF SPARSEELLIPTIC OPTIMAL CONTROL PROBLEMS IN MEASURE SPACE

KONSTANTIN PIEPER† AND BORIS VEXLER‡

Abstract. In this paper an optimal control problem is considered, where the control variablelies in a measure space and the state variable fulfills an elliptic equation. This formulation leadsto a sparse structure of the optimal control. For this problem a finite element discretization basedon [7] is discussed and a priori error estimates are derived, which significantly improve the estimatesfrom [7]. Numerical examples for problems in two and three space dimensions illustrate our results.

Key words. optimal control, sparsity, finite elements, error estimates

AMS subject classifications.

1. Introduction. In this paper we consider the following optimal control prob-lem:

Minimize J(q, u) =1

2‖u− ud‖2L2(Ω) + α‖q‖M(Ω), q ∈M(Ω) (1.1)

subject to {−∆u = q in Ω

u = 0 on ∂Ω.(1.2)

Here, Ω ⊂ Rd (d = 2, 3) is a convex bounded domain with a C2,β-boundary ∂Ω.The control variable q is searched for in the space of regular Borel measures M(Ω),which is identified with the dual of the space of continuous functions vanishing on theboundary C0(Ω). The state variable u is the solution of the state equation (1.2), seethe next section for the precise weak formulation. The desired state ud is in L

2(Ω),see also further assumptions (ud ∈ Lp(Ω) or ud ∈ L∞(Ω)) below. The parameter α isassumed to be positive.

This problem setting with the control from a measure space was considered in [10],where it has been observed that this setting leads to optimal controls with sparsestructure. This is important for many applications, cf., e.g., [11]. For another func-tional analytic concept utilizing the L1(Ω)-norm of the control combined with a L2-regularization and/or with control constraints we refer e.g. to [18, 20, 9].

This paper is mainly concerned with the discretization of the problem (1.1) –(1.2). In [7] a discretization concept for this problem is presented and the followingerror estimates are derived:

J(q̄, ū)− J(q̄h, ūh) = O(h2−

d2

)and ‖ū− ūh‖L2(Ω) = O

(h1−

d4

),

†Lehrstuhl für Mathematische Optimierung, Technische Universität München, Fakultät für Math-ematik, Boltzmannstraße 3, 85748 Garching b. München, Germany ([email protected]). The firstauthor gratefully acknowledges support from the International Research Training Group IGDK1754,funded by the German Science Foundation (DFG).‡Lehrstuhl für Mathematische Optimierung, Technische Universität München, Fakultät für Math-

ematik, Boltzmannstraße 3, 85748 Garching b. München, Germany ([email protected]). The secondauthor gratefully acknowledges support by the DFG Priority Program 1253 “Optimization with Par-tial Differential Equations”.

1

2 KONSTANTIN PIEPER AND BORIS VEXLER

where (q̄, ū) is the unique solution to (1.1) – (1.2), h is the discretization parameterand (q̄h, ūh) is the discrete solution. Our main contribution is the improvement ofthese estimates using the same discretization concept to

J(q̄, ū)− J(q̄h, ūh) = O(h4−d|lnh|γ

)and ‖ū− ūh‖L2(Ω) = O

(h2−

d2 |lnh|

γ2

), (1.3)

with γ = 72 for d = 2 and γ = 1 for d = 3. Moreover we provide an estimate for the

error in the control variable. Although one can only expect q̄h∗⇀ q̄ in M(Ω), see [7],

we derive the following estimate with respect to the H−2(Ω)-norm:

‖q̄ − q̄h‖H−2(Ω) = O(h2−

d2 |lnh|

γ2

).

We obtain these improved estimates with similar assumptions as in [7], but employingerror estimates for the state solution in Lt(Ω) for t < 2, which are of (almost) optimalorder, see Lemma 3.3, combined with a more careful study of the regularity of thestate solutions for a measure valued right hand side. However, the assumption on thedesired state ud needs to be slightly stronger than in [7], see Remark 4.1 below.

The numerical examples (see Section 7) indicate that the estimates (1.3) are sharp.However, we make the following observation: In the two-dimensional case we see thepredicted order of almost O(h) with respect to the state variable in all examples. Butfor the three-dimensional case, the predicted order of (almost) O(h 12 ) is observed onlyin examples where the exact optimal controls contains Dirac measures. For optimalcontrols q̄ with better regularity, we observe convergence rates similar to the two-dimensional case. Motivated by this observation, we show in Section 2, that assuminga bounded desired state ud ∈ L∞(Ω) implies that ū must be bounded as well, whichimmediately rules out controls containing Dirac measures. Another direct consequenceis q̄ ∈ H−1(Ω), which allows us to show an order of convergence of (almost) orderO(h 23 ) for the state error ‖ū− ūh‖L2(Ω) and d = 3. Under the additional assumptionthat q̄ ∈W−1,p(Ω) with p > 2 this rate can be improved further, see Theorem 5.1.

The paper is structured as follows. In the next section we recall the optimalityconditions from [10] and [7], discuss some consequences of them and prove that theoptimal state ū is bounded provided that ud ∈ L∞(Ω). In Section 3 we describe thefinite element discretization and derive some error estimates for the state equation. InSection 4 we prove the main estimates (1.3) and in Section 5 we derive an improvedestimates for d = 3 under an additional regularity assumption. In the last section wepresent numerical examples illustrating our results.

Throughout we will denote by (·, ·) the L2(Ω) inner product and by 〈·, ·〉 theduality product between M(Ω) and C0(Ω).

2. Optimality system and regularity. As the first step we recall the weakformulation of the state equation (1.2). For a given q ∈ M(Ω) the solution u = u(q)is determined by

u ∈ L2(Ω) : (u,−∆ϕ) = 〈q, ϕ〉 for all ϕ ∈ H2(Ω) ∩H10 (Ω).

It is well known, that the above formulation possesses a unique solution, which belongsto W 1,s0 (Ω) for all 1 ≤ s < dd−1 , see, e.g., [6]. Moreover, there holds the followingstability estimate.

Lemma 2.1. For each 0 < ε ≤ 1d−1 let sε be given as

sε =d

d− 1− ε.

FEM FOR SPARSE ELLIPTIC OPTIMAL CONTROL 3

There exists a constant c independent of ε, such that for all q ∈ M(Ω) and thecorresponding solution u of (1.2) the following estimate holds:

‖u‖W 1,sε0 (Ω) ≤c

ε‖q‖M(Ω).

Proof. The estimate for ‖u‖W 1,s0 (Ω) with an s-dependent constant is shown in [6].

To obtain the precise dependence of ε we use the continuous embedding of W1,s′ε0 (Ω)

into C0(Ω), there

1

s′ε+

1

sε= 1, s′ε > d.

From Theorem 8.10 (in the 3. edition) in [2] we obtain

‖v‖C0(Ω) ≤c

ε‖v‖

W1,s′ε0 (Ω)

for all v ∈ W 1,s′ε(Ω) with the constant c independent of ε. Using the result from [1],see also [15], we estimate

‖∇u‖Lsε (Ω) ≤ c supv∈W 1,s

′ε

0 (Ω)

(∇u,∇v)‖∇v‖

Ls′ε (Ω)

= c supv∈W 1,s

′ε

0 (Ω)

〈q, v〉‖∇v‖

Ls′ε (Ω)

≤ cε‖q‖M(Ω).

This completes the proof.Due to the embedding of W 1,s0 (Ω) into L

2(Ω) for 2dd+2 ≤ s <dd−1 the cost func-

tional (1.1) is well-defined. Moreover, the solution operator mapping q ∈ M(Ω) tou = u(q) ∈ L2(Ω) is injective and therefore the cost functional is strictly convex. Us-ing this fact, the existence of a unique solution (q̄, ū) to (1.1) – (1.2) can be directlyobtained, see [10] for details.

The following optimality system is obtained in [10, 7].Theorem 2.2. Let (q̄, ū) be the solution to (1.1) – (1.2). Then there exists a

unique adjoint state z̄ ∈ H2(Ω) ∩H10 (Ω) ↪→ C0(Ω) satisfying{−∆z̄ = ū− ud in Ω

z̄ = 0 on ∂Ω,(2.1)

and

−〈q − q̄, z̄〉+ α‖q̄‖M(Ω) ≤ α‖q‖M(Ω) for all q ∈M(Ω). (2.2)

Furthermore this implies

‖z̄‖C0(Ω) ≤ α, (2.3)

the support of q̄ is contained in the set {x ∈ Ω | |z̄(x)| = α } , and for the Jordan-decomposition q̄ = q̄+ − q̄− we have

supp q̄+ ⊂ {x ∈ Ω | z̄(x) = −α } and supp q̄− ⊂ {x ∈ Ω | z̄(x) = α } . (2.4)


Remark 2.3. The optimality condition (2.2) can be equivalently reformulated as

(u(q)− ū, ū− ud) + α(‖q‖M(Ω) − ‖q̄‖M(Ω)

)≥ 0 for all q ∈M(Ω). (2.5)

The statement of the above theorem directly implies the following corollary onthe structure of the optimal control q̄.

Corollary 2.4. There exist γ > 0 depending on the data of the problem, suchthat

supp q̄ ⊂ Ωγ = {x ∈ Ω | dist(x, ∂Ω) > γ } , (2.6)

and additionally

dist(supp q̄+, supp q̄−) > γ. (2.7)

The first property implies that the support is compact.Proof. The adjoint state z̄ belongs to H2(Ω) ↪→ C0,β(Ω̄) with some β > 0.

This implies (due to the homogeneous Dirichlet boundary conditions) the existenceof γ > 0, such that

|z̄(x)| < α2

on Ω \ Ωγ .

We complete the first part of the proof using the statement on the support of q̄ fromTheorem 2.2. With a similar argument we derive the second statement, since dueto (2.4), the adjoint state attains the values ±α respectively on the support of q̄− andq̄+.

Finally, we will derive an additional regularity for ū if the desired state ud isbounded.

Theorem 2.5. Assume that the desired state ud is in L∞(Ω). Then the optimal

state ū is also in L∞(Ω) and there holds

‖ū‖L∞(Ω) ≤ ‖ud‖L∞(Ω).

A direct consequence of this theorem is an additional regularity for the optimalcontrol q̄ and for the optimal state ū.

Corollary 2.6. Assume that the desired state ud is in L∞(Ω). Then the optimal

state ū lies in H10 (Ω)∩L∞(Ω) and the optimal control q̄ lies in H−1(Ω). There holds

‖∇ū‖2L2(Ω) ≤ ‖q̄‖M(Ω)‖ud‖L∞(Ω) and ‖q̄‖H−1(Ω) = ‖∇ū‖L2(Ω).

In order to prove Theorem 2.5 and Corollary 2.6 we use some results from potentialtheory: First, introduce the Green’s function GΩ : Ω×Ω→ R+ ∪ {+∞} as in e.g. [3]or [14]. Then, for a positive measure µ ∈M(Ω), µ ≥ 0 we define the numeric functionv? : Ω→ R+ ∪ {+∞} by

v? = S(µ) :=∫

Ω

GΩ(·, y) dµ(y), (2.8)

which is subharmonic and thus lower semicontinuous (see again [3]). If we normalizeGΩ by the right constant, we obtain the following simple result.


Lemma 2.7. For a compactly supported µ ∈ M(Ω), µ ≥ 0 the weak solutionv ∈W 1,s0 (Ω) with 1 ≤ s < dd−1 to the problem

−∆v = µ in Ω,v = 0 on ∂Ω,

(2.9)

is equal to v? = S(µ) (Lebesgue-)almost everywhere.Proof. With [3, Theorem 4.3.8] the function v? is a distributional solution of (2.9),

and by a density argument, it is also a weak solution, unique in an almost everywheresense.

With the help of the above lemma, we obtain a pointwise representative of theoptimal solution u? : Ω→ R ∪ {−∞,∞}, defined as

u? := S(q̄+)− S(q̄−) = S(q̄).

Due to (2.6) the measures q̄+ and q̄− are compactly supported, and with (2.7) u? iswell defined with values in R ∪ {−∞,∞}. With Lemma 2.7 we easily derive thatu? = ū almost everywhere.

The next lemma states (roughly speaking), that if the optimal state is boundedon supp q̄, then it is bounded everywhere on Ω by the same constant. For positivemeasures inM(Ω) this statement can be directly obtained from [14, Theorem 1.6’] inthe two-dimensional case. For d = 3, the analogous theorem, see [14, Theorem 1.10],is stated only for Ω = Rd. Therefore, we provide a direct proof.

Lemma 2.8. Let q̄ ∈M(Ω) be the optimal control. If u? = S(q̄) is bounded fromabove by some constant C+ ≥ 0 on supp q+, then it is bounded everywhere by C+.Analogously, if u? is bounded from below by some C− ≤ 0 on supp q−, then u? isbounded from below everywhere by C−.

Proof. Suppose u? ≤ C+ on supp q+. With (2.7) we estimate

S(q̄+) = u? + S(q̄−) ≤ C+ + cγ‖q̄−‖M(Ω) on supp q̄+ ,

where cγ = c log(1γ diam Ω) for d = 2 and cγ =

cγ for d = 3 due to the growth prop-

erties of the Green’s function. Thus, S(q̄+) is bounded on supp q̄+ as well. With [3,Corollary 4.5.2] we can now construct a sequence of compact sets {Ki} with

q̄+(supp q̄+ \Ki)→ 0 for i→∞, (2.10)

such that the functions S(q̄+|Ki) are continuous.Now, we consider the solutions

ui = S(q̄+|Ki)− S(q̄−) ≤ u?.

Recalling that −S(q̄−) is upper semicontinuous, we obtain that each ui is uppersemicontinuous as well. For each x0 on the boundary of Ω\ supp q̄+, which is a subsetof supp q̄+ ∪ ∂Ω, we have ui(x0) ≤ u?(x0) ≤ C+ and with upper semicontinuity

lim supx→x0

ui(x) ≤ C+. (2.11)

Using the fact that ui is subharmonic on Ω\supp q+ and the condition (2.11) we applythe maximum principle for subharmonic functions [3, Theorem 3.1.5], and obtain thatui is bounded by C

+ everywhere on Ω for every i.


To complete the proof, it remains to show the convergence ui(x)→ u?(x) for allx ∈ Ω\ supp q̄+. Let x ∈ Ω\ supp q̄+ be fixed. We denote by δ = dist(x, supp q̄+) > 0.There holds

|ui(x)− u?(x)| = |S(q̄+|Ki)(x)− S(q̄+)(x)| ≤ cδ q̄+(supp q̄+ \Ki)→ 0, i→∞,

where we have again used growth properties of the Green’s function and (2.10).The second statement is proved completely analogously.With these preparations we can give proofs of the claimed results.Proof. [Proof of Theorem 2.5] Assume the contrary, i.e., that we have C, ε > 0,

such that |ud| ≤ C almost everywhere in Ω, but |ū| ≥ C + ε on some set of positiveLebesgue measure. Without loss of generality, we can assume that

|{x ∈ Ω | ū(x) ≥ C + ε }| > 0.

Let u? = S(q̄), which necessarily must be larger than C + ε for some x ∈ supp q+with Lemma 2.8. In the ball Bγ(x) we have with Corollary 2.4 that q̄

−|Bγ(x) = 0 andtherefore S(q̄|Bγ(x)) is lower semicontinuous. We decompose

u? = S(q̄|Bγ(x)) + S(q̄|Ω\Bγ(x))

and obtain that S(q̄|Ω\Bγ(x)) is harmonic and consequently continuous on Bγ(x). Thisimplies the lower semicontinuity of u? on Bγ(x). This means, the set

{ y ∈ Bγ(x) | u?(y) > C + ε }

is open, and we can find a radius r > 0 such that ū ≥ C + ε almost everywhere in theball Br(x).

Note that x ∈ supp q̄+ implies z̄(x) = −α with Theorem 2.2. We define w to bethe solution to

−∆w = ε in Br(x),w = 0 on ∂Br(x),

which is clearly strictly positive at x. Considering the minimum principle for z̃ = z̄−wwhich solves

−∆z̃ = ū− ud − ε ≥ 0 in Br(x),z̃ = z̄ on ∂Br(x),

we see that the minimum value zmin = infx∈Br(x) z̃(x) must be attained for somex′ ∈ ∂Br(x). Comparing with the center x we find

z̄(x′) = z̃(x′) = (z̄ − w)(x′) ≤ (z̄ − w)(x) < z̄(x) = −α,

which is a violation of the bounds on the adjoint state (2.3) and thus a contradiction.

Proof. [Proof of Corollary 2.6] The result can be derived by considering a sequenceof smooth approximations to q̄, testing the corresponding state equation with thesmooth solution and a subsequential weak limit argument.

However, the statement directly follows from a well-known classical result: Since,by the previous theorem, u? is bounded, we can pair u? with q̄ to obtain

‖q̄‖M(Ω)‖u?‖L∞(Ω) ≥ 〈q̄, u?〉 =∫

Ω

u?(x) dq̄(x) =

∫Ω

∫Ω

GΩ(x, y) dq̄(x) dq̄(y).


With [14, Theorem 1.20], this implies ∇u? ∈ L2(Ω) and∫Ω

∫Ω

GΩ(x, y) dq̄(x) dq̄(y) = ‖∇u?‖2L2(Ω),

which implies the first part of the claim. The second assertion is evident.

3. Discretization. For the discretization of the state equation we use linearfinite elements on a family of shape regular quasi-uniform triangulations {Th}h, see,e.g., [4]. The discretization parameter h denotes the maximal diameter of cellsK ∈ Th.We set

Ω̄h =⋃

K∈Th

K̄

and make the usual assumption

|Ω \ Ωh| ≤ ch2.

The finite element space associated with Th is defined as usual by

Vh = { vh ∈ C0(Ω) | vh|K ∈ P1(K) for all K ∈ Th and vh = 0 on Ω \ Ωh } .

For a given q ∈M(Ω) the discrete solution uh = uh(q) is determined by

uh ∈ Vh : (∇uh,∇vh) = 〈q, vh〉 for all vh ∈ Vh. (3.1)

To define the approximation of the optimal control problem (1.1) – (1.2) we follow theapproach from [7] and do not discretize the control space, cf. the variational approachby [13]. The discrete optimal control problem is then given as

Minimize J(qh, uh), qh ∈M(Ω) and subject to (3.1). (3.2)

The existence of a solution can be shown as on the continuous level. The optimalstate ūh is unique. The discrete solution operator mapping q ∈M(Ω) to uh(q) is notinjective and the uniqueness of the optimal control can not be guaranteed. However,one special solution can be identified, which is numerically accessible, see [7] and thediscussion below.

By {xi}, i = 1, 2 . . . , Nh we denote the interior nodes of Ωh and by {ei} ⊂ Vh thecorresponding node basis functions. We introduce the space Mh consisting of linearcombination of Dirac functionals associated with the nodes xi:

Mh =

{qh ∈M(Ω)

∣∣∣∣∣ qh =Nh∑i=1

βi δxi , βi ∈ R, i = 1, 2, . . . , Nh

}

and an operator Λh : M(Ω)→Mh (see [7]) by

Λhq =

Nh∑i=1

〈q, ei〉 δxi .

There holds the following theorem, see [7].


Theorem 3.1. Among the solutions to (3.2) there exists a unique solution q̄h ∈Mh with the corresponding state ūh = uh(q̄h). Any other solution q̃h ∈M(Ω) satisfiesΛhq̃h = q̄h. Moreover there holds

q̄h∗⇀ q̄ in M(Ω) and ‖q̄h‖M(Ω) → ‖q̄‖M(Ω)

for h→ 0.For the solution (q̄h, ūh) from this theorem the following discrete version of the

optimality conditions holds, which can be derived as in the continuous case, cf. [7].Theorem 3.2. Let (q̄h, ūh) ∈Mh×Vh be the discrete solution, see Theorem 3.1.

Then there exists the discrete adjoint state z̄h ∈ Vh fulfilling

(∇vh,∇z̄h) = (ūh − ud, vh) for all vh ∈ Vh

and the optimality condition

−〈q − q̄h, z̄h〉+ α‖q̄h‖M(Ω) ≤ α‖q‖M(Ω) for all q ∈M(Ω). (3.3)

The last condition can be equivalently rewritten as

(uh(q)− ūh, ūh − ud) + α(‖q‖M(Ω) − ‖q̄h‖M(Ω)

)≥ 0 for all q ∈M(Ω), (3.4)

cf. Remark 2.3.In order to prove our main result mentioned in the introduction, we first provide

some estimates for the error u(q)− uh(q) for a fixed control q ∈M(Ω).Lemma 3.3. Let q ∈ M(Ω) with associated continuous and discrete states u =

u(q) and uh = uh(q) be given. Then there holds:

(i) ‖u− uh‖Lp(Ω) ≤ cph2− d

p′ ‖q‖M(Ω), p ∈ (1,∞),1

p+

1

p′= 1

(ii) ‖u− uh‖L1(Ω) ≤ ch2|lnh|r ‖q‖M(Ω)

with r = 2 for d = 2 and r = 114 for d = 3.Proof.(i): For the first estimate in case p = 2 we refer, e.g., to [5]. For a general case,

p ∈ (1,∞) we set e = u− uh and

gp(x) = |e(x)|p−1 sgn(e(x)).

By a direct calculation it follows gp ∈ Lp′(Ω) and

‖gp‖Lp′ (Ω) = ‖e‖p−1Lp(Ω).

We consider a dual problem

w ∈ H10 (Ω) : (∇w,∇v) = (gp, v) for all v ∈ H10 (Ω)

and its Ritz projection

wh ∈ Vh : (∇wh,∇vh) = (gp, vh) for all vh ∈ Vh.

By the elliptic regularity we obtain w ∈W 2,p′(Ω) and the corresponding L∞-estimategives

‖w − wh‖C0(Ω) ≤ cph2− d

p′ ‖∇2w‖Lp′ (Ω) ≤ cph2− d

p′ ‖gp‖Lp′ (Ω) ≤ cph2− d

p′ ‖e‖p−1Lp(Ω).


For the error ‖e‖Lp(Ω) we obtain

‖e‖pLp(Ω) = (e, gp) = (∇e,∇w)

= (∇e,∇(w − wh)) = (∇u,∇(w − wh))= 〈q, w − wh〉 ≤ ‖q‖M(Ω) ‖w − wh‖C0(Ω)≤ cph2−

dp′ ‖q‖M(Ω)‖e‖p−1Lp(Ω),

which gives the desired estimate.(ii): To obtain the second estimate, we set g1 = sgn(e) ∈ L∞(Ω). There holds

‖e‖L1(Ω) = (e, g1).

We consider a dual problem

w ∈ H10 (Ω) : (∇w,∇v) = (g1, v) for all v ∈ H10 (Ω)

and its Ritz projection

wh ∈ Vh : (∇wh,∇vh) = (g1, vh) for all vh ∈ Vh.

Then we obtain using the Galerkin orthogonality for both errors u− uh and w −wh:

‖e‖L1(Ω) = (e, g1) = (∇e,∇w)= (∇e,∇(w − wh)) = (∇u,∇(w − wh))= 〈q, w − wh〉 ≤ ‖q‖M(Ω) ‖w − wh‖C0(Ω).

For the pointwise error in w we use the result from Rannacher and Frehse [12] ford = 2 and Rannacher [16] for d = 3 and obtain:

‖w − wh‖C0(Ω) ≤ ch2|lnh|r ‖g1‖L∞(Ω).

This completes the proof.Assuming higher regularity for q̄, we can also give the following estimate, which

will be needed later on in Section 5 for the improved error estimates.Lemma 3.4. Let q ∈W−1,p(Ω) for 1 < p


Another useful result concerns the growth behavior of discrete solutions in thelimiting cases of the Sobolev embedding theorem

‖uh‖Lt(Ω) ≤ ct‖q‖M(Ω) for all t <d

d− 2.

For the discrete solutions, we have the following result.Lemma 3.5. Let q ∈M(Ω) with the discrete solution uh = uh(q) as above. Then

we have

‖uh‖L∞(Ω) ≤ c|lnh|32 ‖q‖M(Ω) for d = 2,

‖uh‖L3(Ω) ≤ c|lnh|‖q‖M(Ω) for d = 3.

Proof. In the first step we estimate

‖uh‖L∞(Ω) ≤ c|lnh|12 ‖∇uh‖L2(Ω) for d = 2,

by the discrete Sobolev inequality, see [4], and

‖uh‖L3(Ω) ≤ c‖∇uh‖L 32 (Ω) for d = 3,

by the Sobolev embedding. Defining σ = dd−1 (σ = 2 and σ =32 for 2d and 3d

respectively), we proceed in a common way with an inverse estimate and the stabilityof the Ritz projection with respect to the W 1,s-seminorm, see [4],

‖∇uh‖Lσ(Ω) ≤ c hdσ−

ds ‖∇uh‖Ls(Ω)

≤ ch dσ− ds ‖∇u‖Ls(Ω),

for any 1 < s < σ, where the constant c is independent of s. Then we chooses = sε = σ − ε for 0 < ε < σ − 1, which implies that

d

σ− dsε

= − dεσ(σ − ε)

> −ε dσ−1 = −ε(d− 1).

We obtain by Lemma 2.1

‖∇uh‖Lσ(Ω) ≤c

εh−ε(d−1)‖q‖M(Ω).

Choosing now ε = 1|lnh| we obtain

‖∇uh‖Lσ(Ω) ≤ c|lnh|‖q‖M(Ω),

which, together with the first estimate, completes the proof.

4. General error estimates. In the next theorem we provide an error estimatefor the error with respect to the cost functional. To state this theorem we need anassumption on the desired state ud.

Assumption 1. We assume

ud ∈

{L∞(Ω), for d = 2

L3(Ω), for d = 3.


Remark 4.1. Assumption 1 is only slightly stronger than the correspondingassumption in [7], where ud ∈ L4(Ω) in 2d and ud ∈ L

83 (Ω) in 3d is assumed.

Theorem 4.2. Let Assumption 1 be fulfilled. Let moreover (q̄, ū) be the solutionto (1.1) – (1.2) and (q̄h, ūh) ∈ Mh × Vh be the discrete solution, see Theorem 3.1.Then there holds

|J(q̄, ū)− J(q̄h, ūh)| ≤ c h4−d|lnh|γ

with γ = 72 for d = 2 and γ = 1 for d = 3.Proof. By the optimality we obtain

J(q̄, ū) ≤ J(q̄h, u(q̄h)) and J(q̄h, ūh) ≤ J(q̄, uh(q̄)).

Consequently we have

J(q̄, ū)− J(q̄, uh(q̄)) ≤ J(q̄, ū)− J(q̄h, ūh) ≤ J(q̄h, u(q̄h))− J(q̄h, ūh)

Therefore, it remains to estimate the error with respect to the cost functional for afixed q ∈M(Ω), i.e.

|J(q, u(q))− J(q, uh(q))| =∣∣∣∣12‖u(q)− ud‖2L2(Ω) − 12‖uh(q)− ud‖2L2(Ω)

∣∣∣∣and then to apply this estimate for both q = q̄ and q = q̄h.

For fixed q ∈ M(Ω) we now use the notation u = u(q) and uh = uh(q). Thereholds:

J(q, u)− J(q, uh) =1

2‖u− ud‖2L2(Ω) −

1

2‖uh − ud‖2L2(Ω)

=1

2(u− uh, u+ uh − 2ud)

= −(u− uh, ud) +1

2‖u− uh‖2L2(Ω) + (u− uh, uh).

(4.1)

For the second term in (4.1) we obtain by the estimate (i) for p = 2 from Lemma 3.3

‖u− uh‖2L2(Ω) ≤ ch4−d‖q‖2M(Ω).

The other terms are estimated separately in 2d and in 3d.The case d = 2. The first and last terms in (4.1) are estimated using (ii) from

Lemma 3.3:

(u− uh, ud) ≤ ‖u− uh‖L1(Ω) ‖ud‖L∞(Ω) ≤ ch2|lnh|2‖q‖M(Ω),(u− uh, uh) ≤ ‖u− uh‖L1(Ω) ‖uh‖L∞(Ω) ≤ ch2|lnh|2‖q‖M(Ω)‖uh‖L∞(Ω).

Additionally, by Lemma 3.5 we have ‖uh‖L∞(Ω) ≤ |lnh|32 ‖q‖M(Ω).

The case d = 3. Now, we use (i) for p = 32 from Lemma 3.3 for the remainingterms in (4.1) to obtain

(u− uh, ud) ≤ ‖u− uh‖L

32 (Ω)‖ud‖L3(Ω) ≤ ch‖q‖M(Ω),

(u− uh, uh) ≤ ‖u− uh‖L

32 (Ω)‖uh‖L3(Ω) ≤ ch‖q‖M(Ω)‖uh‖L3(Ω).


We apply again Lemma 3.5 and complete the proof.Remark 4.3. Assumption 1 excludes the case, where the desired state ud is given

as a Green’s function. However, for construction of irregular examples with knownexact solutions (see Section 7), it is desirable to choose ud to be the solution of

−∆ud = δx0 in Ωud = 0 on ∂Ω,

with some x0 ∈ Ω. For this choice of ud there holds:

ud ∈ Lp(Ω) for all p ∈ (1,∞) for d = 2

and

ud ∈ L3−ε(Ω) for all ε ∈ (0, 1) for d = 3.

The result of Theorem 4.2 can be directly extended to to this situation. In this casean additional logarithmic term |lnh| will appear.

In the next theorem we prove the main estimate for the error in the state variable,as announced in (1.3).

Theorem 4.4. Let the conditions of Theorem 4.2 be fulfilled. Then there holds

‖ū− ūh‖L2(Ω) ≤ ch2−d2 |lnh|

γ2 .

Proof. We use the optimality condition (2.5), choose q = q̄h and obtain

(u(q̄h)− ū, ū− ud) + α(‖q̄h‖M(Ω) − ‖q̄‖M(Ω)

)≥ 0.

For the corresponding discrete optimality condition (3.4) we choose q = q̄ resulting in

(uh(q̄)− ūh, ūh − ud) + α(‖q̄‖M(Ω) − ‖q̄h‖M(Ω)

)≥ 0.

Adding these two inequalities we arrive at

(u(q̄h)− ū, ū− ud) + (uh(q̄)− ūh, ūh − ud) ≥ 0.

Rearranging the terms we obtain

(ūh − ū, ū− ud) + (u(q̄h)− ūh, ū− ud) + (ū− ūh, ūh − ud) + (uh(q̄)− ū, ūh − ud) ≥ 0

resulting in

‖ū− ūh‖2L2(Ω) ≤ (u(q̄h)− ūh, ū− ud) + (uh(q̄)− ū, ūh − ud)= (u(q̄h)− ūh, ū− uh(q̄)) + (u(q̄h)− ūh, uh(q̄)− ud) + (uh(q̄)− ū, ūh − ud). (4.2)

For the first term in (4.2) we obtain by the estimate (i) for p = 2 from Lemma 3.3

(u(q̄h)−ūh, ū−uh(q̄)) ≤ ‖u(q̄h)−ūh‖L2(Ω) ‖ū−uh(q̄)‖L2(Ω) ≤ ch4−d‖q̄‖M(Ω) ‖q̄h‖M(Ω).

The second and the third terms in (4.2) are estimated by the same procedure as inthe proof of Theorem 4.2 resulting in

‖ū− ūh‖2L2(Ω) ≤ ch4−d|lnh|γ .


This completes the proof.With help of this result, we can also provide an estimate for the error of the

control in H−2(Ω).Theorem 4.5. Let the conditions of Theorem 4.2 be fulfilled. Then there holds

‖q̄ − q̄h‖H−2(Ω) ≤ ch2−d2 |lnh|

γ2 .

Proof. For a given ψ ∈ H2(Ω) ∩H10 (Ω) and the nodal interpolation ihψ ∈ Vh wehave 〈q̄h, ψ〉 = 〈q̄h, ihψ〉 since q̄h is a linear combination of nodal Dirac delta functionsand we we obtain

〈q̄ − q̄h, ψ〉 = 〈q̄, ψ〉 − 〈q̄h, ihψ〉 = 〈q̄ − q̄h, ihψ〉+ 〈q̄, ψ − ihψ〉= (∇(ū− ūh),∇ihψ) + 〈q̄, ψ − ihψ〉= (∇(ū− ūh),∇ψ)− (∇(ū− ūh),∇(ψ − ihψ)) + 〈q̄, ψ − ihψ〉.

For the first term we get by Theorem 4.4

(∇(ū− ūh),∇ψ) = (ū− ūh,−∆ψ) ≤ ch2−d2 |lnh|

γ2 ‖ψ‖H2(Ω).

For the second term we obtain

(∇(ū− ūh),∇(ψ − ihψ)) ≤ c(‖∇ū‖Ls(Ω) + ‖∇ūh‖Ls(Ω)

)‖∇(ψ − ihψ)‖Ls′ (Ω)

for all s′ with 1s′ +1s = 1 and s <

dd−1 . Using the interpolation estimate

‖∇(ψ − ihψ)‖Ls′ (Ω) ≤ ch1− d2 +

ds′ ‖ψ‖H2(Ω),

choosing s = sε =dd−1 − ε for 0 < ε <

12 and exploiting the estimate from Lemma 2.1

we obtain

(∇(ū− ūh),∇(ψ − ihψ)) ≤c

εh2−

d2−4ε

(‖q̄‖M(Ω) + ‖q̄h‖M(Ω)

)‖ψ‖H2(Ω).

The choice ε = 1|lnh| yields:

(∇(ū− ūh),∇(ψ − ihψ)) ≤ ch2−d2 |lnh|‖ψ‖H2(Ω).

For the third term we get

〈q̄, ψ − ihψ〉 ≤ c‖q̄‖M(Ω))‖ψ − ihψ‖C0(Ω) ≤ ch2− d2 ‖ψ‖H2(Ω).

This completes the proof.

5. Improved error estimates. In the following we exploit the additional reg-ularity derived in Section 2 to provide an improved estimate under the assumptionthat ud is bounded.

Theorem 5.1. In the case d = 3, let (q̄, ū) be the solution to (1.1) – (1.2)and (q̄h, ūh) ∈ Mh × Vh be the discrete solution, see Theorem 3.1. Let moreoverud ∈ L∞(Ω), which implies ū ∈ H10 (Ω) ∩ L∞(Ω) and q̄ ∈ H−1(Ω) with Theorems 2.5and 2.6. Then there holds

‖ū− ūh‖L2(Ω) ≤ c h23 |lnh| 2312 .


Furthermore, under the additional assumption q̄ ∈ W−1,p(Ω) for some p > 2, weobtain

‖ū− ūh‖L2(Ω) ≤ c h12 (1+θ)|lnh| 11θ4 +1, where θ =

23 −

1p

1− 1p.

Proof. First, we obtain an L2(Ω) estimate for ū in terms of an L∞(Ω) estimatefor z̄. For that, we use the optimality condition (2.2), choosing q = q̄h

−〈q̄h − q̄, z̄〉+ α‖q̄‖M(Ω) ≤ α‖q̄h‖M(Ω),

and the optimality condition (3.3) choosing q = q̄

−〈q̄ − q̄h, z̄h〉+ α‖q̄h‖M(Ω) ≤ α‖q̄‖M(Ω).

Adding these two inequalities results in

〈q̄h − q̄, z̄ − z̄h〉 ≥ 0.

We introduce z̃h = zh(q̄) ∈ Vh defined by

(∇vh,∇z̃h) = (ũh − ud, vh) for all vh ∈ Vh,

where ũh = uh(q̄) as before. There holds:

0 ≤ 〈q̄h − q̄, z̄ − z̄h〉 = 〈q̄h − q̄, z̄ − z̃h〉+ 〈q̄h − q̄, z̃h − z̄h〉= 〈q̄h − q̄, z̄ − z̃h〉+ (∇(ūh − ũh),∇(z̃h − z̄h))= 〈q̄h − q̄, z̄ − z̃h〉 − ‖ũh − ūh‖2L2(Ω)

and therefore

‖ũh − ūh‖2L2(Ω) ≤ c‖z̄ − z̃h‖L∞(Ω) (5.1)

since ‖q̄‖M(Ω) and ‖q̄h‖M(Ω) are bounded. Note, that it now suffices to estimate theterm on the right in (5.1) to obtain the final result, since

‖ū− ūh‖L2(Ω) ≤ ch ‖q̄‖H−1(Ω) + ‖ũh − ūh‖L2(Ω),

holds with the estimate for ū− ũh from Lemma 3.4 with p = 2.To this end, we introduce ẑh ∈ Vh as the Ritz projection of z̄ determined by

(∇vh,∇ẑh) = (ū− ud, vh) for all vh ∈ Vh,

and split the last term in (5.1) as

‖z̄ − z̃h‖L∞(Ω) ≤ ‖z̄ − ẑh‖L∞(Ω) + ‖ẑh − z̃h‖L∞(Ω).

Using the L∞-estimate from [16] we obtain for the first term

‖z̄ − ẑh‖L∞(Ω) ≤ ch2|lnh|114 ‖ū− ud‖L∞(Ω).

For the second term, define gh to be the Ritz projection of the Green’s function atx0 ∈ Ω, which fulfills

(∇gh,∇vh) = 〈δx0 , vh〉 = vh(x0) for all vh ∈ Vh.


Then, for the error eh = ẑh − z̃h ∈ Vh pointwise at x0 we obtain

eh(x0) = 〈δx0 , eh〉 = (∇gh,∇eh) = (ū− ũh, gh) ≤ ‖ū− ũh‖L 32 (Ω)‖gh‖L3(Ω),

with Hölder’s inequality. The last term is bounded by

‖gh‖L3(Ω) ≤ c|lnh|,

with Lemma 3.5 applied to q = δx0 . The estimate for ū− ũh in L32 (Ω) is obtained by

interpolation, recalling that

‖ū− ũh‖L1(Ω) ≤ ch2|lnh|114 ‖q‖M(Ω) and ‖ū− ũh‖Lp(Ω) ≤ ch‖q̄‖W−1,p(Ω),

with Lemmas 3.3 and 3.4. Now, interpolation between these two estimates yields anestimate in the interpolation space [Lp(Ω), L1(Ω)]θ = L

pθ (Ω) of the form

‖ū− ũh‖Lpθ (Ω) ≤ ch1+θ|lnh|θ114 ‖q̄‖1−θW−1,p(Ω)‖q‖

θM(Ω). (5.2)

For pθ =32 we have to choose θ ∈ (0, 1) as

2

3= θ +

1− θp

, or equivalently θ =

23 −

1p

1− 1p.

Putting everything together, we obtain the second part of the result. The first state-ment is simply the special case for p = 2.

Remark 5.2. In case ud is bounded and q̄ ∈W−1,p(Ω) for all p 0 is known to be given by the projec-tion formula

q̄ε =1ε shα(−z̄ε),

where the Nemizkij-operator shα (soft-shrinkage) can be written as

shα(y) = max(0, y − α)−max(0, α− y),

and z̄ε fulfills the adjoint equation (2.1) with a corresponding state solution ūε solving(1.2). Thus, the control variable can be eliminated to obtain the system

G(z, u) =

(u−∆z − ud

−∆u− 1ε shβ(−z)

)= 0,


which can be solved with a semismooth Newton method, see e.g. [19].We proceed completely analogously for the discrete problem. However, since the

controls are discretized as nodal Diracs measures, it is not immediately clear how tointerpret the regularization term in the discrete setting. For simplicity, we implementthe regularization term as

ε

2‖qh‖2L2h =

ε

2

N∑i=1

d−1i q2i (6.2)

where qi is the coefficient of the control qh ∈ Mh at the nodal Dirac measure δxiand (di)i=1...N is the diagonal of the lumped mass matrix. The discrete regularizedproblem is then given by

minqh∈Mh

1

2‖uh − ud‖2L2(Ω) + α‖qh‖M(Ω) +

ε

2‖qh‖2L2h

s.t. (∇uh,∇vh) = 〈qh, vh〉 for all vh ∈ Vh.(6.3)

A related mass lumping for discretization of L1-control-costs is also employed in [8].The optimality system for (6.3) can then be derived as in the continuous setting.

We only point out, that here we obtain the optimality condition

d−1i qi =1

εshα(−z̄h,ε(xi)) for i = 1 . . . N,

where qi is the coefficient of the optimal control qh,ε ∈Mh at the nodal Dirac δxi . Thecorresponding algorithm for the discrete regularized problem (6.3) was implementedwith [17], and the arising linear systems were solved with a Schur-complement methodand conjugate gradients.

7. Numerical examples. We present some examples to verify the rates of con-vergence established in Sections 4 and 5.

7.1. Example for d = 2. We take Ω = B1(0) as the unit ball and construct aradially symmetric example with the optimal state given as

ū(x) = − 12π

ln(max{ρ, |x|}),

with a kink in the radial direction at ρ ∈ [0, 1). See Figure 7.1 for the representativecases ρ = 12 and ρ = 0. For ρ = 0 the state ū is simply a Green’s function, and theoptimal control is then given by q̄ = δ0. For ρ > 0 we obtain the surface measure(given in terms of the 1-dimensional Hausdorff measure H)

q̄ =1

2πρH1|∂Bρ(0)

which, due to the choice of scaling, has a norm of ‖q̄‖M(Ω) = 1. The optimal dualstate can then be chosen as any element in H2(Ω) ∩ H10 (Ω), such that |z̄| ≤ α andz̄|∂Bρ(0) = −α. We make the specific choice

z̄(x) = h(|x|),

where h ∈ C1([0, 1]) is a piecewise cubic polynomial interpolating h(0) = h(1) = 0,h(ρ) = −α with the choices h′(ρ) = h′(0) = h′(1) = 0. This yields z̄ ∈ C1(Ω), which


0.2 0.4 0.6 0.8 1.0r

0.1

0.2

0.3

0.4

0.5

ū for ρ=12

ū for ρ=0

0.2 0.4 0.6 0.8 1.0r

−α

α

z̄ for ρ=12

z̄ for ρ=0

0.2 0.4 0.6 0.8 1.0r

0.1

0.2

0.3

0.4

0.5

ud for ρ=12, α=0.001

ud for ρ=0, α=0.01

Fig. 7.1. Radially symmetric example for the unit circle in R2 in radial direction r

is piecewise twice continuously differentiable with bounded second derivatives, and amatching desired state ud ∈ L∞(Ω) can be computed in strong formulation as

ud = ∆z̄ + ū,

as depicted in Figure 7.1 for ρ ∈ { 0, 12 }. For the convenience of the reader, the exactformula for ud is given by

ud(r) =

α6 (3 r−2 ρ)

ρ3 −1

2π ln(ρ) for r < ρ

α6 (3 r2−2 rρ−2 r+ρ)

(ρ−1)3r −1

2π ln(r) for r ≥ ρ,

where r = |x|.The convergence rates for a choice of ρ = 12 and ρ = 0 are given in Figure 7.2.

The inital grid (refinement level 0) consists of five cells, a small square in the middleand four additional trapezoids at each edge, glued together at the corners. For bothexamples we plot the error in the cost functional J(q̄, ū)− J(q̄h, ūh) and the L2-errorin the state variable. The dashed lines indicate the orders of convergence O(h2) andO(h), which are what theory predicts for the respective quantities (up to logarithmiccontributions). Since the regularization is present in the numerical computations,we also report the size of the term ε2‖q̄h‖

2L2h

. As a parameter choice rule, at each

refinement level the regularization parameter ε is decreased until

ε

2‖q̄h‖2L2h ≤ cregh

2

is fulfilled, where creg > 0 is a constant chosen heuristically in advance. This isdone to ensure that at least the asymptotic best case convergence behaviour of the


10−10

10−8

10−6

10−4

10−2

100

1 2 3 4 5 6 7 8 9

|J(q̄, ū)− J(q̄h, ūh)|‖ū− ūh‖L2(Ω)

ε2‖q̄h‖2L2

h

(a) Errors for ρ = 12

, α = 0.001

10−8

10−6

10−4

10−2

100

1 2 3 4 5 6 7 8 9

|J(q̄, ū)− J(q̄h, ūh)|‖ū− ūh‖L2(Ω)

ε2‖q̄h‖2L2

h

(b) Errors for ρ = 0, α = 0.01

Fig. 7.2. Convergence rates for the 2d example at different refinement levels.

functional O(|lnh|γh2) should not be altered by the regularization. In Figure 7.2(a)we e.g. observe that the regularization term is an order of a magnitude smaller thanthe exact functional error, such that the reported error in the functional should be atleast accurate in the first significant digit.

We see that the observed rates seem to agree with the rates predicted by theory.In Figure 7.2(a) the rates seem to be even slightly better, however, this is far fromconclusive. In Figure 7.2(b), even though the rate for the functional is somewhatwiggly, we observe the expected rates. The wiggles could be caused by the fact thatthe initial mesh was perturbed slightly, and thus the approximation quality dependsfor a large part on the smallest distance of a grid-point to the origin, where the optimalcontrol q̄ = δ0 is located. If we choose a mesh which has a point at the origin, theexact control is representable at each level, and the wiggles disappear. In the Diraccase, due to the low regularity of ud, it is also clear that the rate of almost O(h) forthe state error is the best theoretically possible.

7.2. Example for d = 3. The construction of an example in three dimensionsis completely analogous, except for the different Green’s function

ū(x) =1

4π

(1

max{ρ, |x|}− 1),

thus we omit a detailed description. The final formula for ud in this case is given by

ud(r) =

α6 (4 r−3 ρ)

ρ3 +1

4π

(1ρ − 1

)for r < ρ

α6 (4 r2−3 rρ−3 r+2 ρ)

(ρ−1)3r +1

4π

(1r − 1

)for r ≥ ρ,

where r = |x|. The computational results can be seen in Figure 7.3. Note that theparameter choice rule for ε is simply the same as before. In this case, the generaltheory predicts an order of convergence close to O(h) for the functional and close toO(h 12 ) for the L2-error of the state. This is clearly observed in the case ρ = 0, wherethe optimal control q̄ is a single Dirac delta function, see Figure 7.3(b). In this case


10−7

10−6

10−5

10−4

10−3

10−2

10−1

1 2 3 4 5 6

|J(q̄, ū)− J(q̄h, ūh)|‖ū− ūh‖L2(Ω)

ε2‖q̄h‖2L2

h

(a) Errors for ρ = 12

, α = 0.001

10−6

10−5

10−4

10−3

10−2

10−1

100

1 2 3 4 5

|J(q̄, ū)− J(q̄h, ūh)|‖ū− ūh‖L2(Ω)

ε2‖q̄h‖2L2

h

(b) Errors for ρ = 0, α = 0.01

Fig. 7.3. Convergence rates for the 3D example at different refinement levels.

the rate for the state error is again the theoretically best possible. However, in thecase ρ = 12 , depicted in 7.3(a), where the optimal control is a surface measure andthe optimal state is Lipschitz continuous, the rates are clearly better than that. Forvisual comparison we plot the rates O(h) and O(h2), which seem to be the closestmatch. With the result of Theorem 5.1 we can show an order of convergence of atleast |lnh| 154 h 56 for the error in the state in this example, since the optimal control isan element of W−1,∞(Ω) in this case.

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