-
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
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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.
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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)
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4 KONSTANTIN PIEPER AND BORIS VEXLER
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.
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FEM FOR SPARSE ELLIPTIC OPTIMAL CONTROL 5
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.
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6 KONSTANTIN PIEPER AND BORIS VEXLER
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).
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FEM FOR SPARSE ELLIPTIC OPTIMAL CONTROL 7
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].
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8 KONSTANTIN PIEPER AND BORIS VEXLER
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).
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FEM FOR SPARSE ELLIPTIC OPTIMAL CONTROL 9
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
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10 KONSTANTIN PIEPER AND BORIS VEXLER
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.
-
FEM FOR SPARSE ELLIPTIC OPTIMAL CONTROL 11
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).
-
12 KONSTANTIN PIEPER AND BORIS VEXLER
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 .
-
FEM FOR SPARSE ELLIPTIC OPTIMAL CONTROL 13
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 .
-
14 KONSTANTIN PIEPER AND BORIS VEXLER
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.
-
FEM FOR SPARSE ELLIPTIC OPTIMAL CONTROL 15
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,
-
16 KONSTANTIN PIEPER AND BORIS VEXLER
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
-
FEM FOR SPARSE ELLIPTIC OPTIMAL CONTROL 17
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
-
18 KONSTANTIN PIEPER AND BORIS VEXLER
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
-
FEM FOR SPARSE ELLIPTIC OPTIMAL CONTROL 19
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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