The output least squares identifiability of the diffusion coefficient from an <span class="mathjax-formula">$H^1$</span>-observation in a 2-D elliptic equation

URL:http://www.emath.fr/cocv/ DOI: 10.1051/cocv:2002028

THE OUTPUT LEAST SQUARES IDENTIFIABILITY OF THE DIFFUSION COEFFICIENT FROM AN H

–OBSERVATION

IN A 2–D ELLIPTIC EQUATION

Guy Chavent

and Karl Kunisch

Abstract. Output least squares stability for the diffusion coefficient in an elliptic equation in dimen- sion two is analyzed. This guarantees Lipschitz stability of the solution of the least squares formulation with respect to perturbations in the data independently of their attainability. The analysis shows the influence of the flow direction on the parameter to be estimated. A scale analysis for multi-scale resolution of the unknown parameter is provided.

Mathematics Subject Classification. 62G05, 35R30, 93E24.

Received October 26, 2001.

1. Introduction

We consider in this paper the identification of the diffusion coefficientain an elliptic equation on a bounded, two-dimensional domain Ω with appropriate boundary conditions:

−div (agradu) =f in Ω. (1.1)

Known are the setCof admissible parameters fora, and a measurement

z=Ou (1.2)

ofu, whereO denotes the observation operator.

Theidentifiability of the diffusion coefficientainC is usually understood as the injectivity of the a→ Oua

mapping onC, whereu_a denotes the solution to (1.1) as a function ofa.

The stronger notion of parameterstability is used for the continuous dependence of the inverse ofa→ Oua. Identifiability and stability ofain (1.1) have been treated in several papers. We mention [17] and [10], where the analysis is based on the method of characteristics for the hyperbolic equation fora, which arises from (1.1) when f and u are given functions. The analysis in [15] is based on variational techniques. All these results refer to the case of distributed observation. Many current research efforts focus on identifiability in the case of boundary observations. Relevant references can be found in [14], for example.

Keywords and phrases:Parameter estimation, diffusion coefficient, inverse problem, identifiability, least squares.

∗Research in part supported by the Fonds zur F¨orderung der wissenschaftlichen Forschung, Austria, under SFB 003, Opti- mierung und Kontrolle.

1Ceremade, Universit´e Paris–Dauphine, Paris Cedex 16, France; e-mail: [email protected]

2Institute of Mathematics, University of Graz, Austria.

c EDP Sciences, SMAI 2002

In practice however the measurementzmay typically not belong to the set of attainable outputsD={Ou_a: a∈C}, and one has to estimateain the least squares sense:

minimize 1

2 |Ou_a−z|² over C. (1.3)

Identifiability and stability have of course to hold if one wants the optimization problem (1.3) to be well behaved, but these properties are far from being sufficient.

A still stronger definition was introduced in [2]: the parameter a∈C is calledOutput Least Squares (OLS-) identifiable if there exists a neighborhood V of D such that for every z ∈ V the least squares problem (1.3) has no local minima and a unique solution in C depending Lipschitz-continuously on z ∈ V. The precise definition of this concept is recalled in Section 2. OLS-identifiability takes into account the situation relevant in practice where due to errors in the data and in the model, the datazmay not be contained in the attainable set D. As a consequence the problem of unique and continuous projection of z onto the nonconvex set D must be considered. This makes OLS-identifiability quite difficult to establish. It was analysed so far only for the determination of coefficients in lower order terms by means of regularized least squares formulations and observation in L²(Ω) [6], and for the diffusion coefficient in a one-dimensional version of equation (1.1) with distributedH¹observation [5].

We consider in this paper the OLS-identifiability of the diffusion coefficient a in the two-dimensional equation (1.1) in the case of a distributed H¹ observation, where Ou_a = grad u_a. Of course, it can seem unrealistic that one can observe or measure the gradient ofuthroughout Ω. However, the results of this paper can be combined with the technique of state-space regularization of [7] to handle the somewhat more realistic case of a distributed L²observation. Let us mention also that our results can easily be extended to equations containing lower order terms.

OLS-identifiability with boundary measurement is a completely open problem.

The paper and the results are organized as follows:

In Section 2 we define the inverse problem under consideration, and set up conditions onf and Ω which will ensure the identifiability of a.

We introduce in Section 3 a div/rot decomposition of vector fields in L²(Ω) which will be used throughout the paper.

Sections 4 and 5 are devoted to the stability analysis of ainC⊂L²(Ω) fromu_a in H¹(Ω): in Section 4 we shall observe that even such a strong observation does not control the variations ofaperpendicular to the flow lines, and conclude that stability does not hold whenCis infinite dimensional. Then we show in Section 5 that stability is restored in finite dimensional subsetsC_n ofC, with a stability constant which blows up to infinity when the dimension ofC_n increases.

In Section 6, we establish the sensitivity, deflection and curvature estimates which are needed for the theory of strictly quasiconvex sets [3, 4] to guarantee that the projection on the sets D_n of attainable outputs is well behaved. When the dimension of C_n increases, the sensitivity decreases toward zero, the deflection remains bounded, and the curvature tends to infinity. This generalizes to the two dimensional case the results of [16].

The main OLS-identifiability result for a∈C_n⊂L²(Ω) from a measurez of grad uin H¹(Ω) is stated and proved in Section 7: at each scalen, the diffusion coefficientais OLS-identifiable inC_n over a neighborhoodV_n of the output setD. When the parameterization ofais refined, i.e. whenngoes to infinity, the neighborhood Vn shrinks aroundD, and the Lipschitz constant of the z→aˆmapping explodes. These theoretical results are coherent with the nice properties of multiscale parameterization which have been observed numerically in [16].

2. The inverse problem

We consider a domain Ω⊂R² such that its boundary∂Ω is partitioned into Γ_D, Γ_N, and Γ_i,i= 1,· · ·, N.

Here Γ_i represent the boundaries of holes, which will be used to model source and sink terms, and Γ_D and Γ_N are a partition of the outer boundary of Ω corresponding to Dirichlet and Neumann boundary conditions.

We define the Hilbert space

V ={v∈H¹(Ω) :v|ΓD = 0, v|Γi =v_i= const, i= 1,· · ·, N}

kvk_V =|∇v|_L², (2.1)

where in case Γ_D=φthe conditionv|ΓD = 0 is replaced byR

Ωv= 0. It is supposed that Ω satisfies Ω is bounded and connected; ∂Ω∈C^1,1; ¯Γ_N,Γ¯_D and

¯Γ_i, i= 1,· · ·, N are pairwise disjoint. (2.2) On V we define the linear formL by

L(v) =R

Ωf v+R

ΓNg v+P_N

i=1Q_iv_i, forv∈V,

wheref ∈L^p(Ω), g∈L^p(Γ_N),for somep >2, Q_i∈R, i= 1,· · · , N, (2.3) with the additional condition that

Z

Ω

f + Z

ΓN

g + XN i=1

Q_i= 0 if Γ_D=φ.

Henceforth we denote byC the set of admissible parametersa. The precise conditions onC we be given further below. In particular they will allow to associate to every a∈C the solutionu=u_a∈V defined by

(Q)

Z

Ω

a∇u∇v=L(v) for all v∈V, which is the variational formulation of the elliptic equation











−div (agradu) =f in Ω u|ΓD = 0, a∂u

∂n|ΓN =g R

Γia∂u

∂n=Q_i, u= unknown constant on Γ_i, i= 1,· · · , N.

(2.4)

We shall be concerned with the inversion of the mappinga→gradu_afromL²(Ω) toL²(Ω) in the least-squares sense:

(P) minimize 1

2|gradu_a−z|²_L2 over a∈C, where z∈L²(Ω) is a given observation.

Definition 2.1. The parameterais OLS-identifiable inC from the measurementzof graduif and only if the nonlinear least squares problem (P) is quadratically (Q-) wellposed in the sense that:

D={gradua∈L²(Ω) :a∈C}

possesses a neighborhoodV such that

1. for everyz∈ V problem (P) has a unique solution ˆa;

2. for everyz∈ V problem (P) has no parasitic local minima;

3. the mapping z→ˆais Lipschitz continuous fromL²(Ω) toL²(Ω).

Our objective is to find conditions on C such that the parametera is output least squares OLS-identifiable onC. For this purpose we require first that the parameters belong to the space

E={a∈C^0,1(Ω) :a|Γi = unknown constant a_i=i= 1,· · ·, N}, (2.5) equipped with the normk · k_C^0,1. The set of admissible parametersC is assumed to satisfy:

C⊂ {a∈ E:a_m≤a(x) a.e. in Ω,kak_C^0,1≤a_M}, (2.6) and

C is convex and closed in L²(Ω), (2.7)

where 0< a_m≤a_M are given constants. Note that the image inC(⊂ E) of the mappingz →ˆais considered in the L²-norm, whereas the set C is endowed with the norm of E. Condition (2.6) ensures that (Q) has a unique solution for all a ∈ C. Moreover, the requirement that a is Lipschitz continuous, together with the modelling of source and sink terms by holes, and the regularity hypotheses (2.2, 2.3) for Ω,f, andgimply that {|ua|_W^2,p:a∈C}is bounded, (see [18], p. 180). Since W^2,p(Ω) is continuously embedded into C¹(Ω) for every p >2, then existu_M andγ_M such that

|ua|L^∞ ≤u_M,|grad u_a|L^∞ ≤γ_M for all a∈C. (2.8) The hypothesis that the parameters satisfy a|Γi = a_i = unknown constant is the 2-D generalization of the hypothesis that ais constant on some neighborhood of each Dirac source term, which was required in the 1-D case to ensure OLS–identifiability [5]. It is also physically not too restrictive, as one can assume that the size of Γ_i’s, which model the well boundaries, are small compared to the size of Ω. The convexity and closedness condition (2.7) are required for the study of OLS–identifiability by the geometric techniques for nonlinear least squares developed in [3, 4].

As a first step towards OLS–identifiability we shall analyse in Section 3 inverse stability estimates of the form (S) |(a−b) gradu_a|_L²≤k|b(gradu_a− gradu_b)|_L²,

for k≥0. As this stability estimate ought to hold for perturbations a−bin any direction b∈ E, we attempt to prove (S) only at pointsa∈C which are identifiable:

Definition 2.2. The parametera∈C is identifiable inE if, for everyb∈ E which admits a solutionu_b∈V to (Q) one has

u_b=u_a implies b=a. (2.9)

However, we shall see in Section 3 that one cannot find for any k >0 an infinite set C satisfying (2.5–2.7) on which (S) holds uniformly. Therefore we reduce in Section 4 our attention to finite dimensional parameter sets:

for this purpose letEn,n∈Nbe a family of subspaces such that









E0={constant functions} ⊂ E1⊂ E2· · · ⊂ E ⊂C^0,1( ¯Ω) dimEn<∞ for each n

[

n∈N

En=L²(Ω), (2.10)

where the closure is taken in L²(Ω), and define for alln:

C_n=C∩ En. (2.11)

In order to have a chance for (S) to hold uniformly on C_n we require that the data of the problem, i.e.

(Ω,Γ_D,Γ_N,Γ_i, f, g, Q_i, a_m, a_M,En), are chosen such that (H)

for all n∈N one has

a∈C_n implies thatais identifiable inE_n.

For the definition of identifiability of a∈C_n in En one simply replacesC,E in Definition 2.2 byC_n,En. Under condition (H) we shall be able to prove in Section 4 the inverse stability estimate (S) on C_n for all n ∈ N, for some k = k_n, with lim

n→∞ k_n = ∞, and to estimate, in Section 5, sensitivity, deflection and curvature of the a→grad u_a mapping. These estimates will be used to prove, in Section 6, that ais OLS–identifiable on C_n for each n, provided the diameter of C in L^∞(Ω), denoted by diam_∞, is small enough. The last section will be devoted to the analysis of the advantages and disadvantages of parameterizing the problem byb= 1/a instead ofa.

Before proceeding to the next section we make sure that our theory is not empty by giving an example for sufficient conditions which ensure that (H) holds. We require two technical lemmas.

Lemma 2.1. The parameter a∈C is identifiable inE if and only if

h∈ E and Z

Ω

hgradu_a grad v= 0 for all v∈V implies h= 0.

As a consequence we observe that ifa∈C is identifiable inE andh6= 0,h∈ E, then necessarilyhgradu_a6= 0.

Lemma 2.2. Fora∈C andh∈ E define u=u_a and v= ^hu_a . Thenv∈V and Z

Ω

hgradu· gradv=1 2 Z

Ω

h²

a |gradu|²+1 2 Z

Ω

h²

a² u f+1 2 Z

ΓN

h² a² u g+

XN i=1

h²_i a²_i u_iQ_i.

Proof. Since C ⊂ E ⊂ C^0,1( ¯Ω) one has v = ^hu_a ∈ H¹(Ω). Moreover v satisfies the boundary conditions defining V and hence v∈V. It follows that

Z

Ω

hgradu· gradv= Z

Ω

h²

a| gradu|²+1 2 Z

Ω

u

a gradu· gradh²− Z

Ω

h²u

a² grada· gradu.

Integrating by parts the second term on the right hand side implies Z

Ω

hgradu· gradv =1 2 Z

Ω

h²

a| gradu|²−1 2 Z

Ω

h² u

a ∆u+ u

a² grada· gradu

+1 2 Z

ΓN

u gh² a² +1

2 PN

i=1 u_iQ_i h²_i a²_i,

which, utilizing −a∆u−grada·gradu=f gives the desired result.

Theorem 2.1. Let the data of the problem satisfy

• f =g= 0;

• Q_i, i= 1,· · · , N are not all zero;

• 0< a_m≤a_M;

• |Γ_i|, i= 1,· · ·, N, are sufficiently small.

Then for all n, alla∈C_n are identifiable in E_n and (H) holds.

Proof. Letn∈Nbe given. Leta∈C_n and u_a denote the solution to (2.4). We argue that gradu(a) cannot vanish on a setIof positive measure. Letγdenote a curve in Ω connecting the inner boundaries Γ_ito Γ_D∪Γ_N such that Ω\γis simply connected and meas γ= 0.

Then I_γ = (Ω\γ)∩I satisfies measI_γ >0. From [1], Theorem 2.1 and Remark it follows that eitheru_a is constant on Ω\γ and hence on Ω oru_a has only isolated critical points,i.e. pointsz satisfying∇u(z). But u_a cannot equal a constant over Ω since this violates the boundary conditions at the wells Γ_iat whichQ_i6= 0. On the other hand the number of isolated critical points in I_γ can be at most countable, and hence measI_γ = 0.

Consequently meas {x:∇ua(x) = 0}= 0.

Suppose that Γ_i surrounds for each i = 1,· · ·, N, a fixed source/sink location x_i. If |Γi| → 0, for all i = 1,· · ·, N, the solutionu_a converges towards the weak solution associated to a right-hand side with Dirac source term P^N

i=1Q_iδ(x−x_i), which is singular atx_i. Henceu_a|Γi =u_a,i → ∞ ifQ_i >0 and u_a|Γ_i → −∞ if Q_i <0. SinceC_n is compact anda→u_a,iis continuous, we conclude that for|Γ_i|sufficiently small the solution u_a satisfies

u_a,iQ_i≥0 for i= 1,· · ·, N, and all a∈C_n.

Henceforth it is assumed that|Γ_i|, i= 1,· · · , N, is sufficiently small. For eacha∈C_n, Lemma 2.2 implies that











for each h∈ En and v= hu_a a Z

Ω

hgradu_a· gradv≥ 1 2 Z

Ω

h²

a|grad u_a|². Since |gradu_a(x)|>0 a.e. on Ω,

Z

Ω

h gradu_a· gradw= 0 for all w∈V

implies, by choosingw= ^hu_a^a, that h²= 0 a.e. on Ω and henceais identifiable in En by Lemma 2.1.

3. Decomposition of L

(Ω)

It will be convenient to denote by (·,·) the scalar products inL²(Ω) andL²(Ω). Then for everya, b∈Cwe obtain from the variational formulation (Q) defining u_a and u_b that

((a−b) gradu_a, gradv) = (b( gradu_b− gradu_a), gradv), (3.1) for allv∈V. This suggests to associate toV an equivalence relation∼of vectorfields inL²(Ω) according to

~

q∼q~⁰ if ~q·gradv=q~⁰·gradv for all v∈V, (3.2)

to denote by

G=L²(Ω)/_∼ the quotient space

G^⊥ the orthogonal complement (3.3)

and by

P andP^⊥ the orthogonal projection inL²(Ω) ontoGandG^⊥. (3.4) The decomposition

L²(Ω) =G⊕G^⊥

is, by construction, adapted to the elliptic problem (Q). We further introduce W ={ψ∈H¹(Ω) :ψ|ΓN = 0}, where the conditionR

Ωψ= 0 is added to the definition ofW if Γ_N =φ. Forϕ∈H¹(Ω) andψ~∈H¹(Ω)×H¹(Ω) we define

rot~ ϕ=







∂ϕ

∂x₂

−∂ϕ

∂x₁





 and rotψ~= ∂ψ₂

∂x₁ −∂ψ₁

∂x₂·

The following representation forGandG^⊥ can be obtained.

Proposition 3.1.

G={gradϕ: ϕ∈V} G^⊥={rotψ : ψ∈W}·

Moreover, for every ~q∈L²(Ω) one has

P ~q= gradϕ, P^⊥~q=rot~ ψ, where ϕ∈V andψ∈W are given by

(gradϕ, gradv) = (~q, gradv) for allv∈V, (rot~ ψ, ~rotv) = (~q, ~rotv) for allv∈W.

Except for the atypical boundary conditions this decomposition is rather standard. For convenience an outline of the proof is given in the Appendix. The identifiability condition can now be reformulated as

Proposition 3.2. A parametera∈C (respectivelyC_n) is identifiable in E (resp. E_n)if and only if:

h6= 0andh∈ E (resp. En) implyP(hgradu_a)6= 0.

The proposition follows directly from Proposition 3.1 and Lemma 2.1.

4. From which direction can an identifiable parameter be recovered in a stable way?

Leta∈C be a given reference parameter and letb∈C be a possibly different parameter. We investigate in this section conditions onbfor which the stability estimate (S) holds.

From (3.1) we have

(a−b) gradu_a∼b( gradu_b− gradu_a), (4.1) so that

k(a−b) gradu_ak_G=kb( gradu_b− gradu_a)k_G ≤ |b( gradu_b− gradu_a|_L². (4.2) We decompose (a−b) gradu_a onG⊕G^⊥:

(a−b) gradu_a= gradϕ+rot~ ψ, (4.3) where ϕ∈V andψ∈W are given according to Proposition 3.1 by

(gradϕ, gradv) = ((a−b) gradu_a, gradv) for all v∈V, (4.4)

(rot~ ψ, ~rotv) = ((a−b) gradu_a, ~rotv) for all v∈W. (4.5) Clearly one has

grad ϕ=P((a−b) gradu_a), ~rotψ=P^⊥((a−b) gradu_a). (4.6) Moreover

k(a−b) gradu_akG=|P((a−b) gradu_a)|_L² which together with (4.2) implies

|P((a−b) gradu_a|_L² ≤ |b(gradub− gradu_a)|_L². (4.7) Defining forM >0 the set

S_M(a) ={b∈C :|P^⊥((b−a) gradu_a)|_L² ≤M|P((b−a) gradu_a)|_L²}, (4.8) we conclude that

|(b−a) gradu_a|_L² ≤(1 +M²)^1/2|P((b−a) gradu_a)|_L², (4.9) for allb∈S_M(a). Combining (4.7) and (4.9) implies

Proposition 4.1. LetM >0 anda∈C be given. Then for allb∈S_M(a)the stability estimate (S) holds with k= (1 +M²)^1/2:

|(b−a) gradu_a|L² ≤(1 +M²)^1/2|b( gradu_b− gradu_a)|L². (4.10)

Hence the directions b−a, withb ∈S_M(a) are those from which the parametera can be recovered within C with a stability constant (1 +M²)^1/2. We investigate now the shape ofS_M(a). The setS_M(a) is the intersection ofC with a wedge having its vertex ata. Clearlya∈S_M(a). Ifais in theE–interior ofC,S_M(a) also contains parameters of the formb=a+tγ(u_a) fortsmall enough, whereγ is any regular function. In fact, in this case the gradients of b−aand u_a are collinear so thatP^⊥((b−a) gradu_a) = 0 (see (4.14) below), and hence (S) holds with M = 0 andk= 1.

Proposition 4.2. Leta∈C be identifiable inE. Then [

M>0

S_M(a) =C. (4.11)

Proof. Letb∈C withb6=a. Asais identifiable it follows from Proposition 3.2 thatP((b−a) gradu_a)6= 0.

Hence b∈S_M(a) forM sufficiently large.

We next interpret the quantities which enter the definition ofS_M(a).

Lemma 4.1. For everya∈C andh∈ E we have

kdiv (hgradu_a)k_H⁻¹ ≤ |P(hgradu_a)|_L² ≤ |hgradu_a|_L², (4.12)

krot~ hgradu_ak_W^∗ =|P^⊥(hgradu_a)|_L²

≤min{CW|rot~ h· gradu_a|,|hgradu_a|}, (4.13) where C_W is the Poincare constant inW.

Proof. From Proposition 3.1 we have

|P(hgradu_a)|_L² =|gradϕ|_L² = sup{( gradϕ, ~q⁰) : q~⁰∈L²,|q~⁰|_L² = 1}·

Butq~⁰ = gradv+rot~ wwithv∈V andw∈W, so that

|P(hgradu_a)|_L² = sup{(gradϕ, gradv+rot~ w) :

v∈V, w∈W,|gradv|²+|rot~ w|²= 1}

= sup{(gradϕ, gradv) :v∈V,|gradv|= 1}

and

|P(hgrad u_a)|_L² = sup Z

Ω

hgradu_a·gradv :v∈V,|grad v|= 1

·

These estimates imply by the Cauchy–Schwartz the second inequality in (4.12). The first follows from H₀¹(Ω)

⊂V. From Proposition 3.1 we obtain as well that

|P^⊥(hgradu_a)|_L² = sup Z

Ω

hgradu_a·rot~ w:w∈W, |gradw|= 1

· (4.14)

By Green’s formula we find:

Z

Ω

hgrad u_a·rot~ w= Z

Ω

rot (hgradu_a)w− Z

∂Ωh∂u_a

∂τ w. (4.15)

Sinceu_a = 0 on Γ_D andu_a = const on each Γ_i, we have ^∂u_∂τ^a = 0 on Γ_D and on Γ_i, i= 1,· · ·, N, andw= 0 on Γ_N. Thus all boundary terms vanish. From (4.14, 4.15) and the fact that rot(hgradu_a) =rot~ h· gradu_a we find

|P^⊥(hgradu_a)|_L²= sup

w∈W,|gradw|=1

Z

Ω

(rot~ h· gradu_a)w, which, by Poincar´e inequality inW shows that

|P^⊥(hgradu_a)|_L²=|rot~ h· gradu_a|_W^∗ ≤C_W|rot~ h· gradu_a|_L².

Combining (4.14) and the last estimate implies (4.13) and the lemma is proved.

Lemma 4.1 implies thatS_M(a) contains allb∈C such thath=b−asatisfies C_W|rot~ hgradu_a| ≤M|div (hgradu_a)|_H⁻¹.

Hence we see that the perturbations from which a can be stably recovered are, speaking loosely those whose gradient tends to be mostly oriented along the flow lines of u_a at each pointx∈Ω. In particular,acannot be recovered stably from perturbations hsuch that div(hgrad u_a) = 0. Whenu_a is harmonic (e.g. iff = 0 and a= const), these unstable perturbations are such that gradh·grad u_a = 0. This corresponds to the intuition that the observation of the pressure field gradu_agives little information on the diffusion parameteraorthogonal to flow lines.

We next show that if a can be recovered stably from b it can also be recovered stably, but with a larger constant, from allc∈Cin someL²– neighborhood ofb:

Theorem 4.1. Let a∈C be identifiable inE,b∈C,b6=a be given, and define:

0≤M = |P^⊥((b−a) gradu_a)|_L²

|P((b−a) gradu_a)|_L² <∞. Then for every M⁰> M there exists >0 such that

S_M0(a)⊃C∩ {c∈L²(Ω) :|c−b|_L² ≤} · (4.16) Proof. The mappings Λ(h) = P(hgradu_a) and Λ^⊥(h) = P^⊥(h∇u_a) are continuous from L²(Ω) into itself.

Hence we get for |c−b|_L²≤

|P^⊥((c−a) gradu_a)| ≤ |P^⊥((b−a) gradu_a)|+kΛ^⊥k,

|P((c−a) gradu_a)| ≥ |P((b−a) gradu_a)| − kΛk.

Since a is identifiable we have P((b−a) grad u_a) 6= 0. Hence for M⁰ > M there exists > 0 such that

|P^⊥((c−a) gradu_a)| |P((c−a) gradu_a|⁻¹≤M⁰ as soon as |c−b| ≤. This implies (4.16). Of coursec=a

cannot belong to this neighborhood of b!

At this point the question arises whether it is possible for the stability estimate (S), or (4.10) to hold uniformly for somek= (1 +M²)^1/2 for alla, b∈C. In other terms we search forC satisfying (2.6, 2.7) and

a, b∈C implies b∈S_M(a). (4.17)

We give a negative answer in the sense that there is no infinite dimensionalCwith nonemptyE-interior satisfying the specified properties.

Suppose that such a C exists, and let a be an element of the E-interior of C. Further let B denote a ball with center a and radiuscontained in C. Then (4.17) would imply that B ⊂S_M(a) and hence as S_M(a) is the intersection of Cwithawedge, that

|P^⊥(hgradu_a)|_L² ≤M|P(hgradu_a)|_L² for all h∈ E. (4.18) We show on a simple example that (4.18) cannot be true in general. For this purpose consider (Q) with Ω the unit square inR²,f = 0,g= 0 on top and bottom,g=−1, on the left andg= 1 on the right lateral boundary, and no internal sources and sinks. The solution corresponding toa= 1 is given byu_a(x, y) =x−¹₂.

• We check thata= 1 is identifiable inE. For everyh∈ E we have from Lemma 2.2 that forv= ^hu_a^a ∈V Z

Ω

hgradu_a gradv=1 2

Z

Ω

h²+1 2 Z

ΓN

h²u g.

Sinceu g≥0 on∂Ω we see that Z

Ω

hgradu_a gradv= 0 for all v∈V implies h= 0, and thus by Lemma 2.1ais identifiable inE.

• Next we consider parameter perturbations which are orthogonal to the flow lines. This results in choosing (x, y denote the coordinates inR²)

h(x, y) =h(y).

We shall require that

h∈C^0,1(0,1), Z ₁

0

h= 0 and h(0) =h(1) = 0. (4.19)

Under these conditions we estimate lower and upper bounds to |P^⊥(hgradu_a)|and|P(hgradu_a)|.

(i) From Lemma 4.1 we have

|P^⊥(hgradu_a)|=krot hgradu_akW^∗ =kh⁰kW^∗, and by (4.19)

kh⁰k_W^∗ = sup Z

Ω

h∂w

∂y: w∈W,|gradw|= 1

· (4.20)

LetH be the primitive ofh:

H(y) = Z _y

0

h(τ)dτ

and note thatH(0) =H(1) = 0. We define

˜

w(x, y) =x(1−x)H(y), so thatwe= 0 on∂Ω = Γ_N, and hencewe∈W, with

|gradwe|²= ¹₃(|H|²+₁₀¹|h|²).

Choosingw= ^we

|gradw|^e in (4.20) gives

kh⁰k_W^∗ ≥

√3|h|² 6

q|H|²+₁₀¹|h|²·

Since |H|²≤ ¹₂|h|²we obtain

|P^⊥(hgradu_a)| ≥ ²₃|h|. (4.21)

(ii) From the proof of Lemma 4.1 we get

|P(hgradu_a)|= sup Z

Ω

h∂v

∂x : v∈V,|gradv|= 1

,

and, integrating by parts with respect tox,

|P(hgradu_a)|= sup Z ₁

0

h(y)(v(1, y)−v(0, y))dy : v∈V,|gradv|= 1

·

Let γ =∂Ω∪ {x= 1}, the right lateral boundary of Ω, and denote byc the continuity constant from V to H^1/2(γ). Then|gradv|= 1 implieskv|γk_H^1/2_(γ)≤c, so that by symmetry

|P(hgradu_a)| ≤2 sup Z ₁

0

h ξ:ξ∈H^1/2(γ),kξk_H^1/2_(γ)≤c

·

Associating to ha functionH ∈H^1/2(γ) satisfying ((H, ξ))_H1/2(γ)=

Z ₁

0

h ξ, for all ξ∈H^1/2(γ), (4.22)

we have

|P(hgradu_a)| ≤2 sup

kξk_H1/2(γ)≤c((H, ξ))_H1/2 = 2ckHk_H1/2. (4.23) From (4.21, 4.23) we see that (4.18) would imply that

|h| ≤3M ckHk_H1/2(γ), (4.24)

for allh∈ E satisfying (4.19). But we can choose a sequence of functionsh_n satisfying h_n ∈C^0,1(0,1),

Z ₁

0

h_n= 0, h_n(0) =h_n(1) = 0, and

|h_n|L² = const,|h_n|_H^−1/2_(γ)→0 for n→ ∞.

From (4.22) we see that kH_nk_H^1/2_(γ) → 0 for n → ∞. This contradicts (4.24) and consequently (4.18) as

well.

It is hence impossible to find an infinite dimensional set C with nonempty interior on which the stability estimate holds uniformly. This motivates the reduction to finite dimensional parameter sets in the remaining sections.

5. Finite dimensional stability estimates

We turn to the finite dimensional setting of Section 2 withE0⊂ E1⊂ · · · ⊂ E satisfying (2.11). We recall the definition C_n =C∩ En and suppose throughout this section that (H) holds. Identifiability ofainEnimplies by Proposition 3.2 that for alla, b∈C_n withb6=awe have

P((b−a) gradu_a)6= 0. (5.1)

Hence we can define for eachn∈N

M_n= sup

a,b∈Cn a6=b

|P^⊥((b−a) gradu_a)|_L²

|P((b−a) gradu_a)|_L² · (5.2)

From Proposition 3.2 we know that (5.1) holds with b−areplaced by anyh∈ En,h6= 0, and hence M_n≤ sup

a∈Cn

sup

h∈En,khkE=1

|P^⊥(hgradu_a)|_L²

|P(hgradu_a)|_L² · (5.3)

As the fraction in (5.3) is a continuous function of h and a, and the suprema are taken on compact sets it follows that M_n is finite for eachn. Since E0 consists of constant functionsM₀= 0 by Lemma 4.1. Moreover sinceEn⊂ En+1, for alln, we have

0 =M₀≤M₁≤ · · · ≤M_n· · · <∞. (5.4) From the example at the end of the previous section we can generally expect that lim

n→∞M_n =∞.

Proposition 4.1 implies the following stability estimates:

Theorem 5.1. Let (2.5–2.7, 2.10, 2.11) and (H) hold. Then for every n∈N

|(b−a) gradu_a|_L² ≤(1 +M_n²)^1/2|b(gradu_b− gradu_a)|_L², for all a, b∈C_n, whereM_n is defined in (5.2).

This theorem gives a rigorous justification to the numerical observation [12, 13, 16] that the sensitivity of the inversion of the a → u_a mapping is a decreasing function of the scale at which the parameter is estimated.

This observation together with the analysis of the nonlinearity of the mappinga→u_a, to be given in the next section, has motivated the introduction of successful multiscale approaches to parameter estimation [9, 16].

We close the section with an estimate ofM_n in the case where (H) is satisfied by virtue of Theorem 2.1.

Theorem 5.2. Let (2.5–2.7, 2.10, 2.11) hold, and suppose that the hypotheses of Theorem 2.1 are satisfied.

Then

M_n≤ sup

a∈Cn

h∈Esupn,h6=0M_n(a, h), where, for a∈C_n andh∈ En,h6= 0:

M_n(a, h) = 2 a_M

a_m

_1/2 |grad (^h_a)u_a|

|^h_a gradu_a| min

1, C_W|rot~ hgradu_a|

|hgradu_a|

· (5.5)

Proof. Choosing v = ^hu_a^a| grad (^hu_a^a)|⁻¹_L2 on the right hand side of the equality above (4.14) we obtain by Theorem 2.1 and Lemma 2.2

|P(hgradu_a)|_L² ≥1 2

|_a1/2^h gradu_a|²_L2

|grad (^hu_a^a)|_L² and hence

|P(hgradu_a)|_L² ≥ a^1/2_m 2a^1/2_M

|^h_a gradu_a| |hgradu_a|

|grad (^h_au_a) ·

Combining this estimate with (4.13) of Lemma 4.1 and (5.3) implies the theorem.

Fora, b∈C_n the stability constant M_n(a, b−a) of (5.5) allows the following interpretation:

• the first factor is related to the size ofC_n;

• the second factor tends to infinity when the dimension ofE_n goes to infinity. It does not depend on the relative orientations of gradhand grad u_a;

• the third factor is bounded by 1 and tends to zero as gradhbecomes collinear with gradu_a.

6. Finite dimensional sensitivity, deflection and curvature estimates

In this section we analyze the geometric quantities associated to the parameter-to-solution mappinga→u_a defined by (Q). Knowledge of these quantities will be required in the next section to prove that the output setD_n is strictly quasiconvex, and hence the OLS-identifiability ofa.

Fora₀, a₁∈C_n andt∈[0,1] we set

h=a₁−a₀, (6.1)

a= (1−t)a₀+t a₁. (6.2)

The geometric quantities are related to the curve t ∈ [0,1] → ∇u_a ∈ L²(Ω) in the range of the mapping a→ ∇u_a. We denote byη the velocity andξ the acceleration,i.e. the first and second derivatives ofu_a with respect to t. The equations characterizingη∈V andξ∈V are found to be:

Z

Ω

agrad η· gradv=− Z

Ω

hgradu_a gradv, for all v∈V, (6.3) Z

Ω

agradξ· gradv=−2 Z

Ω

hgradηgrad v, for all v∈V. (6.4) For givenn∈Nthe objective is to find 0< α_m≤α_M, Θ≥0 andR >0 such that the following inequalities hold uniformly for all a₀, a₁∈C_n andt∈[0,1]:

α_m|h|_L² ≤ |gradη|_L² ≤α_M|h|_L², (6.5)

|grad ξ|_L² ≤Θ|gradη|_L², (6.6)

|gradξ|_L² ≤ 1

R|gradη|²_L2. (6.7)

These inequalities can be interpreted as follows:

• α_m and α_M are lower and upper bounds to the first derivative of a → ∇u_a. They are referred to as minimal and maximal sensitivity;

• Θ is an upper bound to the deflection along the curvet→ ∇u_a (i.e. to the angle between the tangents at two points of the curve);

• _R¹ is an upper bound to the curvature along the curve.

Theorem 6.1. Let (2.5–2.7, 2.10, 2.11) and (H) hold. Then, for every n ∈ N, a₀, a₁ ∈ C_n and t ∈ [0,1], (6.5–6.7) hold with

α_m= γ_m,n

a_M(1 +M_n²)^1/2, α_M = γ_M

a_m, (6.8)

Θ = 2 diam_∞(C_n)

a_m , (6.9)

1

R = 2K_n (1 +M_n²)^1/2 γ_m,n

a_M

a_m, (6.10)

where γ_M is defined in (2.8), and

γ_m,n= inf

a∈Cn inf

h∈En,|h|_L2=1|hgradu_a|_L² >0, (6.11) K_n= sup

h∈En,|h|_L2=1

|h|_L^∞. (6.12)

Proof. Lett+ dt ∈[0,1] and denote bya_t anda_t+dt the corresponding values ofagiven by (6.2). Choosing a=a_t andb=a_t+dtin Theorem 5.1 and letting dttend to zero implies that

|hgradu_a|_L² ≤(1 +M_n²)^1/2|agradη|_L². (6.13) Hence the left inequality of (6.5) is satisfied with α_mdefined by (6.8, 6.11). The strict positivity ofγ_m,nfollows from (H) which ensures that the argument of the inf is strictly positive, and hence the inf itself is strictly positive as it is taken over a compact set. The right inequality of (6.5) is obtained by choosing v=η in (6.3) and using (2.9). Setting v=ξ in (6.4) gives

|a^1/2gradξ|_L² ≤2| h

√a|L^∞ |gradη|,

which implies (6.9), and also (6.10) using (6.11–6.13).

Notice first that γ_m,n can be understood as a lower bound to the local mean value of|gradua| at scale n.

In general u_a will have stationary points where grad u_a(x) = 0, in which case one expects that γ_m,n→0 for n→ ∞. In special cases, as for instance the example of Section 4, see also [15, 17] for further examples, it can happen that |gradu_a(x)| ≥γ_m>0 for alla∈C andx∈Ω, in which caseγ_m,n≥γ_m>0 for all n.

Let us now discuss the behavior of the constants α_m, α_M, Θ and R as n tends to infinity. In case (H) is satisfied through the assumptions of Theorem 2.1 an upper bound toM_n can obtained by Theorem 5.2:

M_n≤2 a_M

a_{m 1/2}

1 + u_M

γ_m,n sup

a∈Cn

sup

h∈En,|^h_a|_L2=1

grad h a

!

· (6.14)

In case the finite dimensional spacesE_nare obtained from a regular family of triangulations of Ω by elements of size ∆x, the right hand side of (6.14) is of the order _∆x¹ forn→ ∞. In this situation the continuity constantK_n of the L² → L^∞ injection (for h ∈ En) is also of the order _∆x¹ as n → ∞. These considerations imply the following