A sensitivity-based extrapolation technique for the numerical solution of state-constrained optimal control problems

DOI:10.1051/cocv/2009016 www.esaim-cocv.org

A SENSITIVITY-BASED EXTRAPOLATION TECHNIQUE

FOR THE NUMERICAL SOLUTION OF STATE-CONSTRAINED OPTIMAL CONTROL PROBLEMS

Michael Hinterm¨ uller

and Irwin Yousept

Abstract. Sensitivity analysis (with respect to the regularization parameter) of the solution of a class of regularized state constrained optimal control problems is performed. The theoretical results are then used to establish an extrapolation-based numerical scheme for solving the regularized problem for vanishing regularization parameter. In this context, the extrapolation technique provides excellent initializations along the sequence of reducing regularization parameters. Finally, the favorable numer- ical behavior of the new method is demonstrated and a comparison to classical continuation methods is provided.

Mathematics Subject Classification.49M15, 49M37, 65K05, 90C33.

Received July 13, 2007. Revised January 28, 2009.

Published online July 2nd, 2009.

1. Introduction

The numerical treatment of optimal control problems for partial differential equations (PDEs) with pointwise state inequality constraints is challenging due to the measure-valuedness of the Lagrange multiplier associated with the state constraints; see [2] for an analytical assessment. A typical instance of such a problem is given by

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

minimizeJ(u, y) := ¹₂y−yd²L²(Ω)+^α₂u²L²(Ω)

over (u, y)∈L²(Ω)×H₀¹(Ω)∩H²(Ω) subject toAy=u in Ω, y= 0 on Γ,

ya ≤y≤yb a.e. in Ω.

(P)

Keywords and phrases. Extrapolation, mixed control-state constraints, PDE-constrained optimization, semismooth Newton algorithm, sensitivity, state constraints.

∗M.H. acknowledges support by the Austrian Ministry for Science and Education and the Austrian Science Fund FWF under START-program Y305 “Interfaces and Free Boundaries”.

∗∗I.Y. is supported by DFG Research Center Matheon, project C9: Numerical simulation and control of sublimation growth of semiconductor bulk single crystals.

1 Department of Mathematics Humboldt-University of Berlin, Unter den Linden 6, 10099 Berlin, Germany.

hint@math.hu-berlin.de

2 Institute of Mathematics and Scientific Computing, University of Graz, 8010 Graz, Austria.

3 Institut f¨ur Mathematik, Technische Universit¨at Berlin, Str. des 17. Juni 136, 10623 Berlin, Germany.

yousept@math.tu-berlin.de

Article published by EDP Sciences c EDP Sciences, SMAI 2009

where Ω ⊂R^N is a bounded and sufficiently regular domain, A is a second order elliptic partial differential operator, andyd,α,ya,yb are given data which will be specified shortly.

In order to have a numerical technique at hand for solving (P) with stable iteration numbers as the mesh size of discretization is reduced, one can use an approach based on mixed control-state constraints as investigated in [12]. In fact, in order to overcome the measure-valuedness of the Lagrange multiplier associated with the pointwise inequality constraints in (P) one addsutoy in the set of pointwise inequality constraints and then uses the result in [12] which yields an L²-property of the Lagrange multiplier associated with the modified set of constraints. The regular multiplier therefore represents a smooth approximation of the measure-valued quantity. Using this technique, (P) becomes

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

minimizeJ(u, y) := ¹₂y−yd²_L2(Ω)+^α₂u²_L2(Ω)

over (u, y)∈L²(Ω)×H₀¹(Ω)

subject toAy=u in Ω, y= 0 on Γ, ya ≤u+y≤yb a.e. in Ω.

(P)

Let (¯y,u¯) denote the solution of (P). For solving (P) efficiently, in [4,8] a semismooth Newton method (SSN) was proposed on a function space as well as on a discrete level. Besides the locally superlinear convergence, the mesh-independence of SSN for >0 was established.

In view of (P) one is interested in studying the behavior of the solution algorithm as→0. For the numerics in the case of vanishing regularization parameter, however, in [4,8] it turned out to be crucial to suitably tune as it tends to zero and, even more importantly, to initialize the algorithm for solving (P) appropriately along the sequence. Ignoring these issues typically makes the numerical algorithm suffer from ill-conditioning which is usually reflected by large iteration numbers and reduced numerical solution accuracy.

In this note we focus on this latter point and propose an numerical approach based on an extrapolation technique in order to overcome the aforementioned problems. For this, we first need to study the quality of the dependence of (¯y,u¯) on. For instance, under a strict complementarity assumption we prove differentiability of the solution to (P) with respect to. Let ( ˙y,u˙) denote the corresponding derivative. We then establish a system of sensitivity equations which characterize ( ˙y,u˙) uniquely. Subsequently, the theoretical findings are employed in our numerical approach.

The remainder of the paper is organized as follows: in the rest of this section we detail the problem under investigation and settle the notation (Sect.1.1), and we recall some results on (P) (Sect.1.2). The subsequent Section 2 is devoted to Lipschitz and differentiability properties of the solution to (P) with respect to >0.

Then, in Section 3 we define our semismooth Newton-type solver based on extrapolation in . We end this paper by a report on the numerical behavior including a comparison to a technique without extrapolation in Section 4. It turns out that our new method compares favorably to classical continuation methods without extrapolation.

1.1. General assumptions and notation

In connection with our model problem (P), throughout this paper we assume that Ω is an open bounded domain inR^N,N ∈ {2,3}, with sufficiently smooth boundary Γ. The upper and lower boundsya, yb∈ C(Ω) on the state variableysatisfyya(x)< yb(x) for allx∈Ω and guarantee that the feasible set of (P) is non-empty.

Moreover, the desired stateyd∈L²(Ω) andα >0 are assumed to be fixed. Byuwe denote the control variable.

The second-order elliptic partial differential operatorA is defined by

Ay(x) =− N i,j=1

Di(aij(x)Djy(x)),

where the coefficient functionsaij ∈C^0,1( ¯Ω) satisfy the ellipticity condition N

i,j=1

aij(x)ξiξj≥θξ²_RN ∀ (ξ, x)∈R^N×Ω

for some constant θ > 0. Furthermore, A stands for the associated adjoint operator. By G we denote the solution operator G : L²(Ω) → H₀¹(Ω)∩H²(Ω) that assigns to every u ∈ L²(Ω) the solution y = y(u) ∈ H₀¹(Ω)∩H²(Ω) of the state equation

Ay =uin Ω, y= 0 on Γ.

We setS=ı₀G, where ı₀denotes the compact embedding operator fromH¹(Ω) toL²(Ω).

Using the above notation and assumptions, the regularized problem (P) can be expressed as follows:

minimizef(u) := ¹₂Su−yd²_L2(Ω)+^α₂u²_L2(Ω) overu∈L²(Ω)

subject toya ≤(S+I)u≤yb a.e. in Ω, (P)

where I denotes the identity operator in L²(Ω). In [10] the name Lavrentiev regularized problem was coined for (P).

1.2. Standard results

For every >0 standard arguments guarantee the existence of a unique solution of (P). Throughout the paper we denote this solution by ¯uwith associated state ¯y. As in [10], first order optimality of (¯y,u¯) can be characterized as follows:

Theorem 1.1 (first-order optimal conditions for (P)). The pair (¯y,u¯)is optimal for(P)if and only if there exist an adjoint statep∈H₀¹(Ω)∩H²(Ω) and Lagrange multipliers μ^a, μ^b∈L²(Ω) such that

A¯y= ¯u inΩ, y¯= 0 onΓ, (1.1)

Ap= ¯y−yd+μ^b−μ^a inΩ, p= 0 onΓ, (1.2)

p+α¯u−(μ^a−μ^b) = 0, (1.3)

¯u+ ¯y≥ya, μ^a ≥0, (μ^a, ¯u+ ¯y−ya)L2(Ω)= 0, (1.4)

¯u+ ¯y≤yb, μ^b≥0, (μ^b, ¯u+ ¯y−yb)L²(Ω)= 0. (1.5) Using the maximum operator, the complementarity system (1.4)–(1.5) can be equivalently expressed as

μ^a = max

0, μ^a −μ^b+γ(ya−¯u−y¯) , (1.6) μ^b = max

0, μ^b−μ^a +γ(¯u+ ¯y−yb) , (1.7)

with an arbitrarily fixed γ >0,cf.[5,13]. Note that γ acts as a weight between the inequality constraints for the primal variable and the corresponding dual variables or Lagrange multipliers. For the choice γ := α/² in (1.6)–(1.7) and using (1.3), a short computation yields

μ^a = max

0,1 p+ α

²(ya−y¯) , (1.8)

μ^b = max

0,−1 p+ α

²(¯y−yb) . (1.9)

This latter system will be useful in our subsequent analysis.

Finally, we mention that in [11], Theorem 3.3, the strong convergence inL²(Ω) of ¯uto ¯u, the optimal control of (P), is proven. From the results in [9], Lemma 2.4, the weak* convergence in C( ¯Ω)^∗ as → 0 of μ^a and μ^b to ¯μ^a and ¯μ^b, the multipliers associated with the pointwise state constraints at the solution of (P), can be inferred. Further, in [4] the H¨older continuity (with exponent ¹₂) of ¯u with respect to >0 is established.

2. Regularity of solutions to (P

)

As pointed out in the introduction, one of our main goals is to establish Lipschitz continuity and differen- tiability of the mapping→y¯. For this purpose, the transformation of (P) into an associated minimization problem with pure control constraints will be useful.

We start our investigation by considering the operator (I+S). It is well known that the linear operator S=ı₀G:L²(Ω)→L²(Ω) is positive-definite. For this reason, the equation

(I+S)z= 0

admits only the trivial solutionz= 0. Thus, by the Fredholm alternative, the compactness property ofSensures the existence of the inverse operator (I+S)⁻¹. Setting (I+S)⁻¹v =u, or equivalentlyu= ¹(v−ı₀y(u)) in (P), we transform (P) into the following optimal control problem with (pointwise) box constraints imposed on the new control variablev:

minimizef(v) := ¹₂Sv−yd²_L2(Ω)+₂^α₂v−Sv²_L2(Ω) overv∈L²(Ω)

subject toya≤v≤yb a.e. in Ω. (P^v)

Here we useS=ı₀G, where the operatorGassigns to eachv∈L²(Ω) the solutiony(v) =y∈H₀¹(Ω)∩H²(Ω) of the following elliptic equation:

Ay+1 y= 1

v in Ω, y= 0 on Γ. (E)

Notice that the underlying ideas of converting (P) into (P^v) and considering the state equation in the form (E) were originally introduced in [10,11]. In what follows, we denote the unique solution of (P^v) by ¯vwith associated optimal state ¯y. Obviously, (P^v) and (P) are equivalent, i.e., ¯v solves (P^v) if and only if ¯u = (I+S)⁻¹v¯

is the optimal solution of (P). We use the auxiliary problem (P^v) in the following analysis.

The first derivative offatv in an arbitrary directions∈L²(Ω) is given by f(v)s=

Sv−yd+ α

²(Sv−v), Ss

L²(Ω)

+ α

²(v−Sv, s)_L2(Ω). By standard arguments, this can be equivalently expressed as

f(v) =q(v)− α

²y(v) + α

²v, (2.1)

where the adjoint stateq=q(v)∈H₀¹(Ω)∩H²(Ω) is defined as the solution of the following adjoint equation:

Aq+1 q=1

y(v)−yd+ α

²(y(v)−v) in Ω, q= 0 on Γ. (2.2)

2.1. Lipschitz continuity

Next we establish a Lipschitz property of y and various other quantities. Since one might by interested in (P) independently of (P), we reduce the regularity requirements on, e.g., Ω, as we do not assume to have H²(Ω)-regularity of the states and adjoints pertinent to (P). We only consider a Lipschitz-domain Ω⊂R^N with N ∈ {2,3}which is sufficient to obtainH¹(Ω)∩ C(Ω)-regularity of the solutions to the adjoint and state equations (see the elliptic regularity results in [1]).

In what follows, letD= (l, u)⊂R++ with 0< l< u. Lemma 2.1. There exists a positive real number cs(l)such that

y₁(v)−y₂(v)H₀¹(Ω)∩C(Ω)≤cs(l)vL²(Ω)|₁−₂| for allv∈L²(Ω) andi∈ D,i= 1,2.

Proof. We note that standard elliptic regularity estimates yield a constantc >0 independent of andv such that

y(v)H¹₀(Ω)∩C(Ω)≤c1

vL²(Ω) (2.3)

for all v ∈ L²(Ω) and ∈ D. Now, let i ∈ D, i = 1,2, andv ∈ L²(Ω). By definition, fori = 1,2 the states y_i(v) =y_i satisfy

Ay_i+ 1 i

y_i = 1 i

v in Ω, y_i = 0 on Γ.

From this we infer

A(y₁−y₂) + 1

₁(y₁−y₂) = 1

₁− 1

₂ (v−y₂) in Ω, y₁−y₂= 0 on Γ, and hence due to (2.3)

y₁(v)−y₂(v)H₀¹(Ω)∩C(Ω) ≤ 1 ₁ − 1

₂cv−y₂(v)L2(Ω)

≤ |₁−₂| c

l2(vL²(Ω)+y₂(v)L²(Ω))

≤ |₁−₂|c ²_l

1 + c

l vL²(Ω). Setting cs(l) := ^c2

l

1 + ^c

l

yields the assertion.

We proceed by proving the Lipschitz continuity of the adjoint statesq. Lemma 2.2. There exists a positive real number cq(D) such that

q₁(v)−q₂(v)_H1

0(Ω)∩C(Ω)≤cq(D)(vL²(Ω)+ 1)|₁−₂|, for allv∈L²(Ω) and₁, ₂∈ D.

Proof. An argument analogous to the one in the proof of Lemma2.1implies that there exists a constantc(D)>0 independent ofandv such that

q(v)H₀¹(Ω)∩C(Ω)≤c(D)

vL²(Ω)+ 1 (2.4)

for all∈ Dandv∈L²(Ω). Let₁, ₂∈ D. By definitionq₁(v) =q₁ andq₂(v) =q₂ solve, fori= 1,2, Aq_i+ 1

i

q_i= 1 i

(y_i(v)−yd+ α

²_i(y_i(v)−v)) in Ω, q_i = 0 on Γ.

Hence, the differenceq₁−q₂ satisfies A(q₁−q₂) + 1

₁(q₁−q₂) =r in Ω, q₁−q₂= 0 on Γ, where

r= 1

₁(y₁(v)−y₂(v)) + 1

₁ − 1

₂ (y₂(v)−yd) + α

³₁(y₁(v)−y₂(v)) +

α ³₁ − α

³₂ (y₂(v)−v) + 1

₂ − 1 ₁ q₂.

Since |³₂−³₁| = |(²₂+²₁+₂₁)(₂−₁)| ≤ 3²_u|₂−₁|, Lemma 2.1 and (2.4) yield a constant cq(D) > 0 independent of₁, ₂ such that

q₁(v)−q₂(v)H₀¹(Ω)∩C(Ω)≤ rL2(Ω)≤cq(D)|₁−₂|(vL2(Ω)+ 1),

which ends the proof.

For arguing the Lipschitz continuity of the controls, we require uniform positive definiteness off, which we establish next.

Lemma 2.3. There exists a positive real number δ(D)such that for all ∈ D, we have

f(v)s²≥δ(D)s²L²(Ω) ∀v, s∈L²(Ω). (2.5) Proof. Letv∈L²(Ω) and∈ D be arbitrarily fixed. Then, it holds by definition thatf(v) =f((S+I)⁻¹v).

Hence, for an arbitrary directions∈L²(Ω), the chain rule implies f(v)s² = f((S+I)⁻¹v)((S+I)⁻¹s)²

= (S(S+I)⁻¹s, S(S+I)⁻¹s)_L2(Ω)+α((S+I)⁻¹s,(S+I)⁻¹s)_L2(Ω)

≥ α((S+I)⁻¹s,(S+I)⁻¹s)L2(Ω)

= α(S+I)⁻¹s²L²(Ω)

≥ α

S+I²_L2,L2

s²L²(Ω)

≥ α

(SL²,L²+u)²s²L²(Ω).

Defining δ(D) :=α/(SL²,L²+u)², the lemma is verified.

Theorem 2.1. The mapping v:D →L²(Ω),→¯v, is Lipschitz-continuous.

Proof. Suppose that₁, ₂∈ D. By definition v(₁) = ¯v₁ and v(₂) = ¯v₂ solve (P^v

1) and (P^v

2), respectively.

Hence, first order optimality yields

f₁(¯v₁)(v−v¯₁) ≥ 0 ∀v∈V_ad, f₂(¯v₂)(v−v¯₂) ≥ 0 ∀v∈V_ad,

whereV_ad={v∈L²(Ω)|ya ≤v≤yb}. Since ¯v₁ and ¯v₂ are feasible,i.e., ¯v₁,v¯₂∈V_ad, the inequalities above imply

(f

1(¯v₁)−f

2(¯v₂))(¯v₂−v¯₁)≥0, (2.6) which is equivalent to

(f

1(¯v₁)−f

1(¯v₂) +f

1(¯v₂)−f

2(¯v₂))(¯v₂−¯v₁)≥0. (2.7) Sincef₁ is Fr´echet-differentiable, there exists ˜v∈L²(Ω) such that

(f₁(¯v₁)−f₁(¯v₂))(¯v₂−¯v₁) =−f₁(˜v)(¯v₂−v¯₁)². Then (2.7) yields

(f₁(¯v₂)−f₂(¯v₂))(¯v₂−¯v₁)≥f₁(˜v)(¯v₂−v¯₁)² (2.8) and further

q₁(¯v₂)−q₂(¯v₂)L²(Ω)+ α

²₁y₁(¯v₂)−y₂(¯v₂)L²(Ω)+ α

²₁²₂|₁−₂| · |₁+₂|(y₂(¯v₂)L²(Ω)+¯v₂L²(Ω))

¯v₂−¯v₁L²(Ω)≥f₁(˜v)(¯v₂−¯v₁)², where we used (2.1). Due to Lemmas2.1–2.3, there exists a constantc(D)>0 independent of₁, ₂such that

c(D)|₂−₁|(¯v₂L²(Ω)+ 1)≥ ¯v₂−v¯₁L²(Ω). (2.9) Notice that the operator v(·) is uniformly bounded in L^∞(Ω), as ¯v ∈ V_ad for all ∈ R++. This together

with (2.9) proves the assertion.

We have the following immediate corollaries establishing the Lipschitz continuity of ¯y and ¯u. Corollary 2.1. The mapping →y¯ is Lipschitz-continuous from DtoH₀¹(Ω)∩ C(Ω).

Proof. Theorem 2.1and Lemma2.1 guarantee the existence of a constantc(D)>0 independent of₁, ₂∈ D such that

y¯₁−y¯₂H₀¹(Ω)∩C(Ω)≤ y₁(¯v₁)−y₂(¯v₁)H₀¹(Ω)∩C(Ω)+y₂(¯v₁−¯v₂)H₀¹(Ω)∩C(Ω)≤c(D)|₁−₂|,

for all₁, ₂∈ D.

Corollary 2.2. The mapping →u¯ is Lipschitz-continuous from DtoL²(Ω).

Proof. The assertion is immediate since, by construction, ¯u= ¹(¯v−y¯) for all >0.

Clearly, Corollaries2.1and2.2further imply the following result.

Corollary 2.3. The mapping →p is Lipschitz-continuous from DtoH₀¹(Ω)∩ C(Ω).

Proof. Utilizing (1.3) in (1.2) and exploiting the Lipschitz continuity of ¯u and ¯y, respectively, yields the

assertion.

2.2. Parameter sensitivity

We continue our investigation by studying the mapping→(¯y, p). Under a strict complementary assump- tion, we establish the differentiability of this mapping with respect to . First of all, we define the following operators:

ga:D → C(Ω), ga() = 1 p− α

²y¯+ α

²ya, (2.10)

gb:D → C(Ω), gb() = −1 p+ α

²y¯− α

²yb. (2.11)

Note thatga andgb appear in the maximum operators in (1.8)–(1.9):

⎧⎪

⎪⎨

⎪⎪

⎩

μ^a = max

0, ¹p+^α2(ya−y¯) = max(0, ga()), μ^b = max

0,−¹p+^α2(¯y−yb) = max(0, gb()).

(2.12)

Therefore, in order to study the dependence of the Lagrange multipliersμ^a, μ^bon the regularization parameter, we need to consider the operatorsga, gb. Suppose that∈ Dis arbitrarily fixed. Then the Lipschitz-continuity of the mapping→(¯u,y¯, p) fromDtoL²(Ω)×H₀¹(Ω)×H₀¹(Ω) ensures the existence of a weak accumulation point ( ˙u,y˙,p˙)∈L²(Ω)×H₀¹(Ω)×H₀¹(Ω) of

u¯_n−u¯

n− ,y¯_n−y¯

n− ,p_n−p

n− (2.13)

as n→∈ Dforn→ ∞. Further, 1

n−(ga(n)−ga()) = 1 n−

1 n

p_n− α

²_ny¯_n+ α ²_nya−1

p+ α ²y¯− α

²ya

= 1

n− 1

(p_n−p)− 1

− 1 n

p_n+ α

²− α ²_n y¯_n

−α

²(¯y_n−y¯)− α

² − α ²_n ya

= 1

p_n−p

n− − 1 n

p_n+α(n+)

²²_n y¯_n− α ²

y¯_n−y¯

n− −α(n+) ²²_n ya

has a weak accumulation

g˙a() = 1 p˙− 1

²p+2α ³y¯− α

²y˙−2α ³ya

in H₀¹(Ω) as n → forn→ ∞. Analogously, there exists a weak accumulation point of ¹

n−(gb(n)−gb()), which is given by

g˙b() =−1 p˙+ 1

²p−2α ³y¯+ α

²y˙+2α ³yb.

By the compact embedding H₀¹(Ω)⊂L²(Ω) the accumulation points ˙ga() and ˙gb() are strong inL²(Ω).

Without any further assumption, in general there is a need to distinguish betweenupperandlower accumu- lation points depending on whethern ↓ orn ↑ , respectively. See [7] for a similar observation in a related context. However, under some type of strict complementarity condition this distinction is not required as all limits overn-sequences yield a unique accumulation point. This motivates the following assumption.

Assumption 2.1. We assume that the solution of (P)satisfies thestrict complementarity condition meas{x∈Ω|ga()(x) = p(x)−^αy¯(x) +^αya(x) = 0 a.e.} = 0,

meas{x∈Ω|gb()(x) =−p(x) +^αy¯(x)−^αyb(x) = 0 a.e.} = 0. (SC) Notice that by (1.8)–(1.9) the Lagrange multipliers for (P) are given by

μ^a = max

0,1 p+ α

²(ya−y¯) =g_a⁺(), μ^b = max

0,−1

p+ α

²(¯y−yb) =g⁺_b(),

whereg⁺_a = max(0, ga) and analogously forg⁺_b. Hence, using (1.3), the condition (SC) requires that meas{x∈Ω :μ^a(x) +γ(ya−¯u−y¯)(x) = 0}= 0,

or equivalently, that the set where the max-operation is non-differentiable is of measure zero; analogously for the set involvingμ^b andyb.

Next we study the behavior of

g_a⁺(n)−g⁺_a()

n− and g⁺_b(n)−g_b⁺() n− as n→forn→ ∞. For this purpose we introduce

Sa,:={x∈Ω :ga()(x)>0}

and analogously Sb,. By S_a,^c and S_b,^c we denote the complement of Sa, and Sb, in Ω, respectively. Recall that Assumption2.1implies meas{x∈Ω :ga()(x) = 0}= 0. For this reason, we have

measSa,= meas{x∈Ω :ga()(x)>0}= meas{x∈Ω :ga()(x)≥0}

and analogously forS_b,. Hence, forv∈L²(Ω) we infer

Ω

g⁺_a(n)−g_a⁺()

_n− vdx= 1 _n−

S_a,^c

(g_a⁺(n)−g_a⁺())vdx

+ 1

n−

S_a,

(g_a⁺(n)−ga(n))vdx+ 1 n−

S_a,

(ga(n)−ga())vdx.

From the strict complementary condition (SC) and the definition of the setSa,we obtain

ga()(x)<0 for a.a. x∈S_a,^c and ga()(x)>0 for a.a. x∈Sa,. (2.14) In addition, Corollaries2.1and2.3yield

nlim→∞ga(n) =ga() inC(Ω). (2.15)

The latter uniform convergence together with (2.14) results in

nlim→∞

g⁺_a(n)(x)−g⁺_a()(x)

n− = 0 for a.a. x∈S_a,^c ,

nlim→∞

g_a⁺(n)(x)−ga(n)(x)

n− = 0 for a.a. x∈Sa,.

Hence, forn→ ∞, the Lebesgue dominated convergence theorem and Assumption2.1yield 1

_n−

S_a,^c

(g_a⁺(n)−g_a⁺())vdx→0, 1

n−

S_a,

(g_a⁺(n)−ga(n))vdx→0, 1

n−

S_a,

(ga(n)−ga())vdx→

Ω

g˙a()χS_a,vdx,

where χS_a, is the characteristic function of Sa,. The analogous result holds true forg⁺_b. These properties of g_a⁺,g_b⁺ are important for the proof of the following theorem.

Theorem 2.2. Suppose that the solution of (P)satisfies Assumption2.1. Then the mappingsy:D →L²(Ω), →¯y, and p:D →L²(Ω),→p, are strongly differentiable at ∈ D.

Proof. Let (δu, δy, δp) be the difference of two weak accumulation points of u¯_n−u¯

n− ,y¯_n−y¯

n− ,p_n−p

n− ,

asn →. Byδμa andδμb we denote the difference of the associated weak accumulation points of ^{ga+(n)−g+a()}

n−

and ^{g+b(n)−gb+()}

n− . Due to first order optimality, (δu, δy, δp, δμ^a, δμ^b) is characterized by

Aδy=δu in Ω, δy= 0 on Γ, (2.16)

Aδp=δy+δμ^b−δμ^a in Ω, δp= 0 on Γ, (2.17)

δp+αδu+(δμ^b−δμ^a) = 0, (2.18)

δμ^a= 1

δp− α

²δy χS_a,, (2.19)

δμ^b=

−1 δp+ α

²δy χS_b,. (2.20)

Inserting equations (2.18)–(2.20) in (2.16) and (2.17) and then rearranging terms, we obtain Aδy+1

(χS_a,+χS_b,)δy = 1

α(−I+χS_a,+χS_b,)δp in Ω, δy= 0 on Γ, (2.21) Aδp+1

(χ_Sa,+χ_Sb,)δp =

I+ α

²χ_Sa,+ α

²χ_Sb, δy in Ω, δp= 0 on Γ. (2.22) Further, let us define the bilinear form associated with the differential operatorA:

a:H₀¹(Ω)×H₀¹(Ω)→R, a(η, τ) =

Ω

N i,j=1

aij(x)Diη(x)Djτ(x) dx.

In view of (2.21)–(2.22),δy, δp∈H₀¹(Ω) are given by the solutions of a(δy, z) +1

(χ_Sa, +χ_Sb,)δy, z

L²(Ω)= 1

α(−I+χ_Sa, +χ_Sb,)δp, z

L²(Ω)

∀z∈H₀¹(Ω), (2.23)

a(z, δp) +1

(χ_Sa, +χ_Sb,)δp, z

L²(Ω)=

(I+ α

²χ_Sa, + α

²χ_Sb,)δy, z

L²(Ω)

∀z∈H₀¹(Ω). (2.24) Insertingz=δpandz=δyin (2.23) and (2.24), respectively, we find that

0≥ 1

α(−I+χ_Sa, +χ_Sb,)δp, δp

L²(Ω)

= a(δy, δp) +1

(χ_Sa, +χ_Sb,)δy, δp

L²(Ω)

= a(δy, δp) +1

(χ_Sa, +χ_Sb,)δp, δy

L²(Ω)

=

I+ α

²χ_Sa, + α ²χ_Sb,

δy, δy

L2(Ω)

≥ 0.

From this we inferδy=δp= 0, and hence by (2.18)–(2.19)δu=δμa=δμb= 0. Thus, asn→, y¯_n−y¯

n− ,p_n−p

n−

has a unique weak accumulation point in H₀¹(Ω)×H₀¹(Ω). By the compact-embedding ofH₀¹(Ω) in L²(Ω), (^{y¯n −y¯}

n− ,^{pn −p}

n− ) admits a unique accumulation point inL²(Ω)×L²(Ω). Finally, since our arguments hold for

any convergent subsequences, a standard argument completes the proof.

Remark 2.1. Based on the optimality conditions, the derivativesy() andp() satisfy the following system:

Ay() = ˙u in Ω, y() = 0 on Γ, (2.25)

Ap() =y() + ˙μb,−μ˙a, in Ω, p() = 0 on Γ, (2.26)

p() +αu˙+( ˙μb,−μ˙a,) = 0; (2.27)

μ˙a,= 1

p()− 1

²p+2α ³y¯− α

²y()−2α

³ya χ_Sa,; (2.28)

μ˙b,=

−1

p() + 1

²p−2α ³y¯+ α

²y() +2α

³yb χ_Sb,. (2.29)

Remark 2.2. It is also of interest to study the case where Assumption2.1 does not hold true. In this case, differentiability with respect to cannot be expected. However, the existence of weak accumulation points in H₀¹(Ω) of, e.g., ¯y_n−y¯/(n−) for n → still holds true by the Lipschitz property of ¯y with respect to according to Corollary2.1. On the other hand, one can no longer expect uniqueness of these weak accumulation points. In our numerics, the term on the right hand side of (3.8) then has to be interpreted as a difference approximation of the directional derivative of ¯y_n in direction (n+1−n).

3. Extrapolation-based algorithm

Next we introduce a semismooth Newton (SSN) algorithm which utilizes the theoretical results of the previous section within an extrapolation framework. Conceptually, we exploit the differentiability property of the solution of (P) in order to predict a solution of (P₂) given an approximate solution of (P₁) with ₁ > ₂. Hence,