Convergent algorithm based on Carleman estimates for the recovery of a potential in the wave equation.

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the recovery of a potential in the wave equation.

Lucie Baudouin, Maya de Buhan, Sylvain Ervedoza

To cite this version:

Lucie Baudouin, Maya de Buhan, Sylvain Ervedoza. Convergent algorithm based on Carleman esti-

mates for the recovery of a potential in the wave equation.. SIAM Journal on Numerical Analysis,

Society for Industrial and Applied Mathematics, 2017, 55 (4), pp.1578-1613. �hal-01352772�

FOR THE RECOVERY OF A POTENTIAL IN THE WAVE

2

EQUATION.

^∗

3

LUCIE BAUDOUIN^†, MAYA DE BUHAN^‡, AND SYLVAIN ERVEDOZA^§ 4

Abstract. This article develops the numerical and theoretical study of the reconstruction 5

algorithm of a potential in a wave equation from boundary measurements, using a cost functional built 6

on weighted energy terms coming from a Carleman estimate. More precisely, this inverse problem 7

for the wave equation consists in the determination of an unknown time-independent potential from 8

a single measurement of the Neumann derivative of the solution on a part of the boundary. While its 9

uniqueness and stability properties are already well known and studied, a constructive and globally 10

convergent algorithm based on Carleman estimates for the wave operator was recently proposed in 11

[BdBE13]. However, the numerical implementation of this strategy still presents several challenges, 12

that we propose to address here.

13

Key words. wave equation, inverse problem, reconstruction, Carleman estimates.

14

AMS subject classifications. 93B07, 93C20, 35R30.

15

1. Introduction and algorithms.

16

1.1. Setting and previous results. Let Ω be a smooth bounded domain of

17

R

^d

, d ≥ 1 and T > 0. This article focuses on the reconstruction of the potential in a

18

wave equation according to the following inverse problem:

19

Given the source terms f and f

∂

and the initial data (w

0

, w

1

), con-

20

sidering the solution of

21

(1)







∂

_t²

W − ∆W + QW = f, in (0, T ) × Ω,

W = f

∂

, on (0, T ) × ∂Ω,

W (0) = w

0

, ∂

t

W (0) = w

1

, in Ω,

22

can we determine the unknown potential Q = Q(x), assumed to

23

depend only on x ∈ Ω, from the additional knowledge of the flux of

24

the solution through a part Γ

0

of the boundary ∂Ω, namely

25

(2) M = ∂

_n

W, on (0, T ) × Γ

₀

?

26

Beyond the preliminary questions about the uniqueness and stability of this inverse

27

problem, already very well documented as we will detail below, we are interested in

28

the actual reconstruction of the potential Q from the extra information given by the

29

measurement of the flux M of the solution on a part of the boundary. This issue was

30

already addressed theoretically in our previous work [BdBE13] based on Carleman

31

estimates. However, the algorithm proposed in [BdBE13], proved to be convergent,

32

cannot be implemented in practice as it involves minimization processes of function-

33

als containing too large exponential terms. Therefore, our goal is to address here the

34

∗Submitted to the editors DATE.

Funding:Partially supported by the Agence Nationale de la Recherche (ANR, France), Project MEDIMAX number ANR-13-MONU-0012. The authors wish to thank Institut Henri Poincar´e (Paris, France) and the Research In Paris funding.

†LAAS-CNRS, Universit´e de Toulouse, CNRS, Toulouse, France (baudouin@laas.fr)

‡CNRS, UMR 8145, MAP5, Universit Paris Descartes, Sorbonne Paris Cit, France (maya.de- buhan@parisdescartes.fr)

§Institut de Math´ematiques de Toulouse, Universit´e de Toulouse, CNRS, Toulouse, France (ervedoza@math.univ-toulouse.fr)

1

numerical challenges induced by that approach.

35 36

Before going further, let us recall that if Q ∈ L

^∞

(Ω), f ∈ L

¹

(0, T ; L

²

(Ω)),

37

f

∂

∈ H

¹

((0, T ) × ∂Ω), w

0

∈ H

¹

(Ω) and w

1

∈ L

²

(Ω), and assuming the compati-

38

bility condition f

∂

(0, x) = w

0

(x) for all x ∈ ∂Ω, the Cauchy problem (1) is well-posed

39

in C

⁰

([0, T ]; H

¹

(Ω))∩C

¹

([0, T ]; L

²

(Ω)), and the normal derivative ∂

n

W is well-defined

40

as an element of L

²

((0, T ) × ∂Ω), see e.g. [Lio88, LLT86].

41 42

Our results will require the following geometric conditions (sometimes called “mul-

43

tiplier condition” or “Γ-condition”):

44

∃ x

0

6∈ Ω, such that

45

Γ

₀

⊃ {x ∈ ∂Ω, (x − x

₀

) · ~ n(x) ≥ 0},

46

(3)

T > sup

x∈Ω

|x − x

0

|.

(4)

47

Space and time conditions (3)–(4) are natural from the observability point of view, and

48

appear naturally in the context of the multiplier techniques developed in [Ho86, Lio88].

49

They are more restrictive than the well-known observability results [BLR92] by Bar-

50

dos Lebeau Rauch based on the behavior of the rays of geometric optics, but the

51

geometric conditions (3)–(4) yield much more robust results, and this will be of pri-

52

mary importance in our approach.

53 54

In fact, under the regularity assumption

55

(5) W ∈ H

¹

(0, T ; L

^∞

(Ω)),

56

the positivity condition

57

(6) ∃α > 0 such that |w

0

| ≥ α in Ω,

58

the knowledge of an a priori bound m > 0 such that

59

(7) kQk

_L∞(Ω)

≤ m, i.e. Q ∈ L

^∞_≤m

(Ω) = {q ∈ L

^∞

(Ω), kqk

_L^∞_(Ω)

≤ m},

60

and the multiplier conditions (3)–(4), the results in [Baufr] (and in [Yam99] under

61

more regularity hypothesis) state the Lipschitz stability of the inverse problem con-

62

sisting in the determination of the potential Q in (1) from the measurement of the

63

flux M in (2).

64 65

We will introduce our work by describing what was done in our former article

66

[BdBE13], in order to highlight stage by stage the main challenges when performing

67

numerical implementations.

68

In [BdBE13], we proposed a prospective algorithm to recover the potential Q from

69

the measurement M on (0, T ) × Γ

0

, that we briefly recall below. We assume that

70

conditions (3)–(4) are satisfied for some x

0

∈ / Ω, and we set β ∈ (0, 1) such that

71

(8) βT > sup

x∈Ω

|x − x

₀

|.

72

We then define, for (t, x) ∈ (−T, T ) × Ω, the Carleman weight functions

73

(9) ϕ(t, x) = |x − x

0

|

²

− βt

²

, and for λ > 0, ψ(t, x) = e

^{λ(ϕ(t,x)+C0)}

,

74

where C

0

> 0 is chosen such that ϕ + C

0

≥ 1 in (−T, T ) × Ω and λ > 0 is large

75

enough. The chore of the algorithm in [BdBE13] is the minimization of a functional

76

K

s,q

[µ] given for s > 0, q ∈ L

^∞_≤m

(Ω) and µ ∈ L

²

((0, T ) × Γ

0

) by

77

(10) K

s,q

[µ](z) = 1 2

Z

T 0

Z

Ω

e

^2sψ

|∂

_t²

z −∆z +qz|

²

dxdt + s 2

Z

T 0

Z

Γ0

e

^2sψ

|∂

n

z −µ|

²

dσdt,

78

set on the trajectories z ∈ L

²

(0, T ; H

₀¹

(Ω)) such that ∂

²_t

z − ∆z + qz ∈ L

²

((0, T ) × Ω),

79

∂

_n

z ∈ L

²

((0, T ) × Γ

₀

) and z(0, ·) = 0 in Ω. Note in particular that [BdBE13] shows

80

that there exists a unique minimizer of the above functional under the aforementioned

81

assumptions. The algorithm then reads as follows:

Algorithm 1 (see [BdBE13])

Initialization: q

⁰

= 0 (or any guess in L

^∞_≤m

(Ω)).

Iteration: From k to k + 1

• Step 1 - Given q

^k

, we set µ

^k

= ∂

_t

∂

_n

w[q

^k

] − ∂

_n

W [Q]

on (0, T ) × Γ

₀

, where w[q

^k

] denotes the solution of (1) with the potential q

^k

and ∂

n

W [Q] is the measurement given in (2).

• Step 2 - Minimize K

_s,qk

[µ

^k

] (defined in (10)) on the trajectories z ∈ L

²

(0, T ; H

₀¹

(Ω)) such that ∂

_t²

z − ∆z + q

^k

z ∈ L

²

((0, T ) ×Ω), ∂

n

z ∈ L

²

((0, T ) × Γ

0

) and z(0, ·) = 0 in Ω.

Let Z

^k

be the unique minimizer of the functional K

_s,qk

[µ

^k

].

• Step 3 - Set

˜

q

^k+1

= q

^k

+ ∂

t

Z

^k

(0)

w

₀

, in Ω, where w

0

is the initial condition in (1) (recall assumption (6)).

• Step 4 - Finally, set

q

^k+1

= T

m

(˜ q

^k+1

), with T

m

(q) =

q, if |q| ≤ m, sign(q)m, if |q| > m, where m is the a priori bound in (7).

82

Algorithm 1 comes along with the following convergence result:

83

Theorem 1 ([BdBE13, Theorem 1.5]). Under assumptions (3)-(4)-(5)-(6)-(7)-

84

(8), there exist constants C > 0, s

₀

> 0 and λ > 0 such that for all s ≥ s

₀

, Algorithm

85

1 is well-defined and the iterates q

^k

constructed by Algorithm 1 satisfy, for all k ∈ N ,

86

(11) Z

Ω

|q

^k+1

− Q|

²

e

^2sψ(0)

dx ≤ C kW [Q]k

²_H1(0,T;L^∞(Ω))

s

^1/2

α

²

Z

Ω

|q

^k

− Q|

²

e

^2sψ(0)

dx.

87

In particular, for s large enough, the sequence q

^k

strongly converges towards Q as

88

k → ∞ in L

²

(Ω).

89

This algorithm presents the advantage of being convergent for any initial guess q

⁰

∈ L

^∞_≤m

(Ω) without any a priori guess except for the knowledge of m. This is why we call this algorithm globally convergent. However, while this algorithm is theoretically satisfactory as at each iteration, it simply consists in the minimization of the strictly convex and coercive quadratic functional K

_s,q

, it nevertheless contains several flaws and drawbacks in its numerical implementation. In particular, we underline that the functional K

_s,q

involves two exponentials, namely

exp(sψ) = exp(s exp(λ(ϕ + C

0

))),

with a choice of parameters s and λ large enough and whose sizes are difficult to estimate. In particular, for s = λ = 3 - which are not so large of course - Ω = (0, 1), x

0

' 0

⁻

, T ' 1

⁺

and β ' 1

⁻

, the ratio

max

(0,T)×Ω

{exp(2sψ)}

min

(0,T)×Ω

{exp(2sψ)}

is of the order of 10

³⁴⁰

! The numerical implementation of Algorithm 1 therefore

90

seems doomed.

91

The goal of this article is to improve the above algorithm so that it can fruitfully be

92

implemented. This will be achieved following several stages: working on the construc-

93

tion of the cost functional (specifically on the Carleman weight function), considering

94

the preconditioning of the cost functional, and adapting the new cost functional to

95

the discrete setting used for the numerics.

96

Before going further, let us mention that the inverse problem under consideration

97

has been well-studied in the literature, starting with the uniqueness result in the

98

celebrated article [BK81], see also [Kli92], which introduced the use of Carleman

99

estimates for these studies. Later on, stability issues were obtained for the wave

100

equation, first based on the so-called observability properties of the wave equation

101

[PY96, PY97] and then refined with the use of Carleman estimates, among which

102

[IY01a, IY01b, IY03, KY06]. In fact, a great part of the literature in this area, con-

103

cerning uniqueness, stability and reconstruction of coefficient inverse problems for

104

evolution partial differential equations can be found in the survey article [Kli13] and

105

we refer the interested reader to it. A slightly different approach can also be found in

106

the recent article [SU13] based on more geometric insights.

107

Let us also emphasize that we are interested in the case in which one performs only

108

one measurement. The question of determining coefficients from the Dirichlet to

109

Neumann map is different and we refer for instance to the boundary control method

110

proposed in [Bel97] or to methods based on the complex geometric optics, see [Isa91].

111

Here, as we said, we will focus on the reconstruction of the potential in the wave equa-

112

tion (1) from the flux M in (2). This question has been studied only recently, though

113

the first investigation [KI95] appears in 1995, and we shall in particular point out

114

the most recent works of Beilina and Klibanov [KB12], [BK15], who study the recon-

115

struction of a coefficient in a hyperbolic equation from the use of a Carleman weight

116

function for the design of the cost functional. However, these techniques differ from

117

ours as they work on the functions obtained after a Laplace transform of the equation.

118 119

In what follows, we propose to develop a numerical algorithm in the spirit of

120

the one in [BdBE13], study its convergence and his implementation. Before going

121

further, let us also mention the fact that one can find in [CFCM13] some numerical

122

experiments based on the minimization of a quadratic functional similar to the one in

123

(10), but with s and λ rather small, namely s = 1 and λ = 0.1, see [CFCM13, Section

124

4]. Our goal is to overcome this restriction on the size of the Carleman parameters,

125

as we request them to be large for the convergence of the algorithm.

126

1.2. New weight functions, new cost functionals, and a new algorithm.

127

In a first stage, we aim at removing one exponential from the cost functional K

s,q

in

128

(10). Similarly to [BdBE13], looking again for a cost functional based on a Carleman

129

estimate for the wave equation, we will work with the Carleman weight function

130

exp(sϕ) instead of exp(s exp(λ(ϕ + C

0

))). This requires an adaptation of the proof

131

of [BdBE13] with such a weight function and the use of the Carleman estimates

132

developed in [LRS86] (see also [IY01b]), that we will briefly recall in Section 2.

133

In particular, instead of minimizing K

s,q

[µ] introduced in (10) as in Step 2 of

134

Algorithm 1, we will perform a minimization process on a new functional J

s,q

[˜ µ],

135

to be defined later in (13), based on the simplified weight function exp(sϕ). Before

136

introducing that functional, we shall define the following restricted set O:

137 138

(12) O = {(t, x) ∈ (0, T ) × Ω, βt > |x − x

0

|}

139

= {(t, x) ∈ (0, T ) × Ω, |∂

t

ϕ(t, x)| ≥ |∇ϕ(t, x)|},

140141

which is depicted in Figure 1.

142

0 1

ϕ = 0, slop e

√1β

| x − x

⁰

| = βt , slop e

1 β

O

x

0

x

T t

Ω

Fig. 1: Illustration of domain O in the case Ω = (0, 1).

For s > 0, q ∈ L

^∞

(Ω) and ˜ µ ∈ L

²

((0, T ) × Γ

₀

), we then introduce the functional

143

J

_s,q

[˜ µ] defined by

144 145

(13) J

_s,q

[˜ µ](z) = 1 2

Z

T 0

Z

Ω

e

^2sϕ

|∂

_t²

z − ∆z + qz|

²

dxdt

146

+ s 2

Z

T 0

Z

Γ0

e

^2sϕ

|∂

n

z − µ| ˜

²

dσdt + s

³

2

Z Z

O

e

^2sϕ

|z|

²

dxdt,

147 148

to be compared with K

s,q

[µ] in (10), on the trajectories z ∈ C

⁰

([0, T ]; H

₀¹

(Ω)) ∩ C

¹

([0, T ]; L

²

(Ω)) such that ∂

_t²

z − ∆z + qz ∈ L

²

((0, T ) × Ω) and z(0, ·) = 0 in Ω.

This functional J

_s,q

[˜ µ] is quadratic, and as we will show later in Section 2.3, under conditions (3)–(4)–(8), it is strictly convex and coercive, therefore enjoying similar properties as the functional K

_s,q

[µ]. Nevertheless, let us once more emphasize that the functional J

_s,q

[˜ µ] is less stiff than the functional K

_s,q

[µ] as now the weights are of the form exp(2sϕ) instead of exp(2sψ) = exp(2s exp(λ(ϕ+C

₀

))) in (10). This already indicates the possible gain we could have by working with the functional J

s,q

[˜ µ] in (13) instead of K

s,q

[µ] in (10).

It may appear surprising to note ˜ µ instead of µ. These slightly different notations come from the fact that the functional K

s,q

[µ] tries to find an optimal solution Z of

∂

_t²

Z − ∆Z + qZ ' 0 in (0, T ) × Ω, and ∂

n

Z ' µ in (0, T ) × Γ

0

,

while the functional J

s,q

[˜ µ] tries to find an optimal solution ˜ Z of

∂

_t²

Z ˜ − ∆ ˜ Z + q Z ˜ ' 0 in (0, T ) × Ω, ∂

n

Z ˜ ' µ ˜ in (0, T ) × Γ

0

, and Z ˜ ' 0 in O.

Therefore, as ˜ Z is sought after such that it is small in O, it is natural to introduce a

149

smooth cut-off function η ∈ C

²

( R ) such that 0 ≤ η ≤ 1 and

150

(14) η(τ) = 0, if τ ≤ 0, and η(τ) = 1, if τ ≥ d

²₀

:= d(x

0

, Ω)

²

,

151

(recall that d

²₀

> 0 according to Assumption 3) see Figure 2. Next, the idea is that if

˜

µ = η(ϕ)µ, in (0, T ) × Γ

₀

.

and if Z denotes the minimizer of the functional K

s,q

[µ] in (10), then the minimizer

152

Z ˜ of J

s,q

[˜ µ] in (13) should be close to η(ϕ)Z in (0, T ) × Ω and in particular at t = 0

153

this should yield, due to the choice of η in (14), ∂

t

Z(0) ˜ ' ∂

t

Z (0) in Ω.

0.2 0.4 0.6 0.8 1 0.5

1

0

ϕ=0 ϕ=0.04

ϕ=0.4 ϕ=0.9

x0

T

0 d²₀ ϕ

1 η

0 ϕ=0

ϕ=d²⁰

x0

T

η◦ϕ= 0

η◦ϕ= 1

Ω

1

Fig. 2: Isovalues of the function ϕ (x

0

= −0.2, β = 1). Definition and application of the cut-off function η.

154 155

We are then led to propose a revised version of our reconstruction algorithm,

156

detailed in Algorithm 2 given below.

157

Of course, if one compares Algorithm 2 with Algorithm 1, the major difference

158

is in Step 2 in which one minimizes the functional J

_s,qk

[˜ µ] in (13) instead of the

159

functional K

_s,qk

[µ] in (10). And as we have said above, the two functionals should

160

have minimizers that are close at t = 0. In fact, similarly as Theorem 1, we will obtain

161

the following result:

162

Theorem 2. Under assumptions (3)-(4)-(5)-(6)-(7)-(8), there exist positive con-

163

stants C and s

0

such that for all s ≥ s

0

, Algorithm 2 is well-defined and the iterates

164

q

^k

constructed by Algorithm 2 satisfy, for all k ∈ N ,

165

(17) Z

Ω

|q

^k+1

− Q|

²

e

^2sϕ(0)

dx ≤ C kW [Q]k

²_H1(0,T;L^∞(Ω))

s

^1/2

α

²

Z

Ω

|q

^k

− Q|

²

e

^2sϕ(0)

dx.

166

In particular, for s large enough, the sequence q

^k

strongly converges towards Q as

167

k → ∞ in L

²

(Ω).

168

The proof of Theorem 2 is given in Section 2 and closely follows the one of Theorem 1

169

in [BdBE13]. The main difference is that the starting point of our analysis, instead

170

Algorithm 2

Initialization: q

⁰

= 0 (or any guess q

⁰

∈ L

^∞_≤m

(Ω)).

Iteration: From k to k + 1

• Step 1 - Given q

^k

, we set ˜ µ

^k

= η(ϕ)∂

t

∂

n

w[q

^k

] − ∂

n

W [Q]

on (0, T ) × Γ

0

, where w[q

^k

] denotes the solution of

(15)







∂

²_t

w − ∆w + q

^k

w = f, in (0, T ) × Ω,

w = f

∂

, on (0, T ) × ∂Ω,

w(0) = w

0

, ∂

t

w(0) = w

1

, in Ω,

corresponding to (1) with the potential q

^k

and ∂

_n

W [Q] is the measurement in (2).

• Step 2 - We minimize the functional J

_s,qk

[˜ µ

^k

] defined in (13), for some s > 0 that will be chosen independently of k, on the trajectories z ∈ C

⁰

([−T, T ]; H

₀¹

(Ω)) ∩ C

¹

([−T, T ]; L

²

(Ω)) such that ∂

²_t

z − ∆z + q

^k

z ∈ L

²

((0, T ) × Ω), ∂

n

z ∈ L

²

((0, T ) × Γ

0

) and z(0, ·) = 0 in Ω. Let Z e

^k

be the unique minimizer of the functional J

_s,qk

[˜ µ

^k

].

• Step 3 - Set

(16) q ˜

^k+1

= q

^k

+ ∂

t

Z e

^k

(0) w

0

, in Ω, where w

0

is the initial condition in (15) (or (1)).

• Step 4 - Finally, set

q

^k+1

= T

m

(˜ q

^k+1

), with T

m

(q) =

q, if |q| ≤ m, sign(q)m, if |q| ≥ m, where m is the a priori bound in (7).

of being the Carleman estimate in [Im02], is the Carleman estimate in [LRS86].

171

The main improvement with respect to Algorithm 1 is the fact that the functional

172

J

s,q

[˜ µ] in (13) contains weight functions with only one exponential, making the prob-

173

lem less difficult to implement. However, it is still numerically challenging to use

174

such functionals, especially as the convergence of Algorithm 2 gets better for large

175

parameter s. We propose below two ideas to make it numerically tractable.

176

1.3. Preconditioning, processing and discretizing the cost functional.

177

When considering the functional J

s,q

[˜ µ] in (13), one easily sees that exponentials

178

factors can be removed if considering the unknown ze

^sϕ

instead of z. Such transfor-

179

mation corresponds to a preconditioning of the functional J

s,q

[˜ µ]. Indeed, that way,

180

exponential factors do not appear anymore when computing the gradient of the cost

181

functional J

_s,q

[˜ µ]. Nevertheless, there are still exponentials factors appearing in the

182

measurements. We therefore also develop a progressive algorithm in the resolution of

183

the minimization process. The idea is to consider intervals in which the weight func-

184

tion ϕ does not significantly change, allowing to preserve numerical accuracy despite

185

the possible large values of s. Details will be given in Section 3.

186 187

When implementing the above strategy numerically, one has to discretize the

188

wave equation under consideration, and to adapt the functional J

s,q

[˜ µ] to the discrete

189

setting. As it is well-known [Tre82, Zua05], most of the numerical schemes exhibit

190

some pathologies at high-frequency, namely discrete rays propagating at velocity 0 or

191

blow up of observability estimates. Therefore, we need to take some care to adapt the

192

functional J

s,q

[˜ µ] to the discrete setting. In particular, following ideas well-developed

193

in the context of the observability of discrete waves (see [Zua05]), we will introduce a

194

naive discrete version of J

s,q

[˜ µ] and penalize the high-frequencies.

195

To simplify the presentation of these penalized frequency functionals, we will introduce

196

it in full details on a space semi-discrete and time continuous 1d wave equations, where

197

the space semi-discretization is done using the finite-difference method on a uniform

198

mesh. In this case, our approach, even at the discrete level, can be made completely

199

rigorous by adapting the arguments in the continuous setting and the discrete Carle-

200

man estimates obtained in [BE11] (recently extended to a multi-dimensional setting

201

in [BEO15]). We refer to Section 4 for extensive details.

202 203

Section 5 then presents numerical results illustrating our method on several ex-

204

amples. In particular, we will illustrate the good convergence of the algorithm when

205

the parameter s is large. We shall also discuss the cases in which the measurement

206

is blurred by some noise and the case in which the initial datum w

0

is not positive

207

everywhere.

208 209

Outline. Section 2 is devoted to the proof of the convergence of Algorithm 2. In

210

Section 3 we explain how the minimization process of the functional J

s,q

in (13) can

211

be strongly simplified. Section 4 then makes precise the new difficulties arising when

212

discretizing the functional J

s,q

, and Section 5 presents several numerical experiments.

213

2. Study of Algorithm 2.

214

2.1. Main ingredients. The goal of this section is to prove Theorem 2. As

215

mentioned in the introduction, the proof will closely follows the one of Theorem 1

216

in [BdBE13]. The main difference is that, instead of using the Carleman estimate

217

developed in [Im02, Baufr], we will base our proof on the following one:

218

Theorem 3. Assume the multiplier conditions (3)-(4) and β ∈ (0, 1) as in (8).

219

Define the weight function ϕ as in (9). Then there exist s

0

> 0 and a positive constant

220

M such that for all s ≥ s

0

:

221 222

(18) s Z

T

−T

Z

Ω

e

^2sϕ

|∂

t

z|

²

+ |∇z|

²

+ s

²

|z|

²

dxdt ≤ M Z

T

−T

Z

Ω

e

^2sϕ

|∂

²_t

z − ∆z|

²

dxdt

223

+ M s Z

T

−T

Z

Γ0

e

^2sϕ

|∂

_n

z|

²

dσdt + M s

³

Z Z

(|t|,x)∈O

e

^2sϕ

|z|

²

dxdt,

224 225

for all z ∈ C

⁰

([−T, T ]; H

₀¹

(Ω)) ∩C

¹

([−T, T ]; L

²

(Ω)) with ∂

_t²

z −∆z ∈ L

²

((−T, T )×Ω),

226

where the set O satisfies (12).

227

Furthermore, if z(0, ·) = 0 in Ω, one can add to the left hand-side of (18), the following

228

term:

229

(19) s

^1/2

Z

Ω

e

^2sϕ(0)

|∂

t

z(0)|

²

dx.

230

The Carleman estimate of Theorem 3 is quite classical and can be found in the liter-

231

ature in several places, among which [LRS86, Isa06, Zha00, FYZ07, Bel08]. For the

232

convenience of the reader, we briefly sketch the proof in Section 2.2. However, the

233

proof of the fact that the term (19) can be added in the left hand side of (18) when

234

z(0, ·) = 0 in Ω is not explicitly written in the aforementioned references, although

235

this is one of the important point of the proof of the stability result in [IY01a, IY01b].

236

Nevertheless, the idea can be adapted easily from [BdBE13], as we will detail below.

237 238

Before giving the details of the proof of Theorem 2, let us first briefly explain the

239

main idea of the design of Algorithm 2, which turns out to be very similar to the one

240

of Algorithm 1. Indeed, both Algorithms 1 and 2 are constructed from the fact that

241

if W [Q] is the solution of equation (1) and w[q

^k

] solves (15), then

242

(20) z

^k

= ∂

t

w[q

^k

] − W [Q]

243

satisfies

244

(21)







∂

_t²

z

^k

− ∆z

^k

+ q

^k

z

^k

= g

^k

, in (0, T ) × Ω,

z

^k

= 0, on (0, T ) × ∂Ω,

z

^k

(0) = 0, ∂

t

z

^k

(0) = z

₁^k

, in Ω,

245

where g

^k

= (Q − q

^k

)∂

t

W [Q], z

₁^k

= (Q − q

^k

)w

0

, and we have µ

^k

= ∂

n

z

^k

on (0, T ) ×Γ

0

.

246

In system (21), the source g

^k

and the initial data z

₁^k

are both unknown, and

247

we are actually interested in finding a good approximation of z

₁^k

, which encodes the

248

information on Q − q

^k

. In order to do so, we will try to fit “at best” the flux ∂

n

z with

249

µ

^k

on the boundary, approximating the unknown source term g

^k

by 0.

250

This strategy works as we can prove that the source term g

^k

brings less informa-

251

tion than µ

^k

does, and this is where the choice of the Carleman parameter s will play

252

a crucial role. This is actually the milestone of the construction of Algorithm 1 and

253

its convergence result [BdBE13]. Here, when considering the functional J

_s,q

[η(ϕ)µ]

254

defined in (13), we rather try to approximate ˜ z

^k

= η(ϕ)z

^k

, which enjoys the following

255

properties:

256

• ∂

t

z ˜

^k

(0, ·) = η(ϕ(0))∂

t

z

^k

(0, ·) = (Q−q

^k

)w

0

encodes the information on Q−q

^k

;

257

• z ˜

^k

= η(ϕ)z

^k

vanishes in domain O defined by (12) and on the boundary in

258

time t = T ;

259

• ∂

n

z ˜

^k

= ˜ µ

^k

in (0, T ) × Γ

0

.

260

These ideas are actually behind the proofs of the inverse problem stability by com-

261

pactness uniqueness arguments as in [PY96, PY97, Yam99] or by Carleman estimates

262

given in [IY01a, IY01b, IY03, Baufr].

263

2.2. Sketch of the proof of the Carleman estimate. Since a lot of different

264

references, several of them mentioned right above, present detailed proof of Carleman

265

estimates for the wave equation, we only give here the main calculations yielding the

266

result presented in Theorem 3.

267

Proof. Set y(t, x) = z(t, x)e

^sϕ(t,x)

for all (t, x) ∈ (−T, T ) × Ω, and introduce the

268

conjugate operator L

s

defined by L

s

y = e

^sϕ

(∂

_t²

− ∆)(e

^−sϕ

y). Easy computations

269

give

270 271

(22) L

s

y = ∂

_t²

y − ∆y + s

²

(|∂

t

ϕ|

²

− |∇ϕ|

²

)y

| {z }

=P1y

− 2s∂

t

y∂

t

ϕ + 2s∇y · ∇ϕ + αsy

| {z }

=P₂y 272

− s(∂

_t²

ϕ − ∆ϕ)y − αsy

| {z }

=Ry 273

274

where we have set α = 2d − 2, d being the space dimension. Based on the estimate

275

2 Z

T

−T

Z

Ω

P

1

yP

2

y dxdt ≤ Z

T

−T

Z

Ω

|P

1

y|

²

+ |P

2

y|

²

dxdt + 2 Z

T

−T

Z

Ω

P

1

yP

2

y dxdt

276

≤ 2 Z

T

−T

Z

Ω

| L

s

y|

²

dxdt + 2 Z

T

−T

Z

Ω

|Ry|

²

dxdt, (23)

277

the main part of the proof consists in the computation and bound from below of the cross-term

I = Z

T

−T

Z

Ω

P

1

y P

2

y dxdt.

Tedious computations and integrations by parts yield

278

I = s Z

T

−T

Z

Ω

|∂

t

y|

²

(∂

_t²

ϕ + ∆ϕ − α) dxdt + s Z

T

−T

Z

Ω

|∇y|

²

(∂

_t²

ϕ − ∆ϕ + α + 4) dxdt

279

+ s

³

Z

T

−T

Z

Ω

|y|

²

∂

_t

∂

_t

ϕ(|∂

t

ϕ|

²

− |∇ϕ|

²

)

+ α(|∂

t

ϕ|

²

− |∇ϕ|

²

)

280

−∇ · ∇ϕ(|∂

_t

ϕ|

²

− |∇ϕ|

²

) dxdt

281

− s Z

Ω

|∂

t

y|

²

+ |∇y|

²

∂

t

ϕ dx

T

−T

+ 2s Z

Ω

∂

t

y (∇y · ∇ϕ) dx

T

−T 282

− s

³

Z

Ω

y

²

(|∂

t

ϕ|

²

− |∇ϕ|

²

)∂

t

ϕ dx

^T

−T

+ αs Z

Ω

∂

t

y y dx

^T

−T 283

− s Z

T

−T

Z

∂Ω

|∂

n

y|

²

∂

n

ϕ dσdt.

284

Let us now briefly explain how each term can be estimated.

285 286

• We focus on the terms in s|∂

_t

y|

²

and s|∇y|

²

in order to insure that they are

287

strictly positive. Taking α = 2d − 2, this means

288

∂

_t²

ϕ + ∆ϕ − α = −2β + 2d − α = 2(1 − β) and

289

∂

_t²

ϕ − ∆ϕ + α + 4 = −2β − 2d + α + 4 = 2(1 − β),

290

that are positive thanks to the assumption β ∈ (0, 1).

291 292

• The terms in s

³

|y|

²

can be rewritten as follows (since ∇

²

ϕ = 2Id):

293

∂

t

∂

t

ϕ(|∂

t

ϕ|

²

− |∇ϕ|

²

)

+ α(|∂

t

ϕ|

²

− |∇ϕ|

²

) − ∇ · ∇ϕ(|∂

t

ϕ|

²

− |∇ϕ|

²

)

294

= (∂

_t²

ϕ − ∆ϕ + α)(|∂

t

ϕ|

²

− |∇ϕ|

²

) + 2|∂

t

ϕ|

²

∂

_t²

ϕ + 2∇

²

ϕ · ∇ϕ · ∇ϕ

295

= (−6β − 2d + α)(|∂

t

ϕ|

²

− |∇ϕ|

²

) + 4(1 − β)|∇ϕ|

²

296

= −(2 + 6β)(|∂

t

ϕ|

²

− |∇ϕ|

²

) + 4(1 − β )|∇ϕ|

²

.

297298

This quantity is bounded from below by a strictly positive constant in the region of (−T, T ) × Ω in which

|∂

t

ϕ(t, x)|

²

− |∇ϕ(t, x)|

²

≤ 0 ⇐⇒ βt ≤ |x − x

0

|,

i.e. the complementary of the set

(t, x) ∈ (−T, T ) × Ω with (|t|, x) ∈ O where O

299

satisfies (12).

300 301

• We now estimate the boundary terms in time

¹

appearing at time t = T and

302

t = −T . We focus on the terms at time T , as the ones at time −T can be handled

303

similarly. Let us first collect them:

304 305

I

T

:= 2sβT Z

Ω

|∂

t

y(T )|

²

+ |∇y(T )|

²

dx + 8s

³

βT Z

Ω

|y(T )|

²

(β

²

T

²

− |x − x

0

|

²

) dx

306

+ 4s Z

Ω

∂

t

y(T )

∇y(T ) · (x − x

0

) + α 4 y(T)

dx.

307 308

The first and second terms are obviously positive (under Condition (8) for the second

309

one), so we only need to check that they are sufficiently positive to absorb the last

310

term, whose sign is unknown. We remark that

311

Z

Ω

∇y(T ) · (x − x

0

) + α 4 y(T )

2

dx

312

= Z

Ω

|∇y(T ) · (x − x

0

)|

²

dx + α 4 Z

Ω

(x − x

0

) · ∇ |y(T )|

²

dx + α

²

16

Z

Ω

|y(T )|

²

dx

313

= Z

Ω

|∇y(T ) · (x − x

0

)|

²

dx + α

²

16 − αd 4

Z

Ω

|y(T )|

²

dx

314

≤ sup

Ω

|x − x

₀

|

²

Z

Ω

|∇y(T )|

²

dx,

315

since α = 2d − 2 gives α

²

− 4αd = −4(d − 1)(d + 1) ≤ 0. This inequality allows to

316

deduce, by Cauchy-Schwarz inequality, that

317 318

4s Z

Ω

∂

t

y(T )

∇y(T ) · (x − x

0

) + α 4 y(T )

dx

319

≤ 2s sup

Ω

{|x − x

0

|}

Z

Ω

|∂

t

y(T )|

²

+ |∇y(T )|

²

dx

.

320 321

Using again Condition (8), we easily obtain I

T

≥ 0.

322 323

Gathering these informations, and using the geometric condition (3) on Γ

₀

, it

324

yields that there exists a constant M > 0 independent of s such that

325 326

Z

T

−T

Z

Ω

P

1

yP

2

y dxdt ≥ M s Z

T

−T

Z

Ω

|∂

t

y|

²

+ |∇y|

²

+ s

²

|y|

²

dxdt

327

− M s Z

T

−T

Z

Γ₀

|∂

n

y|

²

dσdt − M s

³

Z Z

(|t|,x)∈O

|y|

²

dxdt.

328 329

1The authors acknowledge Xiaoyu Fu for having pointed out to us the fact that these boundary terms have positive signs.

From (23), we easily derive

330 331

s Z

T

−T

Z

Ω

|∂

t

y|

²

+ |∇y|

²

+ s

²

|y|

²

dxdt +

Z

T

−T

Z

Ω

|P

1

y|

²

+ |P

2

y|

²

dxdt

332

≤ M Z

T

−T

Z

Ω

| L

s

y|

²

dxdt + M s

²

Z

T

−T

Z

Ω

|y|

²

dxdt

333

+ M s Z

T

−T

Z

Γ0

|∂

n

y|

²

dσdt + M s

³

Z Z

(|t|,x)∈O

|y|

²

dxdt.

334 335

We take now s

₀

large enough in order to make sure that the term in s

²

|y|

²

of the right

336

hand side is absorbed by the dominant term in s

³

|y|

²

of the left hand side as soon as

337

s ≥ s

0

and we obtain

338 339

(24) s Z

T

−T

Z

Ω

|∂

t

y|

²

+ |∇y|

²

+ s

²

|y|

²

dxdt +

Z

T

−T

Z

Ω

|P

1

y|

²

+ |P

2

y|

²

dxdt

340

≤ M Z

T

−T

Z

Ω

| L

s

y|

²

dxdt + M s Z

T

−T

Z

Γ0

|∂

n

y|

²

dσdt + M s

³

Z Z

(|t|,x)∈O

|y|

²

dxdt

341 342

We then deduce (18) by substituting y = ze

^sϕ

.

343 344

Furthermore, under the additional condition z(0, ·) = 0 in Ω, we get y(0, ·) = 0

345

in Ω. We then choose ρ : t 7→ ρ(t) a smooth function such that ρ(0) = 1 and ρ

346

vanishes close to t = −T. We multiply P

1

y by ρ∂

t

y and integrate over (−T, 0) × Ω,

347

to get

348

Z

0

−T

Z

Ω

P

1

y ρ∂

t

y dxdt = Z

0

−T

Z

Ω

∂

_t²

y − ∆y + s

²

((∂

t

ϕ)

²

− |∇ϕ|

²

)y

ρ∂

t

y dxdt

349

= 1 2

Z

0

−T

Z

Ω

ρ∂

t

|∂

t

y|

²

+ |∇y|

²

dxdt + s

²

2

Z

0

−T

Z

Ω

ρ(|∂

t

ϕ|

²

− |∇ϕ|

²

)∂

t

(y

²

) dxdt

350

= 1 2 Z

Ω

|∂

_t

y(0)|

²

dx − 1 2

Z

0

−T

Z

Ω

∂

_t

ρ |∂

_t

y|

²

+ |∇y|

²

+ s

²

∂

_t

ρ(|∂

_t

ϕ|

²

− |∇ϕ|

²

) y

²

dxdt

351

≥ 1 2 Z

Ω

|∂

t

y(0)|

²

dx − M Z

0

−T

Z

Ω

|∂

t

y|

²

+ |∇y|

²

+ s

²

|y|

²

dxdt.

352

By Cauchy-Schwarz inequality, this implies s

^1/2

Z

Ω

|∂

t

y(0)|

²

dx ≤ Z

T

−T

Z

Ω

|P

1

y|

²

dxdt + M s Z

T

−T

Z

Ω

|∂

t

y|

²

+ |∇y|

²

+ s

²

|y|

²

dxdt.

Using (24) and y = ze

^sϕ

, we easily deduce the estimate of term (19) and conclude the

353

proof of Theorem 3.

354

From this proof of Theorem 3, we can directly exhibit (see (24)) the following “con-

355

jugate” Carleman estimate, of practical interest later on:

356

Corollary 4. Assume the multiplier condition (3)-(4) and β ∈ (0, 1) as in (8).

357

Define the weight function ϕ as in (9). Then there exist constants M > 0 and s

0

> 0

358

such that for all s ≥ s

0

,

359 360

(25) s Z

T

−T

Z

Ω

|∂

_t

y|

²

+ |∇y|

²

+ s

²

|y|

²

dxdt +

Z

T

−T

Z

Ω

|P

₁

y|

²

+ |P

₂

y|

²

dxdt

361

≤ M Z

T

−T

Z

Ω

| L

s

y|

²

dxdt + M s Z

T

−T

Z

Γ0

|∂

n

y|

²

dσdt + M s

³

Z Z

(|t|,x)∈O

|y|

²

dxdt

362 363

for all y ∈ C

⁰

([−T, T ]; H

₀¹

(Ω)) ∩ C

¹

([−T, T ]; L

²

(Ω)), with L

s

y ∈ L

²

((−T, T ) × Ω),

364

where L

s

, P

₁

and P

₂

are defined in (22).

365

Furthermore, if y(0, ·) = 0 in Ω, one can add the term s

^1/2

Z

Ω

|∂

t

y(0)|

²

dx to the left

366

hand-side of (25).

367

2.3. Proof of the convergence theorem.

368

Proof of Theorem 2. Let us first begin by showing that Algorithm 2 is well-

369

defined. We introduce

370 371

T

_q

= n

z ∈ C

⁰

([0, T ]; H

₀¹

(Ω)) ∩ C

¹

([0, T ]; L

²

(Ω)),

372

with ∂

_t²

z − ∆z + qz ∈ L

²

((0, T ) × Ω) and z(0, ·) = 0 in Ω o ,

373374

endowed with the norm

375 376

kzk

²_obs,s,q

= Z

T

0

Z

Ω

e

^2sϕ

|∂

_t²

z − ∆z + qz|

²

dxdt + s Z

T

0

Z

Γ₀

e

^2sϕ

|∂

n

z|

²

dσdt

377

+ s

³

Z Z

O

e

^2sϕ

|z|

²

dxdt.

378 379

The proof that this quantity is a norm on T

q

stems from the Carleman estimate of

380

Theorem 3 applied to z

e

(t, x) = z(t, x) for t ∈ [0, T ] and z

e

(t, x) = −z(−t, x) for

381

t ∈ [−T, 0], x ∈ Ω. Indeed, (18) applied to z

e

yields for all s ≥ s

0

,

382

s

³

Z

T

0

Z

Ω

e

^2sϕ

|z|

²

dxdt ≤ 2M Z

T

0

Z

Ω

e

^2sϕ

|∂

_t²

z − ∆z + qz|

²

dxdt

383

+ 2M kqk

²_L∞(Ω)

Z

T 0

Z

Ω

e

^2sϕ

|z|

²

dxdt

384

+ M s Z

T

0

Z

Γ0

e

^2sϕ

|∂

n

z|

²

dσdt + M s

³

Z Z

O

e

^2sϕ

|z|

²

dxdt,

385

so that k · k

_obs,s,q

is a norm on T

_q

provided s is large enough, and then for all s > 0 as

386

the weight functions are bounded on [0, T ] ×Ω. This immediately implies that J

s,q

[˜ µ]

387

defined in (13) is coercive and strictly convex on the set T

q

, so that it admits a unique

388

minimizer and as a consequence, Algorithm 2 is well-defined.

389

Moreover, this shows that the class T

q

, which was a priori dependent of q, is in

390

fact independent of q (for q ∈ L

^∞

(Ω)) and is simply given by

391 392

T = n

z ∈ C

⁰

([0, T ]; H

₀¹

(Ω)) ∩ C

¹

([0, T ]; L

²

(Ω)),

393

with ∂

_t²

z − ∆z ∈ L

²

((0, T ) × Ω) and z(0, ·) = 0 in Ω o .

394395

In order to show estimate (17), instead of considering only functionals of the

396

form J

s,q

[˜ µ], we introduce slightly more general functionals J

s,q

[˜ µ, g] given for s > 0,

397

q ∈ L

^∞

(Ω), ˜ µ ∈ L

²

((0, T ) × Γ

0

), g ∈ L

²

((0, T ) × Ω) and for all z ∈ T , by:

398 399

(26) J

_s,q

[˜ µ, g](z) = 1 2

Z

T 0

Z

Ω

e

^2sϕ

|∂

_t²

z − ∆z + qz − g|

²

dxdt

400

+ s 2

Z

T 0

Z

Γ₀

e

^2sϕ

|∂

n

z − µ| ˜

²

dσdt + s

³

2

Z Z

O

e

^2sϕ

|z|

²

dxdt.

401 402

With the same argument as above, the functional J

_s,q

[˜ µ, g] is coercive in the norm

403

k·k

_obs,s,q

and strictly convex, so that it admits a unique minimizer for each ˜ µ ∈

404

L

²

((0, T ) × Γ

₀

) and g ∈ L

²

((0, T ) × Ω).

405

We then observe that ˜ z

^k

:= η(ϕ)z

^k

, where z

^k

satisfies (20) (recall the definitions

406

of η in (14) and ϕ in (9), pictured in Figure 2), is the minimizer of J

_s,qk

[˜ µ

^k

, ˜ g

^k

] with

407

(27) ˜ g

^k

= η(ϕ)(Q − q

^k

)∂

t

W [Q] + [∂

_t²

− ∆, η(ϕ)]z

^k

,

408

since it solves:

409

(28)







∂

_t²

z ˜

^k

− ∆˜ z

^k

+ q

^k

z ˜

^k

= ˜ g

^k

, in (0, T ) × Ω,

˜

z

^k

= 0, on (0, T ) × ∂Ω,

˜

z

^k

(0) = 0, ∂

t

˜ z

^k

(0) = η(ϕ(0, ·))z

^k₁

, in Ω,

410

and ∂

n

z ˜

^k

= ˜ µ

^k

= η(ϕ)∂

t

∂

n

w[q

^k

] − ∂

n

W [Q]

on (0, T ) × Γ

0

.

411

We shall then compare Z e

^k

and ˜ z

^k

, the minimizers of the functionals J

_s,qk

[˜ µ

^k

, 0]

412

and J

_s,qk

[˜ µ

^k

, ˜ g

^k

] respectively, especially at the time t = 0 corresponding to the set in

413

which the information on (Q − q

^k

) is encoded. The result is stated as follows:

414

Proposition 5. Assume the geometric and time conditions (3)-(4) on Γ

0

and T ,

415

that β is chosen as in (8), and let µ ∈ L

²

((0, T ) × Γ

0

) and g

^a

, g

^b

∈ L

²

((0, T ) × Ω).

416

Assume also that q belongs to L

^∞_≤m

(Ω) for m > 0.

417

Let Z

^j

be the unique minimizer of the functional J

_s,q

[µ, g

^j

] on T for j ∈ {a, b}. Then

418

there exist positive constants s

₀

(m) and M = M (m) such that for s ≥ s

₀

(m) we have:

419

(29) s

^1/2

Z

Ω

e

^2sϕ(0)

|∂

_t

Z

^a

(0) − ∂

_t

Z

^b

(0)|

²

dx ≤ M Z

T

0

Z

Ω

e

^2sϕ

|g

^a

− g

^b

|

²

dxdt.

420

where ϕ and s

₀

(m) are chosen so that Theorem 3 holds.

421

We postpone the proof of Proposition 5 to the end of the section and first show how

422

it can be used for the proof of Theorem 2.

423 424

Recall now that ∂

_t

z ˜

^k

(0, ·) = (Q − q

^k

)w

⁰

. Setting ˜ q

^k+1

as in (16), we get from

425

Proposition 5 applied to Z

^a

= Z e

^k

and Z

^b

= ˜ z

^k

that

426

(30) s

^1/2

Z

Ω

e

^2sϕ(0)

|˜ q

^k+1

− Q|

²

|w

⁰

|

²

dx ≤ M Z

T

0

Z

Ω

e

^2sϕ

|˜ g

^k

|

²

dxdt.

427

The next step is to get an estimates of ˜ g

^k

defined by (27). Using the fact that

428

η(ϕ), ∂

_t²

− ∆

z

^k

has support in a region where ϕ ≤ d

²₀

:= d(x

0

, Ω)

²

, we obtain

429

Z

T 0

Z

Ω

e

^2sϕ

|˜ g

^k

|

²

dxdt ≤ M Z

T

0

Z

Ω

e

^2sϕ

|η(ϕ)(Q − q

^k

)∂

t

W [Q]|

²

dxdt

430

+ M

Z

T 0

Z

Ω

e

^2sϕ

|

η(ϕ), ∂

_t²

− ∆

z

^k

|

²

dxdt

431

≤ M kW [Q]k

²_H1(0,T;L^∞(Ω))

Z

Ω

e

^2sϕ(0)

|q

^k

− Q|

²

dx

432

+ M e

^2sd20

Z

T

0

Z

Ω

|∇z

^k

|

²

+ |∂

_t

z

^k

|

²

+ |z

^k

|

²

dxdt.

433

Usual a priori energy estimates for z

^k

solution of equation (21) also yields

434 435

(31) kz

^k

k

_L∞(0,T;H¹₀(Ω))

+ k∂

_t

z

^k

k

_L∞(0,T;L²(Ω))

≤ M kz

₁^k

k

_L2(Ω)

+ kg

^k

k

_L1(0,T;L²(Ω)) 436

≤ M kQ − q

^k

k

_L2(Ω)

kw

₀

k

_L∞(Ω)

+ k∂

_t

W [Q]k

_L1(0,T;L^∞(Ω))

≤ M kW [Q]k

_H1(0,T;L^∞(Ω))

kQ − q

^k

k

L²(Ω)

,

437 438

so that combining the above estimates, we get

439 440

s

^1/2

Z

Ω

e

^2sϕ(0)

|˜ q

^k+1

− Q|

²

|w

⁰

|

²

dx ≤ M kW [Q]k

²_H1(0,T;L^∞(Ω))

Z

Ω

e

^2sϕ(0)

|q

^k

− Q|

²

dx

441

+ M kW [Q]k

²_H1(0,T;L^∞(Ω))

e

^2sd20

kQ − q

^k

k

²_L2(Ω)

.

442443

Using ϕ(0, x) ≥ d

²₀

for all x in Ω and Assumption (6), we deduce

444

(32) s

^1/2

α

²

Z

Ω

e

^2sϕ(0)

|˜ q

^k+1

−Q|

²

dx ≤ M kW [Q]k

²_H1(0,T;L^∞(Ω))

Z

Ω

e

^2sϕ(0)

|q

^k

−Q|

²

dx.

445

Now, using the a priori assumption (7), i.e. Q ∈ L

^∞_≤m

(Ω), we easily check that this

446

estimate cannot deteriorate in step 4 of Algorithm 2, which is there only to ensure

447

that the sequence q

^k

stays in L

^∞_≤m

(Ω) for all k ∈ N . This completes the proof of

448

Theorem 2.

449

It only remains to prove the former proposition.

450

Proof of Proposition 5. Let us write the Euler Lagrange equations satisfied by

451

Z

^j

, for j ∈ {a, b}. For all z ∈ T , we have

452 453

(33) Z

T

0

Z

Ω

e

^2sϕ

(∂

_t²

Z

^j

− ∆Z

^j

+ qZ

^j

− g

^j

)(∂

_t²

z − ∆z + qz) dxdt

454

+ s Z

T

0

Z

Γ₀

e

^2sϕ

(∂

n

Z

^j

− µ)∂

n

z dσdt + s

³

Z Z

O

e

^2sϕ

Z

^j

z dxdt = 0.

455 456

Applying (33) for j = a and j = b to z = Z = Z

^a

− Z

^b

and subtracting the two

457

identities, we obtain:

458 459

Z

T 0

Z

Ω

e

^2sϕ

|∂

²_t

Z −∆Z +qZ |

²

dxdt+s Z

T

0

Z

Γ₀

e

^2sϕ

|∂

_n

Z |

²

dσdt+s

³

Z Z

O

e

^2sϕ

|Z |

²

dxdt

460

= Z

T

0

Z

Ω

e

^2sϕ

(g

^a

− g

^b

)(∂

_t²

Z − ∆Z + qZ ) dxdt.

461 462

This implies

463 464

(34) 1 2

Z

T 0

Z

Ω

e

^2sϕ

|∂

_t²

Z − ∆Z + qZ |

²

dxdt + s Z

T

0

Z

Γ0

e

^2sϕ

|∂

n

Z |

²

dσdt

465

+ s

³

Z Z

O

e

^2sϕ

|Z|

²

dxdt ≤ 1 2

Z

T 0

Z

Ω

e

^2sϕ

|g

^a

− g

^b

|

²

dxdt.

466 467

Since the left hand side of (34) is precisely the right hand side of the Carleman

468

estimate (18), applying Theorem 3 to Z , we immediately deduce (29).

469

3. Technical issues on the minimization of the cost functional. The goal

470

of this section is to give several details about the actual construction of an efficient

471

numerical algorithm based on Algorithm 2. The main step in Algorithm 2 is to

472

minimize the functional J

s,q

[˜ µ], that we recall here for convenience,

473

J

s,q

[˜ µ](z) = 1 2

Z

T 0

Z

Ω

e

^2sϕ

|∂

²_t

z − ∆z + qz|

²

dxdt + s 2

Z

T 0

Z

Γ₀

e

^2sϕ

|∂

n

z − µ| ˜

²

dσdt

474

+ s

³

2

Z Z

O

e

^2sϕ

|z|

²

dxdt,

475

and which is minimized on the set

476 477

(35) T = n

z ∈ C

⁰

([0, T ]; H

₀¹

(Ω)) ∩ C

¹

([0, T ]; L

²

(Ω)),

478

with ∂

_t²

z − ∆z ∈ L

²

((0, T ) × Ω) and z(0) = 0 in Ω o

479

.

480

Due to the presence of large exponential factors in the functional, the minimization

481

of J

_s,q

[˜ µ] is not a straightforward task from the numerical point of view, even if, as

482

we emphasized earlier, the minimization of J

_s,q

[˜ µ] is much less stiffer than the one of

483

K

s,q

[µ] defined in (10) [BdBE13]. We therefore propose the two following ideas:

484

• Work on the conjugate variable y = ze

^sϕ

. This change of unknown acts as a

485

preconditioner. Details are given in Section 3.1.

486

• A progressive algorithm to minimize the functional J

s,q

[˜ µ] in subdomains in

487

which the variations of the exponential factors are small, see Section 3.2.

488

3.1. Conjugate variable. For z in T , we set y = ze

^sϕ

, so that y satisfies the following equation:



 



 



∂

_t²

y − ∆y + qy − 2s∂

t

ϕ∂

t

y + 2s∇ϕ · ∇y

−s(∂

_t²

ϕ − ∆ϕ)y + s

²

(|∂

t

ϕ|

²

− |∇ϕ|

²

)y = e

^sϕ

(∂

_t²

− ∆ + q)z, in (0, T ) × Ω,

y = 0, on (0, T ) × ∂Ω,

y(0) = 0, ∂

t

y(0) = z

1

e

^sϕ(0)

, in Ω,

where ∂

_t

ϕ = −2βt, ∇ϕ = 2(x − x

₀

), ∂

_t²

ϕ = −2β and ∆ϕ = 2d. We set L

s,q

defined

489

by L

s,q

= e

^sϕ

(∂

_t²

− ∆ + q)e

^−sϕ

:

490

L

s,q

y = ∂

²_t

y − ∆y + qy − 2s∂

t

ϕ∂

t

y + 2s∇ϕ · ∇y − s(∂

_t²

ϕ − ∆ϕ)y

491

+s

²

(|∂

t

ϕ|

²

− |∇ϕ|

²

)y

492

= ∂

²_t

y − ∆y + qy + 4sβt∂

_t

y + 4s(x − x

₀

) · ∇y + 2s(β + d)y

493

+4s

²

(β

²

t

²

− |x − x

0

|

²

)y.

(36)

494