Eric Marcon
email: eric.marcon@ecofog.gf

Practical Estimation of Diversity from Abundance Data

Keywords: Biodiversity, HCDT entropy, Phylodiversity

Measuring biodiversity requires empirical techniques to effectively estimate it from real data. The well-known underestimation of the number of species applies to low orders of diversity in general. I test nine estimators including three new ones on geometric and lognormal distributions that represent realistic, hyper-diverse communities. The best two estimators allow a good estimation of diversity of orders over 0.5, even when the sampling effort is low. I provide criteria to choose the estimator and the necessary code in the R package entropart.

Introduction

Measuring biodiversity requires both a robust theoretical framework [START_REF] Patil | Diversity as a concept and its measurement[END_REF] and empirical techniques to effectively estimate the theoretical variables with real data [START_REF] Beck | Comparing measures of species diversity from incomplete inventories: an update[END_REF]. In this paper I focus on species-neutral measures of diversity based on HCDT entropy [START_REF] Havrda | Quantification method of classification processes. Concept of structural aentropy[END_REF][START_REF] Daróczy | Generalized information functions[END_REF][START_REF] Tsallis | Possible generalization of Boltzmann-Gibbs statistics[END_REF] that fulfill the first requirement. Entropy measures the average surprise brought by observing individuals of a community. Surprise is a decreasing function of probability dropping to 0 when probability is 1. HCDT entropy uses a parameterized surprise function that is the deformed logarithm of order q of the reciprocal of probability (Marcon et al., 2014a). Traditional measures of diversity, namely the number of species as well as Shannon's and Simpson's indices, are special cases of the HCDT entropy for values of q equal to 0, 1 and 2. HCDT entropy should be transformed into Hill numbers [START_REF] Hill | Diversity and Evenness: A Unifying Notation and Its Consequences[END_REF] for better interpretation of the value of diversity as an effective number of species [START_REF] Jost | Entropy and diversity[END_REF]. Hill numbers are simply the deformed exponential of HCDT entropy (Marcon et al., 2014a). Rather than focusing on a single value of q, a profile of diversity, i.e. a plot of diversity against q, can be built [START_REF] Tothmeresz | Comparison of different methods for diversity ordering[END_REF]. Low values of q (starting from 0) give much importance to rare species, whilst higher values (usually up to 2) focus on abundant species. Negative values of q are not used because of poor mathematical properties of their entropy [START_REF] Beck | Generalised Information and Entropy Measures in Physics[END_REF], and values over 2 generally bring little more information. Ordering communities in terms of diversity requires that their profile do not cross [START_REF] Tothmeresz | Comparison of different methods for diversity ordering[END_REF]; else, declaring a community more diverse than another only holds for a range of values of q reflecting the importance given to rare or frequent species [START_REF] Lande | When species accumulation curves intersect: implications for ranking diversity using small samples[END_REF].

To plot those profiles, diversity must be estimated from the data. Estimation bias (I follow the terminology of [START_REF] Dauby | Sampled-based estimation of diversity sensu stricto by transforming Hurlbert diversities into effective number of species[END_REF]) is a well-known issue (Marcon et al., 2014a). Real data are almost always samples of larger communities, so some species may have been missed. The induced bias on the Simpson entropy is smaller than on the Shannon entropy because the former assigns lower weights to rare species, i.e. the sampling bias is even more important when q decreases. Another estimation bias has been widely studied by physicists who generally consider that all species of a given community are known and their probabilities quantified. Their main issue is not at all missing species but the non-linearity of entropy measures (see [START_REF] Bonachela | Entropy estimates of small data sets[END_REF], for a short review). Estimating probabilities at power q > 0 by the power of their estimator is an important source of underestimation of entropy. The need for corrections has generated a considerable literature in ecological statistics and statistical physics.

In this paper, I test the performance of the stateof-the-art estimators when applied to the kind of data ecologists have to deal with. I start with simulated distributions that have the advantage of being easily manipulated to generate various sampling intensities and evaluate the bias and root mean square error (RMSE) of the estimators. I address the classical models of the literature, namely the lognormal and the geometric distributions. The lognormal distribution describes, at least roughly, many hyperdiverse ecosystems even though the link between its statistical success and the underlying ecological mechanisms is poorly documented [START_REF] Tokeshi | Species Abundance Patterns and Community Structure[END_REF]. The geometric distribution is a far more difficult case because it is very uneven: the frequency of rare species is several orders of magnitude smaller than that of the frequent ones, making it impossible to observe with reasonable sampling effort [START_REF] Haegeman | Robust estimation of microbial diversity in theory and in practice[END_REF]. I apply the best-known and best-performing estimators, including three new ones, to those distributions and two actual forest data sets. My purpose is to provide recommendations about the estimation technique to chose when facing different types of data and draw general conclusions about the possible accuracy of diversity estimation.

Phyloentropy is the sum of HCDT entropy along an ultrametric tree (Marcon and Hérault, 2015a) so estimating it reduces to estimating HCDT entropy. Phylodiversity is the deformed exponential of phyloentropy. In short, estimating phylodiversity relies on the methods presented here so I will focus on species-neutral diversity for clarity. I used the package entropart (Marcon and Hérault, 2015b) for R (R Development Core Team, 2015) for all tests. The R code necessary to reproduce all results is in the electronic appendix.

Methods

Consider a community of species indexed by s = 1, 2, . . . , S. n s is the number of individuals of species s sampled in the community, n = ∑ s n s the total number of sampled individuals. The (unknown) probability p s for an individual to belong to species s is estimated by ps = n s /n.

The number of species represented by ν individuals in the sample of size n is s n ν , so s n 0 if the (unknown) number of unobserved species considering the sampling effort. s n ν is considered as a realization of the random variable S n ν so it is used to estimate its expectation E(S n ν ). π ν is the sum of the probabilities p s of species represented by ν individuals.

The deformed logarithm formalism [START_REF] Tsallis | What are the numbers that experiments provide?[END_REF] is very convenient to manipulate entropies. The deformed logarithm of order q is defined as:

ln q x = x 1-q -1 1 -q (1)
It converges to the natural logarithm when q → 1.

The inverse function of ln q x is the deformed exponential:

e x q = [1 + (1 -q)x] 1 1-q (2)

Sample coverage

The sample coverage [START_REF] Good | On the Population Frequency of Species and the Estimation of Population Parameters[END_REF] is the probability for an individual in the community to belong to a species observed in the sample. It equals the sums of the probabilities of the observed species. It is an essential tool for diversity estimation because it is included in some estimators (e.g. [START_REF] Chao | Nonparametric estimation of Shannon's index of diversity when there are unseen species in sample[END_REF] and it allows the evaluation of the completeness of sampling [START_REF] Chao | Coverage-based rarefaction and extrapolation: standardizing samples by completeness rather than size[END_REF]. Its estimator given by Good is:

Ĉ = 1 - s n 1 n (3)
It is biased [START_REF] Zhang | Turing's formula revisited[END_REF], because:

C = 1 - E(S n 1 ) -π 1 n (4)
Good's estimator neglects the term π 1 , the sum of the probabilities of singletons. It was built from Turing's frequency formula relating the average probability of species observed ν times to the number of species observed ν + 1 and ν times. This formula has been improved by [START_REF] Chao | Program SPADE: Species Prediction And Diversity Estimation. Program and user's guide[END_REF][START_REF] Chiu | An Improved Nonparametric Lower Bound of Species Richness via a Modified Good-Turing Frequency Formula[END_REF] to estimate π 1 . Estimating the number of species by the Chao1 estimator [START_REF] Chao | Nonparametric estimation of the number of classes in a population[END_REF], [START_REF] Chao | Program SPADE: Species Prediction And Diversity Estimation. Program and user's guide[END_REF] obtained an improved estimator of the sample coverage:

Ĉ = 1 - s n 1 n (n -1)s n 1 (n -1)s n 1 + 2s n 2 (5)
This estimator has been further used by [START_REF] Chao | Estimating diversity and entropy profiles via discovery rates of new species[END_REF] to derive an estimator of entropy (see below).

An almost unbiased estimator has been derived using the information provided by the whole distribution [START_REF] Chao | A generalized Good's nonparametric coverage estimator[END_REF][START_REF] Zhang | Turing's formula revisited[END_REF]:

Ĉ = 1 - n ∑ ν=1 (-1) ν+1 n ν -1 s n ν (6)
I use it in this paper.

Estimators of entropy

The existing estimators and the new ones proposed here can be classified into four main methods. The simplest one just consists of plugging the estimator of p s , i.e. ps = n s /n, into the definition of diversity to evaluate to obtain a so-called plug-in estimator, sometimes named naive estimator. The plug-in estimator of HCDT entropy of order q is:

q Ĥ = ∑ s ps ln q 1 p s (7)
The plug-in estimator is useless in hyper-diverse communities because it severely underestimates diversity because of unobserved species and of the non-linearity of estimators.

Recent progress have been made in estimating the actual distribution of the probability of species by fitting a model of their distribution to the data. The distribution of the unobserved species can be added if their number is estimated and a distribution form is chosen. Chao et al. (2015) used a two-parameter model based on the estimation of the generalized sample coverage (not detailed here), estimated the total richness with the Chao1 estimator and modeled the unobserved species as a geometric distribution to unveil the complete rankabundance distribution of an observed community. They applied the plug-in estimator this distribution: I'll call it the "Chao-unveiled" estimator.

The Chao1 estimator has been built according to the same theoretical approach as that of the unveiled rankabundance distribution. It is a lower-bound estimator of the number of species. It has been improved by [START_REF] Chiu | An Improved Nonparametric Lower Bound of Species Richness via a Modified Good-Turing Frequency Formula[END_REF] who slightly reduced its negative bias with the iChao1 estimator, integrating species represented by 3 and 4 individuals. I define the "iChao-unveiled" estimator as a variation on the "Chao-unveiled" estimator, where richness is estimated by the iChao1 estimator.

The jackknife estimator [START_REF] Burnham | Robust Estimation of Population Size When Capture Probabilities Vary Among Animals[END_REF]) has shown good performances to estimate richness when the sampling effort is too low for the Chao1 estimator to perform well [START_REF] Brose | Estimating species richness: Sensitivity to sample coverage and insensitivity to spatial patterns[END_REF] even though it actually lacks theoretical support [START_REF] Cormack | Log-Linear Models for Capture-Recapture[END_REF]. I test the use of the jackknife estimator, whose order is selected according to the data, to define the "jackknife-unveiled" estimator. Using the jackknife estimator to unveil the tail of the abundance distribution was not the intention of Chao et al. (2015) because it is not consistent with their theoretical framework. It must be seen here as a merely empirical tool.

The second method relies on the [START_REF] Horvitz | A generalization of sampling without replacement from a finite universe[END_REF] estimator of the weighted sum of a function of its elements, say ∑ s p s f (s) when some of them are not observed. An unbiased estimator of the sum is obtained when each term is divided by its probability to be observed 1 -(1p s ) n . [START_REF] Chao | Nonparametric estimation of Shannon's index of diversity when there are unseen species in sample[END_REF] proposed to combine it with the estimator of the sample coverage: conditionally to the set of observed species, an unbiased estimator [START_REF] Ashbridge | Coverage-adjusted estimators for mark-recapture in heterogeneous populations[END_REF] of p s is ps = Ĉ ps . Chao and Shen estimated the Shannon entropy; the method has then been extended to HCDT entropy (Marcon et al., 2014a) and similarity-based diversity [START_REF] Marcon | The Decomposition of Similarity-Based Diversity and its Bias Correction[END_REF]:

q H = ∑ s=1 Ĉ ps ln q 1 Ĉ ps 1 -1 -Ĉ ps n (8)
A further progress can be done by replacing the conditional estimator of probabilities ps = Ĉ ps by that of Chao et al. (2015). Since the improved probability estimator depends on the generalized sample coverage, I'll call the improved Chao-Shen estimator the "generalized coverage" estimator.

The third method has been derived by [START_REF] Grassberger | Finite sample corrections to entropy and dimension estimates[END_REF] who gave a reduced-bias estimator of the value of an integer at power q. p q s is written as n q s /n q and n q s is estimated (Marcon et al., 2014a) as:

n q s = Γ (n s + 1) Γ (n s -q + 1) + (-1) n Γ (1 + q) sin πq π (n + 1) (9)
The estimator of p q s is simply p q s = n q s /n q . It is plugged into the formula of entropy to obtain the Grassberger estimator:

q H = 1 -∑ s p q s q -1 (10)
The last method has been the subject of an important literature in the last ten years. A review can be found in [START_REF] Chao | Entropy and the species accumulation curve: a novel entropy estimator via discovery rates of new species[END_REF], Appendix A. It relies on the estimation of h q = ∑ s p q s . h q can be written as the following sum:

h q = ∞ ∑ r=0 q -1 r (-1) r ζ r ( 11
)
ζ r is the generalized Simpson entropy ∑ s p s (1p s ) r defined by [START_REF] Zhang | Re-parameterization of multinomial distributions and diversity indices[END_REF]. The first n elements of the sum, denoted hq , can be estimated with no bias [START_REF] Zhang | Entropic Representation and Estimation of Diversity Indices[END_REF]:

hq = S ∑ s=1 ps n-n s ∑ v=1 v ∏ i=1 i -q i v ∏ j=1 1 - n s -1 n -j (12)
Zhang (2013) shows that the bias due to ignoring the remaining terms is asymptotically normal and decays exponentially fast. I'll call the [START_REF] Zhang | Entropic Representation and Estimation of Diversity Indices[END_REF] estimator the one based on hq :

q H = 1 -hq q -1 (13)
Some attempts have been made to estimate the remaining bias [START_REF] Zhang | Bias adjustment for a nonparametric entropy estimator[END_REF]. The most achieved one is that of [START_REF] Chao | Estimating diversity and entropy profiles via discovery rates of new species[END_REF], completing [START_REF] Chao | Entropy and the species accumulation curve: a novel entropy estimator via discovery rates of new species[END_REF]. It relies on two assumptions: the total number of species is estimated by the Chao1 estimator and the actual probabilities of unobserved species can be estimated all equal. A consequence is that the estimator of the average probability of species sampled once also equals the probability estimator of unobserved species.

Its value is noted

A. It is 2s n 2 /[(n -1)s n 1 + 2s n 2 ] if single- tons and doubletons are present or 2/[(n -1) (s n 1 -1) + 2]
if doubletons are missing. The Chao-Wang-Jost estimator of HCDT entropy is:

q H = 1 q -1 [1 -hq - s n 1 n (1 -A) 1-n A q-1 - n-1 ∑ r=0 q -1 r (A -1) r ] (14)
In absence of singletons and doubletons, A is set to 1 and the estimator is identical to that of Zhang and Grabchak.

Confidence intervals

Two methods allow the evaluation of confidence intervals: asymptotic, closed forms are available for some estimators, or bootstrapping is required in the general case. [START_REF] Esty | A Normal Limit Law for a Nonparametric Estimator of the Coverage of a Random Sample[END_REF], completed by [START_REF] Zhang | Turing's formula revisited[END_REF], showed that the estimator of sample coverage (eq. 6) is asymptotically normal with the following confidence interval:

C = Ĉ ± t n 1-α/2 s n 1 1 - s n 1 n + 2s n 2 n ( 15
)
Where t n 1-α/2 is the quantile of a Student distribution with n degrees of freedom at the risk threshold α, here 1.96 for all sample sizes and α = 5%.

The Zhang-Grabchak estimator is also asymptotically normal and comes with an asymptotic confidence interval [START_REF] Zhang | Entropic Representation and Estimation of Diversity Indices[END_REF] implemented in the package EntropyEstimation [START_REF] Cao | EntropyEstimation: Estimation of Entropy and Related Quantities[END_REF].

The theoretical distribution of other estimators is unknown. They must be built by bootstrap techniques: the observed community is re-sampled, say 1000 times, and entropy is calculated each time. The α/2 and 1α/2 quantiles of the distribution of entropy are the bounds of the confidence interval. The issue of re-sampling a community is the same as that of sampling it: rare species are often eliminated, so the entropy is underestimated. Starting from the whole community, a first estimation bias is caused by sampling it. The estimators presented here aim at correcting it. When this observed community is re-sampled, a second estimation bias appears. Estimating the entropy of re-sampled communities with bias correction yields, on average, the entropy of the observed community estimated by the plug-in estimator [START_REF] Marcon | The Decomposition of Shannon's Entropy and a Confidence Interval for Beta Diversity[END_REF]: if the estimator works well, it eliminates the second estimation bias but it cannot address the first one. The solution to this problem is simply to recenter the entropy distribution of re-sampled communities around the value of the entropy of the observed community [START_REF] Marcon | The Decomposition of Shannon's Entropy and a Confidence Interval for Beta Diversity[END_REF][START_REF] Chao | Estimating diversity and entropy profiles via discovery rates of new species[END_REF].

The re-sampling technique may just consist of drawing individuals in the observed community with replacement, or, equivalently, drawing a community in a multinomial distribution respecting the size and probability distribution of the observed community (Marcon et al., 2014a). A more sophisticated technique has been proposed by [START_REF] Chao | Estimating diversity and entropy profiles via discovery rates of new species[END_REF]. Given the sample size, the probability distribution of observed species can be estimated more accurately than by the estimator ps = Ĉ ps which underestimates the probability of rare species (Chao et al., 2015). A better estimate of the probabilities is used (actually, a simplified version of that of the unveiled estimators above) and completed by an estimation of the number of unobserved species, whose probabilities are assumed identical. Despite these extra efforts, the distribution of the entropy of re-sampled community still has to be recentered.

From entropy to diversity

All entropy estimations are finally transformed into diversity values to be interpretable [START_REF] Jost | Entropy and diversity[END_REF]. It is not correct to recenter the confidence interval of diversity estimations because of the non-linearity of the transformation of entropy into diversity [START_REF] Marcon | The Decomposition of Shannon's Entropy and a Confidence Interval for Beta Diversity[END_REF]. The correct process consists of evaluating entropy with its confidence interval and make the final exponential transformation of all values into diversity.

Typical distributions

Comparing the performance of estimators requires simulations of realistic communities. I chose to focus on two opposed models making sense in ecology. The lognormal distribution [START_REF] Preston | The Commonness, And Rarity, of Species[END_REF] fits well species-rich communities for several reasons, including populations dynamics [START_REF] Engen | Population dynamic models generating the lognormal species abundance distribution[END_REF], niche apportionment [START_REF] Bulmer | On Fitting the Poisson Lognormal Distribution to Species-Abundance Data[END_REF], or even statistical physics arguments [START_REF] Pueyo | The maximum entropy formalism and the idiosyncratic theory of biodiversity[END_REF][START_REF] Dewar | Statistical mechanics unifies different ecological patterns[END_REF]. It is often well fitted empirically [START_REF] Tokeshi | Niche Apportionment or Random Assortment: Species Abundance Patterns Revisited[END_REF] even though it has been questioned theoretically [START_REF] Williamson | The lognormal distribution is not an appropriate null hypothesis for the species-abundance distribution[END_REF]. The local community distribution according to the neutral theory [START_REF] Volkov | Neutral theory and relative species abundance in ecology[END_REF] is not lognormal but departs from it very moderately. The logarithm of the species probabilities follows a Gaussian distribution.

The geometric series model [START_REF] Motomura | On the statistical treatment of communities[END_REF][START_REF] Whittaker | Evolution and Measurement of Species Diversity[END_REF] generates far more uneven species distributions. In this model, the first species is represented by a part p of the total resources. The second one has the same part p of the remaining resources, and so on. Finally, probabilities are normalized to be proportional to the resources taken.

I generated four artificial communities following those distributions. Figure 1 presents a lognormal one, with log-standard-deviation equal to 2 (typical of the distribution of tree species in a rainforest) and a geometric distribution with parameter p = 0.1. Both contain 300 species. The other two distributions have identical parameters except for the number of species augmented to 600.

Evaluation of the performance of estimators

The performance of each estimator was calculated as its average relative bias on all values of q (i.e. the average difference between the mean simulated entropy and its true value) and it Root Mean Square Error (RMSE, i.e. the square root of the sum of the squared bias and the variance, divided by the true value). The true entropy of each reference distribution was calculated with the known values of p s . For each reference distribution, 1000 random samples of the chosen size were drawn in a multinomial distribution respecting the reference probabilities p s . Entropy was calculated for q between 0 and 2. The average entropy and its first and last 2.5% quantiles were retained to build the profile and its confidence envelope (which is quite different from that of the estimation of real communities). Finally, entropy was transformed into diversity to be plotted.

Results

I drew multinomial samples of various sizes in the chosen species distributions, simulating a real, independent sampling of individuals. Sample sizes are between 200 and 5000 individuals to cover a range from obvious undersampling to a high-effort inventory: 5000 individuals correspond to 9 to 10 ha of forest.

Sample coverage

I first evaluated the performance of the estimator of sample coverage. 2 communities of each size between 200 and 5000 individuals were sampled in each typical distribution. The real and estimated sample coverages are compared on figure 2. The estimation of sample coverage is very efficient. A model II linear regression [START_REF] Legendre | lmodel2: Model II Regression. R package version 1.7[END_REF] validated the accuracy of the estimation.

Conditionally to the sample size, the relation vanishes but the average estimation is very close to the average actual value: as predicted by the theory, the estimation bias is very small (Figures 7 and8).

Entropy and diversity

I estimated the entropy profiles of the lognormal and geometric distributions of 300 and 600 species, sampled at 4 different intensities (200, 500, 1000 and 5000 individuals), by 9 estimators: Chao-Shen, Grassberger, Chao-Wang-Jost, Zhang-Grabchak, Generalized Coverage, the three unveiled and the Plug-in. The diversity profiles are plotted in the electronic appendices.

The root mean square error of the estimators is shown on figure 3 for the lognormal and the geometric distributions with 300 species when 1000 individuals are sampled, a typical tropical forest inventory of trees.

Unsurprisingly, the plug-in estimator is severely biased and has the poorest results in the tests. The Chao-Wang-Jost estimator systematically outperforms the Zhang-Grabchak estimator (which actually performs little better than the plugin-estimator here) by construction. Its complementary estimation is not paid by increased variance. The Grassberger estimator is totally inefficient for low values of q as already noticed by Marcon et al. (2014a). The generalized coverage estimator outperforms Chao-Shen because of its better estimation of conditional probabilities. The Chao-unveiled estimator is almost confused with the Chao-Wang-Jost estimator. Both are outperformed by the iChao-unveiled estimator because it improves the estimation of the number of Values of q over 0.6 are not shown because all estimators perform similarly well. The legend lists the estimators in the increasing order of RMSE for q > 0.1, where the estimator with the lowest RMSE is the jackknife-unveiled one, closely followed by the iChao-unveiled and Chao-Wang-Jost. Close to q = 0, the jackknife-unveiled estimator has a higher variance making it the least reliable estimator. species. The jackknife-unveiled estimator is more flexible than the previous ones to estimate the number of species. The order of the jackknife estimator it uses changes between simulations, causing an excessive variance for q < 0.1. It performs best for higher orders of diversity.

Results are consistent whatever the model. The general pattern is a poor estimation of low orders of diversity, and a quite accurate estimation of high orders, as previously shown by [START_REF] Haegeman | Robust estimation of microbial diversity in theory and in practice[END_REF]. The RMSE varies a lot according to the model.

I am continuing the analysis with the best two estimators: Chao-Wang-Jost and jackknife-unveiled, ignoring the iChao-unveiled estimator which takes place between them but is too similar to the jackknife-unveiled to bring decisive arguments for the discussion. Figure 4 shows their profiles for a 1000-individual sample of a lognormal distribution of 300 species, with their confidence intervals.

Discussion

The underlying distribution of species is the most important determinant of the success of diversity estimation: the estimation bias of heavy-tailed distributions decays more slowly when the sample size is increased [START_REF] Zhang | Bias adjustment for a nonparametric entropy estimator[END_REF]. Estimating the low-order diversity of a sample from a geometric distribution is all but impossible [START_REF] Haegeman | Robust estimation of microbial diversity in theory and in practice[END_REF] but the low-order diversity of lognormal communities can be estimated meaningfully when the sample size is sufficient. Empirically, it is not possible to discriminate a severely-censused geometric distribution and a lognormal one [START_REF] Tokeshi | Species Abundance Patterns and Community Structure[END_REF]: both models fit well since most of the difference is contained by the unobserved tails of the distributions. So, theoretical, ecological arguments about the actual distribution of the community are necessary to decide whether an estimation of diversity is reliable.

Diversity of order over 0.5 is pretty well estimated in the context of this paper. Haegeman et al. showed that this remains true for q ≥ 1, even when geometric communities of millions of species with parameter 0.5 (the most abundant species takes half the resources, the second one a quarter and so on) are addressed.

3.1 The sample coverage is not always the good indicator of the quality of estimation The sample coverage can not be used as a proxy for how much an estimate of diversity can be relied upon. At the same sampling effort, the sample coverage appears to be higher for the geometric distribution (Figures 7 and8). Far more species are not sampled than in a lognormal distribution, but their total probability is smaller. For example, samples of 200 individuals drawn in 300-species geometric and lognormal communities yield an average estimation of 54 and 149 species by the jackknife-unveiled estimator, but the respective sample coverages are over 95% for the geometric distribution versus around 81% for the lognormal one.

The estimation bias is thus much greater for low orders of diversity even though the sample coverage is higher. [START_REF] Chao | Coverage-based rarefaction and extrapolation: standardizing samples by completeness rather than size[END_REF] argue in favor of the sample coverage as a better measure of the sampling effort than the sample size. I agree as long as the underlying distribution of communities is the same: then, standardizing the sampling effort by the sample coverage is pertinent.

Comparing the diversity of real communities with

different distributions remains untractable When the number of species of the theoretical distributions is doubled, everything else equal, the sampling bias increases (compare figures 14 and 18). With the same sampling effort, the coverage of the lognormal distribution decreases (compare figures 7 and 8, left columns). Doubling the effort brings both the sample coverage and the bias back to their previous level, with a reduced variance (compare figures 18c and 14e). This is a very simple and intuitive behavior, but it is completely different with the geometric distribution: the sample coverage does not change when richness is doubled (compare figures 7 and 8, right columns) because the probabilities of the 300 rarest species are negligible. Doubling the sample size does not restore the bias level (compare figures 18d and 14f). An extensive and rigorous analysis of the influence of the parameters of the theoretical distributions (beyond manipulating the number of species) is not the scope of this paper, but this simple example shows that no general and simple rules are available to compare the low-order diversity of communities of different nature.

3.3 Estimating the number of species is the critical step The lower q, the more difficult the estimation is, but the estimation of the number of species has been long studied and simple rules of decision have been proposed [START_REF] Burnham | Robust Estimation of Population Size When Capture Probabilities Vary Among Animals[END_REF][START_REF] Brose | Estimating species richness: Sensitivity to sample coverage and insensitivity to spatial patterns[END_REF] to chose the most appropriate order of the jackknife estimator. Burnham and Overton derived a selection procedure to obtain the order allowing to minimize the RMSE of the estimation of the number of species. It is implemented in the package SPECIES [START_REF] Wang | SPECIES: An R Package for Species Richness Estimation[END_REF] for R. Brose et al. showed (empirically) hat the first-order jackknife is selected when the sample completeness (terminology by [START_REF] Beck | Comparing measures of species diversity from incomplete inventories: an update[END_REF], i.e. the proportion of observed species (S -s n 0 )/S is over 3/4 (precisely 74% in their paper). When it is less, higher orders have less bias but more variance. It is easy to estimate the number of species of an actual sample this way and compare it to the Chao1 estimator. If both coincide, the Chao-Wang-Jost estimator will perform well for the whole profile: its value at q = 0 is that of Chao1. Else, the jackknifeunveiled estimator will be the best choice since its value at q = 0 is the optimal-order jackknife. If one does not want to rely on the jackknife estimator for some reason, such as its poor theoretical support, the iChao-unveiled estimator is a reasonable compromise as a lower bound estimation.

3.4 Better, but probably not much better, estimators may be derived The most promising ways of research according to the present results are a better estimation of the remaining bias of the Zhang-Grabchak estimator and the improvement of the distribution modeling of the unveiled estimators. The first approach is that of the Chao-Wang-Jost estimator, which is limited by its estimation of the number of species (the lower bound, Chao1 estimator). The price for releasing this constraint is losing the elegant, closed form of the estimator allowed by appropriate approximations of the infinite sum of the unknown elements of eq. ( 11) for a numeric approximation.

The distribution of species is modeled with two parameters in the unveiled estimators. This can be refined by extending the technique presented by Chao et al. (2015) to higher orders of sample coverage. In both cases, better fitting the data to reduce the bias has its limits because the variance of estimation is likely to increase [START_REF] Bonachela | Entropy estimates of small data sets[END_REF]. So, the estimators presented here may not be far from the optimum trade-off (less

Application to real data

I now estimate the diversity of two real forest plots. The first case is Barro Colorado Island's 50-ha plot of tropical forest [START_REF] Hubbell | Barro Colorado Forest Census Plot Data[END_REF], whose inventory data of trees over 10 cm diameter at breast height are available in the package vegan [START_REF] Oksanen | vegan: Community Ecology Package[END_REF] for R. 225 species have been sampled, with a quite good fit to a lognormal distribution. The sample size is over 20000 individuals, the sample coverage is over 99.9%. Estimating the number of species with the Chao1 (239 species) or the jackknife 1 (244) estimators gives very similar results. This is an unusually large dataset, whose diversity estimation (Figure 5) is quite easy.

The best estimator is Chao-Wang-Jost since the Chao1 estimator is appropriate for the number of species. The 95% confidence interval of the estimation is built by resampling according to the technique by [START_REF] Chao | Estimating diversity and entropy profiles via discovery rates of new species[END_REF]. It is very small due to the abundance of data.

The second example takes place at the other extreme of sampling intensity. A 1-ha plot (plot 18) of tropical forest in the experimental forest of Paracou [START_REF] Gourlet-Fleury | Ecology & Management of a Neotropical Rainforest. Lessons Drawnfrom Paracou, a Long-Term Experimental Research Site in French Guiana[END_REF], French Guiana, has been inventoried. Data are available in the package entropart for R. Only 481 trees over 10 cm diameter at breast height have been sampled. They belong to 149 species. The sample coverage is 84.6±4.4%. The estimated number of species is 254 according to Chao1, but the appropriate Jackknife estimator (of order 3) returns 309 species. Clearly, the sampling effort is not sufficient for an accurate estimation: the sample coverage is too low and the estimation of the The jackknife-unveiled estimator is the best choice. Its confidence interval is very wide up to q = 0.3. Over q = 0.5, the simulations above showed that the estimator has a very low variability, so the confidence interval is due to the uncertainty of the sampling only. At lower orders of diversity, the estimator's uncertainty amplifies it so the estimation is not reliable. In this case, the very little accuracy of the jackknife-unveiled estimator (the number of species is estimated between 237 and 439) is preferable to the far smaller confidence interval provided by a less variable but more biased estimator such as Chao-Wang-Jost that would probably not contain the actual values of low-order diversity (Figure 14e).

Conclusion

I have tried to evaluate the performance of diversity estimation in real conditions with simulation studies covering a reasonable set of models. Unsurprisingly, estimating diversity is more difficult when the species distribution has a heavier tail and the number of species is greater. As of the state of the art, the recommendation is to apply the Chao-Wang-Jost, the iChao-unveiled or the jackknife-unveiled estimator and consider diversity of order lower than 0.5 with caution.

When the sampling effort is high enough to allow a correct estimation of the number of species with the Chao1 estimator, the estimation by Chao-Wang-Jost is quite good down to q = 0. If this is not the case, the jackknife-unveiled estimator provides better results but with a higher variability. A conservative compromise for a first estimation of diversity, before choosing between Chao-Wang-Jost and jackknife-unveiled, is the iChaounveiled estimator.

The entropart package for R allows computing speciesneutral diversity and phylodiversity with all the estimators presented here. xlab="Order of Diversity", ylab="RMSE", xlim=c(0, 0.6)) ltype <-1 Cases <-vector() for (Correction in Corrections) { lines(q.seq, eval(as.name(paste("Egeom300_1000", Correction, sep="")))$RMSE, lty=ltype) ltype <-ltype+1 Cases <-append(Cases, Correction) } legend("topright", legend = Cases, lty = 1:ltype)

#################################################### ####### Real data ################################## # BCI

Figure 1 .

 1 Figure 1. Rank-Abundance curves of 300 species following a lognormal (top curve) or a geometric distribution (straight line). The red lines are the fitted models.

Figure 2 .

 2 Figure 2. Estimated vs real sample coverage of simulated samples of a (a) lognormal or (b) geometric distribution of 300 species. Sample sizes are between 200 and 5000 individuals. The line represents the fit of a model II (Major Axis method) linear regression.

Figure 3 .

 3 Figure3. Estimated relative RMSE of the estimators of diversity based on 1000 samples of 1000 individuals of each typical distribution (lognormal and geometric) of 300 species. The RMSE is normalized by the actual diversity. It is quite high for low orders of diversity, especially for the geometric distribution. Values of q over 0.6 are not shown because all estimators perform similarly well. The legend lists the estimators in the increasing order of RMSE for q > 0.1, where the estimator with the lowest RMSE is the jackknife-unveiled one, closely followed by the iChao-unveiled and Chao-Wang-Jost. Close to q = 0, the jackknife-unveiled estimator has a higher variance making it the least reliable estimator.

Figure 4 .

 4 Figure 4. Diversity profiles estimated from 1000 random samples of 1000 individuals from a lognormal community of 300 species. The bold, black line represents the real diversity (starting from 0 D = 269). The jackknife-unveiled estimator is plotted by the blue, dashed line ( 0 D ≈ 250). Its confidence interval (blue dots) is very wide. The Chao-Wang-Jost estimator (red, bold line: 0 D = 209) is more biased downard but its confidence interval (red, solid lines) is much smaller.

Figure 5 .

 5 Figure5. Estimated diversity profile of the tree species of the BCI 50-ha plot. The shaded zone is the 95% confidence interval of the estimation.

Figure 6 .

 6 Figure6. Estimated diversity profiles of the tree species of the Paracou 1-ha plot #18. The shaded zone is the 95% confidence interval of the estimation.

Figure 7 .

 7 Figure 7. Estimated vs real sample coverage of simulated samples. The dotted lines are the average values. The distributions contain 300 species. The plain line represents equality.

Figure 8 .

 8 Figure 8. Estimated vs real sample coverage of simulated samples. The dotted lines are the average values. The distributions contain 600 species. The plain line represents equality.

  data(BCI) NsBCI <-as.AbdVector(colSums(BCI)) plot(NsBCI, Distribution="lnorm") NBCI <-sum(NsBCI)AbdBCI <-AbdFreqCount(NsBCI) # Sample coverage (CBCI <-Coverage(NsBCI)) S1BCI <-AbdBCI[which(AbdBCI[, 1] == 1), 2] S2BCI <-AbdBCI[which(AbdBCI[, 1] == 2), 2] # Confidence interval of the sample coverage estimation ICBCI <-qt(1-Alpha/2, NBCI)*sqrt(S1BCI*(1-S1BCI/NBCI)+2*S2BCI)AbdP18[which(AbdP18[, 1] == 2), 2] # Confidenceinterval of the sample coverage estimation ICP18 <-qt(1-Alpha/2, NP18)*sqrt(S1P18*(1-S1P18/NP18)+2*S2P18)/NP18 # Number of species Richness(NsP18, Correction="Chao1") Richness(NsP18, Correction="Jackknife") # Diversity profile. Very variable, 1000 simulations required. DPP18 <-CommunityProfile(Diversity, NsP18, Correction="UnveilJ", NumberOfSimulations = 1000, Alpha=Alpha) plot(DPP18)

  Geometric distribution. The estimated sample coverage is 1.046 times the real one minus 0.046, with an R 2 value around 69%

	Estimated Sample Coverage	0.75 0.85 0.95					Estimated Sample Coverage	0.92 0.94 0.96 0.98 1.00
		0.80	0.85	0.90	0.95	1.00		0.94	0.96	0.98	1.00
			Real Sample Coverage					Real Sample Coverage
	(a) Lognormal distribution. The estimated sample coverage is	(b)
	0.991 times the real one plus 0.009, with an R 2 value around
	94%.

Acknowledgments

This work has benefited from an"Investissement d'Avenir" grant managed by Agence Nationale de la Recherche (CEBA, ref. ANR-10-LABX-25-01).

I wish to thank the participants of the Anaee France Biodiversity group meeting in Moulis (February 2015) for fruitful discussions that motivated this paper.

Appendix 1: Sample coverage estimation

Figures 7 and 8 compares the estimated and the real sample coverages of 1000 samples of sizes between 200 and 5000 individuals from a lognormal and a geometric distribution of 300 or 600 species The average estimated sample coverage virtually equals the average real coverage even when the sampling size is small.

Appendix 2: Estimated diversity profiles

Figures 9 to 17 show the estimation of diversity profiles of communities of 300 species. Each figure presents an estimator. The diversity of the lognormal and of the geometric community is estimated, for sample sizes from 200 to 5000 individuals. 1000 simulations were performed to produce a 95% confidence envelope of the profiles. Figures 18 and?? present the same profiles for 600-species communities, limited to the best two estimators.

Appendix 3: R code

The necessary code to reproduce all results of the paper is given here. library("entropart") library("lmodel2") library("vegan")

Distributions <-c("lnorm", "geom") Richness <-c(300, 600) SampleSizes <-c(200, 500, 1000, 5000) Corrections <-c("UnveilJ", "UnveiliC", "UnveilC", "ChaoWangJost", "GenCov", "ChaoShen", "Grassberger", "ZhangGrabchak", "None") q.seq <-c(seq(0, .1, 0.025), .15, seq(.2, .6, 0.1)) NumberOfSimulations <-1000 Alpha <-0.05 ylim=c(1e-15, 1e-1), cex=.5) points(Pgeom300, cex=.5) lNs <-log(Pgeom300) Rank <-1:length(Pgeom300) reg <-lm(lNs~Rank) lines (Rank,exp(reg$coefficients[1] abline(a=0, b=1) # Type II regression between real and observed C print(paste(Distribution, "Distribution", S, "species")) print(reg <-lmodel2(RealC~EstimC, nperm=1000))

# Function to build the confidence envelope of the estimation CommunityProfileBySize <-function(FUN, Ps, q.seq, Size, NumberOfSimulations, Correction, Alpha) { # Create a MetaCommunity made of simulated communities MCSim <-rCommunity(NumberOfSimulations, size=Size, NorP=Ps, CheckArguments = FALSE) ProgressBar <-txtProgressBar(min=0, max=NumberOfSimulations) Sims <-matrix(nrow=NumberOfSimulations, ncol=length(q.seq)) # Loops are required for the progress bar, instead of: # Sims <-apply(MCSim$Nsi, 2, function(Nsi) CommunityProfile(FUN, Nsi, q.seq, ...)$y) for (i in 1:NumberOfSimulations) { Sims[i, ] <-sapply(q.seq, function(q) FUN(MCSim$Nsi[, i], q, Correction=Correction, CheckArguments = FALSE)) setTxtProgressBar(ProgressBar, i) } Means <-apply(Sims, 2, mean) Vars <-apply(Sims, 2, var) # Quantiles of simulations for each q EstEnvelope <-apply(Sims, 2, quantile, probs = c(Alpha/2, 1-Alpha/2)) colnames(EstEnvelope) <-q.seq Profile <-list(x = q.seq, y = Means, low = EstEnvelope[1,], high = EstEnvelope[2,], var = Vars, Coverage = MCSim$SampleCoverage.communities ) class(Profile) <-"CommunityProfile" return (Profile) } # Calculate diversity profiles for (Distribution in Distributions) { for (S in Richness) { Ps <-eval(as.name(paste("P", Distribution, S, sep=""))) # Real Profile Values <-sapply(q.seq, function(q) Tsallis(Ps, q, CheckArguments = FALSE)) # Graphical parameters yLim <-c(0.9*min(Values), 1.1*max(Values))

for (Size in SampleSizes) { for (Correction in Corrections) { # Entropy profile CommunityProfileBySize(bcTsallis, Ps, q.seq, Size, NumberOfSimulations, Correction, Alpha) -> Env # Transform entropy into diversity DEnv <-expq.CommunityProfile(Env) DValues <-sapply(1:length(q.seq), function(i) expq(Values[i], q.seq[i])) # Save the data Env$Real <-Values # Calculate RMSE Env$RMSE <-sqrt((Env$Real-Env$y)^2 + Env$var)/Env$Real assign(paste("E", Distribution, S, "_", Size, Correction, sep=""), Env) } } } } # Plot diversity profiles for (Distribution in Distributions) { for (S in Richness) { for (Size in SampleSizes) { for (Correction in Corrections) { DEnv <-eval(as.name(paste("E", Distribution, S, "_", Size, Correction, sep=""))) plot(DEnv, ylim=c(min(DEnv$Real)*.9, max(DEnv$Real)*1.1), LineWidth=1, main="") lines(q.seq, DEnv$Real, lwd=2)

ylim=c(min(Elnorm300_1000ChaoWangJost$Real)*.9, max(Elnorm300_1000ChaoWangJost$Real)*1.1), main="", xlab = "Order of Diversity", ylab = "Diversity", xlim=c(0, 0.6), type="n") CEnvelope(Elnorm300_1000ChaoWangJost, LineWidth=2, main="", xlim=c(0, 0.6), ShadeColor=NA, col="red") lines(Elnorm300_1000ChaoWangJost$x, Elnorm300_1000ChaoWangJost$high, col="red", lty=1) lines(Elnorm300_1000ChaoWangJost$x, Elnorm300_1000ChaoWangJost$low, col="red", lty=1) CEnvelope(Elnorm300_1000UnveilJ, lty=2, LineWidth=2, col="blue", BorderColor="blue", ShadeColor=NA) lines(q.seq, Elnorm300_1000ChaoWangJost$Real, lwd=2, lty=1)

plot(q.seq, seq(0, 1, length.out=length(q.seq)), type="n", xlab="Order of Diversity", ylab="RMSE", xlim=c(0, 0.6)) ltype <-1 Cases <-vector() for (Correction in Corrections) { lines(q.seq, eval(as.name(paste("Elnorm300_1000", Correction, sep="")))$RMSE, lty=ltype) ltype <-ltype+1 Cases <-append(Cases, Correction) } legend("topright", legend = Cases, lty = 1:ltype) # Figure RMSE geom300 plot(q.seq, seq(0, 1, length.out=length(q.seq)), type="n",


Mourad

Mourad Bellassoued

Michel Cristofol

Eric Soccorsi

Identification of the twisting function for the dynamic Schrödinger operator in a quantum waveguide

come

Identification of the twisting function for the dynamic

Schrödinger operator in a quantum waveguide

Introduction 1.Statement and origin of the problem

The problem. Let ω be a bounded open subset of R 2 with C 2 -boundary. Furthermore we assume that ω contains the origin of R 2 . Set Ω := ω × R. For x := (x 1 , x 2 , x 3 ) ∈ Ω we write x := (x τ , x 3 ) with x τ := (x 1 , x 2 ). Put

∇ τ := (∂ 1 , ∂ 2 ) T , ∆ τ := ∂ 2 1 + ∂ 2 2 , ∂ ϕ := x 1 ∂ 2 -x 2 ∂ 1 .
We assume that θ = θ(x 3 ) ∈ C 1 (R) and denote by θ the derivative of θ wrt to x 3 . Define H θ as the self-adjoint operator in L 2 (Ω)

H θ := -∆ τ -( θ(x 3 )∂ ϕ + ∂ 3 ) 2 . (1.1)
with Dirichlet boundary conditions on Γ := ∂Ω. Given T > 0, q 0 ∈ and h ∈, we then consider the Schrödinger equation    -ıq ′ (t, x) + H θq(t, x) = 0, (t, x) ∈ (0, T ) × Ω q(t, x) = h(t, x), (t, x) ∈ (0, T ) × Γ q(0, x) = q 0 (x)

x ∈ Ω, (1.2) where the q ′ stands for ∂q ∂t .

Twisted waveguide. Define the twisted domain

Ω θ := {r θ (x 3 )(x), x ∈ Ω},
where

r θ = r θ (x 3 ) :=   cos θ(x 3 ) sin θ(x 3 ) 0 -sin θ(x 3 ) cos θ(x 3 ) 0 0 0 1   ,
and introduce the transform

(U θ ψ)(x) := ψ(r θ (x)), x ∈ Ω, ψ ∈ L 2 (Ω θ ), 1
which is unitary from L 2 (Ω θ ) onto L 2 (Ω). Then we have H θ = U θ (-∆)U * θ where ∆ denotes the Laplacian operator in L 2 (Ω θ ). Moreover if q is the solution to (1.2) then u := U * θ q obeys the system    -ıu ′ (t, x) -∆u(t, x) = 0, (t, x) ∈ (0, T ) × Ω θ u(t, x) = (U * θ h(t, .))(x), (t, x) ∈ (0, T ) × ∂Ω θ u(0, x) = (U * θ q 0 )(x),

x ∈ Ω θ .

Hence (1.2) describes, modulo a unitary transform, the quantum motion of a1 charged particle in the twisted waveguide Ω θ .

In this paper we examine the inverse problem of determining the second derivative θ of the twisting function θ from some suitable localized, either interior or boundary, measurement of the solution q to (1.2).

Main results

Set I d := (-d, d) and Ω(d) := ω × I d for all d > 0. For ℓ > 0 and ε > 0 we define the set of admissible twistings as Θ ℓ,ε = {θ ∈ C 2 0 (R), θ ∈ C 0 (I ℓ ) and θ C 1 (I ℓ ) ≤ ε}.

The first result establishes Lipschitz stability in the inverse problem of identifying θ from the measurement of q and its time derivative q ′ in Ω 0,L := ω 0 × I L for some L > ℓ. This claim is valid for any arbitrarily small ω 0 with positive R 2 -Lebesgue measure such that (0, 0) / ∈ ω 0 (1.3) and ∂ω 0 ∩ ∂ω = ∅.

(1.4) Theorem 1.1. Let T > 0, ℓ > 0, ω and Ω be the same as in §1.1.

Let ω 0 fulfill (1.3)-(1.4) and pick q0 ∈ H 2 (Ω) ∩ H 1 0 (Ω) in such a way that ∃q > 0, |∂ ϕ q0 (x)| ≥ q, a.e. x ∈ Ω 0 := ω 0 × R. (1.5) Let q 0 ∈ H 2 (Ω) ∩ H 1 0 (Ω)
, let θ (resp. θ) belong to Θ ε and note q (resp. q) the solution to (1.2) (resp. to (1.2) where θ and q0 are substituted for θ and q 0 ). Then, for every L > ℓ there exists a constant C > 0, depending only on L, T, ℓ, ε and ω 0 , such that

θ -θ 2 L 2 (I ℓ ) ≤ C q ′ -q′ 2 L 2 (0,T ;H 1 (Ω 0 (L))) + q 0 -q0 2 H 2 (Ω 0 ) .
In contrast with Theorem 1.1 only boundary, instead of volume internal measurement of the solution are required for the following result to hold. Theorem 1.2. Let T > 0, L > 0, ω and Ω be the same as in §1.1. Pick q0 ∈ H 2 (Ω) obeying (1.5) for some open subset ω 0 ω fulfilling (C 2 ). Let θ (resp. θ) belong to Θ ε and note q (resp. q) the solution to (1.2) (resp. to (1.2) where θ and q0 are substituted for θ and q 0 ). Then there is a constant C ′ > 0, depending only on T, L, ω 0 and ℓ, such that θ -

θ 2 L 2 (I L ) ≤ C ′ ∂ (g 0 ) ν (q ′ -q′ ) 2 L 2 (0,T ;L 2 (Γ 0 )) + q 0 -q0 2 H 2 (Ω) .

Outline

The derivation of Theorems

Carleman inequality for heterogeneous Schrödinger operators

Let n ≥ 2 and M be a n-dimensional domain with smooth boundary ∂M and smooth metric g. This section is devoted to establishing a Carleman estimate for the dynamic Schrödinger operator

B := ı∂ t + ∆ g , (2.6)
acting in L 2 (M T ) where M T := (-T, T ) × M. Here ∆ g denotes the Laplace-Beltrami operator associated to the smooth metric g. In local coordinates, it is given by

∆ g := 1 √ det g n j,k=1 ∂ j det g g jk ∂ k , (2.7)
where g -1 := (g jk ) 1≤j,k≤n is the inverse of g := (g jk ) 1≤j,k≤n . We notice from (1.1) that H θ coincides with ∆ g in the particular case where n = 3 and

g =   1 + θ2 x 2 2 -θ2 x 1 x 2 -θx 2 -θ2 x 1 x 2 1 + θ2 x 2 1 θx 1 -θx 2 θx 1 1   .
(2.8)

Notations

For each x ∈ M we define the inner product

g(X, Y ) = X, Y g := n j,k=1 g j,k (x)α j β k , X = (α j ) 1≤j≤n , Y = (β j ) 1≤j≤n ∈ C n , (2.9)
and the norm |X| g := X, X g 1/2 .

Carleman estimate

Assume that there exists ϑ ∈ C 4 (M; R * + ) such that

D 2 ϑ(ξ, ξ)(x) + | ∇ g ϑ(x), ξ g | 2 > 0, x ∈ M, ξ ∈ R n \ {0}, (2.10)
where ∇ g ϑ := g -1 ∇ϑ and D 2 is the Hessian of ϑ wrt the metric g, and suppose that ϑ has no critical points on M, min

x∈Ω |∇ g ϑ(x)| 2 g > 0.
(2.11)

For further reference let us notice that:

(i) First, (2.10) may be rephrased into the following equivalent statement

D 2 ϑ(ξ, ξ)(x) + | ∇ g ϑ(x), ξ g | 2 > 0, x ∈ M, ξ ∈ C n \ {0}, (2.12) since D 2 ϑ(ξ, ξ) = D 2 ϑ(ξ R , ξ R ) + D 2 ϑ(ξ I , ξ I )
where ξ R := Re (ξ) and ξ I := Im (ξ);

(ii) Second, (2.11) yields the existence of some constant β > 0 satisfying

min x∈M |∇ g ϑ(x)| 2 g ≥ β. (2.13)
Moreover we impose on ϑ the following condition

ϑ(x) ≥ 2 3 sup x∈M ϑ(x), x ∈ M, (2.14)
which can be fulfilled by substituting ϑ + C for ϑ, for any sufficiently large real number C. Further define

ψ(t, x) := e γϑ(x) (T -t)(T + t) , η(t, x) := e 2γ ϑ ∞ -e γϑ(x) (T -t)(T + t) , (x, t) ∈ M T , (2.15)
and introduce M := e -sη Be sη = M 1 + M 2 , where

M 1 := ı∂ t + ∆ g + s 2 |∇ g η| 2 g (2.16) M 2 := ıs∂ t η + 2s∇η • ∇ g + s(∆ g η).
(2.17)

Next we consider a subboundary Γ 0 ⊂ ∂M satisfying

{x ∈ Γ, ∇ g ϑ(x), ν(x) g ≥ 0} ⊂ Γ 0 , (2.18)
where ν denotes the outward unit normal to ∂M wrt the metric g. Finally we define

∂ ν w := ∇ g w • ν on Σ 0 := (-T, T ) × Γ 0 .
Proposition 2.1. Assume (2.10)-(2.11) and (2.18). Let ψ, η be given by (2.15), and M j , j = 1, 2, be defined by (2.16)-(2.17). Then there exist s 0 > 0 and C 0 = C 0 (M, T, g C 2 (M) ) > 0 such that the following Carleman estimate

I(w) := s e -sη ψ 1/2 ∇ g w 2 L 2 (M T ) n + s 3 e -sη ψ 3/2 w 2 L 2 (M T ) + j=1,2 M j e -sη w 2 L 2 (M T ) ≤ C 0 e -sη Bw 2 L 2 (M T ) + s e -sη ψ∂ ν w 2 L 2 (Σ 0 )
, holds true for every s ≥ s 0 whenever w ∈ L 2 (-T, T ; H 1 0 (M)) and Bw ∈ L 2 (M T ).

Twisted waveguide

For L ∈ (ℓ, +∞) fixed, set r := (ℓ + L)/2 and let M be any smooth bounded open subset of Ω(L) containing Ω(r):

Ω(ℓ) ⊂ Ω(r) ⊂ M ⊂ Ω(L). Pick a = (a 1 , a 2 , a 3 ) ∈ R 3 \ M such that d 3 = a 3 -L > 0, (2.19) and put ϑ(x) = |x -a| 2 , x ∈ M. (2.20)
For further reference we notice from (2.15) and (2.19)-(2.20) that η 0 = η(0, .) fulfills

η ′ 0 (x) = 2 (a 3 -x 3 )e |x-a| 2 T 2 ≥ 2d 3 T 2 , x ∈ M. (2.21)
Moreover, in the framework of Theorem 1.2 we shall impose in addition to (2.19) that a τ = (a 1 , a 2 ) ∈ R 2 \ω be taken far enough from ω in such a way that

d τ = dist(a τ , ω) = inf xτ ∈ω |x τ -a τ | (2.22)
is so large relative to L and d 3 that

ω ∩ B a τ , d τ + 4L(L + d 3 ) d τ ω. (2.23) By [1][Lemma 4.1] we have 4D 2 ϑ(g -1 ξ, g -1 ξ) = {g, {g, ϑ}}(x, ξ)
, where g(x, ξ) := n j,k=1 g j,k (x)ξ j ξ j for x ∈ M and ξ ∈ R n . This, combined with (2.8) and the fact that

g jk = δ jk + θ(x 3 )b jk , 1 ≤ j, k ≤ 3, yields D 2 ϑ(g -1 ξ, g -1 ξ) = Hess(ϑ)(ξ, ξ) + O( θ C 1 (I L ) |ξ| 2 ),
where Hess(ϑ) denotes the Hessian of ϑ wrt the Euclidian metric. Therefore we have

D 2 ϑ(g -1 ξ, g -1 ξ) = 2|ξ| 2 + O( θ C 1 (I ℓ ) |ξ| 2 ), ξ ∈ R n ,
by an elementary calculation, whence D 2 ϑ(ξ, ξ) ≥ C|ξ| 2 for all ξ ∈ R n and some constant C > 0, upon taking θ C 1 (I ℓ ) sufficiently small. Namely, there exists ε > 0, depending only on ω, ℓ and a such that (2.10) holds true for every θ ∈ Θ ℓ,ε .

Inverse problem

In this section we investigate the inverse problem of identifying θ from partial knowledge of the solution q to (1.2). We shall prove that θ can be determined uniquely and stably from some suitable localized, either interior or boundary, measurement of q, as stated in Theorem 1.1 and Theorem 1.2. In both cases the method used in the derivation of the corresponding stability estimate preliminarily requires that the problem under study be centered around t = 0, then appropriately "linearized" and differentiated w.r.t. the time variable t.

Time symmetrization and the linearized problem

Time symmetrization. Following the idea of [2] we extend (1.2) to negative values of the time variable t in order to center the problem around t = 0. Indeed, for any real valued initial data q 0 , the solution q to (1.2) extended to Q := (-T, T ) × Ω by setting

q(t, x) := q(-t, x) for a.e. (t, x) ∈ R * -× Ω, (3.24)
is easily seen to be solution to the system

   -ıq ′ (t, x) + H θq(t, x) = 0, (t, x) ∈ Q q(t, x) = h(t, x), (t, x) ∈ Σ := (-T, T ) × Γ q(0, x) = q 0 (x), x ∈ Ω, (3.25)
provided we have preliminarily set

h(t, x) := h(-t, x) for a.e. (t, x) ∈ R * -× Γ.
The "linearized" problem. Let q denote the solution to (3.25) where θ and q 0 are replaced by θ and q0 , a straightforward computation shows that y := qq satisfies the "linearized" system

   -ıy ′ (t, x) + H θy(t, x) = R(t, x), (t, x) ∈ Q y(t, x) = 0, (t, x) ∈ Σ y(0, x) = (q 0 -q0 )(x), x ∈ Ω, (3.26)
where

α := θ -θ and R := (H θ -H θ) q = (α(( θ + θ)∂ ϕ + 2∂ 3 ) + α)∂ ϕ q. (3.27)
Further, by differentiating (3.26) w.r.t. t, we get that z := y ′ is solution to

   -ız ′ (t, x) + H θz(t, x) = R ′ (t, x), (t, x) ∈ Q z(t, x) = 0, (t, x) ∈ Σ z(0, x) = ı(R(0, x) -H θy(0, x)), x ∈ Ω, (3.28) with R ′ := (α(( θ + θ)∂ ϕ + 2∂ 3 ) + α)∂ ϕ q′ , (3.29)
according to (3.27).

Proof of Theorem 1.1

Let ω 1 be any non empty open subset of ω 0 and pick two cut-off functions ρ ∈ C 2 (ω; [0, 1]) and µ ∈ C 2 (R; [0, 1]) satisfying

ρ(x τ ) := 1 in ω 1 0 in ω\ω 0 and µ(x 3 ) := 1 in I ℓ 0 in R\I r , (3.30)
in such a way that χ := ρµ ∈ C 2 (Ω; [0, 1]) fulfills

χ(x) = 1 in Ω 1 (ℓ) 0 in Ω \ Ω 0 (r). (3.31)
By left multiplying (3.28) by χ, we get that z = χz is solution of the system

   -ız ′ (t, x) + H θz(t, x) = χR ′ (t, x) + [H θ, χ]z, (t, x) ∈ M T z(t, x) = 0, (t, x) ∈ ∂M T := (-T, T ) × ∂M z(0, x) = -ıχ(R(0, x) -H θy(0, x)),
x ∈ M.

(3.32) From this and Proposition 2.1 then follows that

I(z) = s e -sη ψ 1/2 ∇ g z 2 L 2 (M T ) 3 + s 3 e -sη ψ 3/2 z 2 L 2 (M T ) + j=1,2 M j e -sη z 2 L 2 (M T ) ≤ C 0 e -sη (χR ′ (t, x) + [H θ, χ]z) 2 L 2 (M T ) , s ≥ s 0 . In light of (3.29) and the expression [H θ, χ] = -(2∇ g χ • ∇ + ∆ g χ), this yields I(z) ≤ C 1 m=0,1 e -sη χα (m) 2 L 2 (-T,T ;L 2 (Ω 0 (ℓ))) + e -sη ∇ m z 2 L 2 (-T,T ;L 2 (Ω 0 (r)\Ω 1 (ℓ))) , s ≥ s 0 ,
(3.33) for some constant C 1 = C 1 (ω, L, T, ε) > 0, where α (0) (resp. α (1) ) stands for α (resp. α). Having established the Carleman estimate (3.33), the remaining of the proof now decomposes into four steps.

The first step involves relating α to z(0, .) with the aid of (3.32). Namely, by multiplying the initial condition in (3.32) by e -sη 0 , squaring and integrating over M, we deduce from (3.27) that e -sη 0 ∂ ϕ q0 χ α 2 L 2 (Ω 0 (ℓ)) is majorized (up to some positive multiplicative constant C 2 depending only on ω, L, T and ε) for all s > 0 by the sum e -sη 0 z(0, .)

2 L 2 (M) + e -sη 0 H θy(0, .) 2 L 2 (Ω 0 (r)) + e -sη 0 χα 2 L 2 (Ω 0 (ℓ)) . Taking into account that |∂ ϕ q0 |χ ≥ qχ, this yields e -sη 0 χ α 2 L 2 (Ω 0 (ℓ)) ≤ C 2 e -sη 0 z(0, .) 2 L 2 (M) + e -sη 0 H θy(0, .) 2 L 2 (Ω 0 (r)) + e -sη 0 χα 2 L 2 (M) , (3.34)
where C 2 = C 2 (ω, L, T, ε) > 0. The next step is to derive a suitable upper bound on e -sη 0 z(0, .) L 2 (M) involving e -sη χ α L 2 (Ω 0 (ℓ)) , from (3.33).

Lemma 3.1. There exists a constant C 3 = C 3 (ω, L, T, ε) > 0 such that we have

e -sη 0 z(0, .) 2 L 2 (M) ≤ C 3 s -3/2 m=0,1 e -sη χα (m) 2 L 2 (-T,T ;L 2 (Ω 0 (ℓ))) + e -sη ∇ m z 2 L 2 (-T,T ;L 2 (Ω 0 (r)\Ω 1 (ℓ))) ,
for all s ≥ s 0 .

Proof. Set ζ := e -sη z and recall that ζ(-T, .) = 0 so that

I := e -sη 0 z(0, .) 2 L 2 (M) = 0 -T M ∂ τ |ζ(t, x)| 2 dxdt = 2Re 0 -T M ζ τ (t, x)ζ(t, x)dxdt .
In view of (2.16) this identity may be rewritten as

I = 2Im 0 -T M (M 1 ζ)ζ(t, x)dtdx , so we get |I| ≤ 2 M 1 e -sη z L 2 (M T ) e -sη z L 2 (M T )
from the Cauchy-Schwarz inequality. As inf (t,x)∈Q ψ(t, x) > 0 according to (2.15), there is necessarily a constant c > 0, depending only on T , such that we have

|I| ≤ cs -3/2 s 3 e -sη ψ 3/2 z 2 L 2 (M T ) + M 1 e -sη z 2 L 2 (M T ) ≤ cs -3/2 I(z), s > 0.
This, combined with (3.33), proves the result.

The third step consists of showing that e -sη 0 χα L 2 (M) can be made negligible w.r.t. e -sη 0 χ α L 2 (M) upon taking s sufficiently large. Lemma 3.2. We have

e -sη 0 χα L 2 (Ω 0 (ℓ)) ≤ T √ d 3 s e -sη 0 χ α L 2 (Ω 0 (ℓ)) , s > 0,
where d 3 is defined in (2.19).

Proof. Bearing in mind that α(±ℓ) = 0, an integration by parts yields

Ω 0 (ℓ) e -2sη 0 χ 2 α αdx = Ω 0 (ℓ) e -2sη 0 ρ 2 α αdx = - 1 2 Ω 0 (ℓ) ∂ x 3 (e -2sη 0 )ρ 2 α 2 dx.
Therefore,

Ω 0 (ℓ) e -2sη 0 χ 2 α αdx = Ω 0 (ℓ) e -2sη 0 s η0 ρ 2 α 2 dx. = s Ω 0 (ℓ) e -2sη 0 η0 χ 2 α 2 dx. (3.35)
Further, as

Ω 0 (ℓ) e -2sη 0 χ 2 α αdx ≤ d 3 T 2 s Ω 0 (ℓ) e -2sη 0 χ 2 α 2 dx + T 2 d 3 s Ω 0 (ℓ) e -2sη 0 χ 2 α2 dx, s > 0,
from Young inequality, (3.35) yields

s Ω 0 (ℓ) e -2sη 0 η0 - d 3 T 2 χ 2 α 2 dx ≤ T 2 d 3 s Ω 0 (ℓ) e -2sη 0 χ 2 α2 dx, s > 0.
This, together with (2.21), completes the proof.

In view of (3.34) and Lemmae 3.1-3.2 we thus may find some constant C 4 = C 4 (ω, L, T, ε) > 0 such that the following estimate

e -sη 0 χ α 2 L 2 (Ω 0 (ℓ)) ≤ C 4 e -sη 0 H θy(0, .) 2 L 2 (Ω 0 (r)) + m=0,1 e -sη ∇ m z 2 L 2 (-T,T ;L 2 (Ω 0 (r)\Ω 1 (ℓ)))
, holds true upon choosing s sufficiently large. Taking into account that e -sη L ∞ (M T ) ≤ 1 and inf x∈Ω 0 (ℓ) e -sη 0 (x) > 0 for any s > 0, this entails

χ α 2 L 2 (Ω 0 (ℓ)) ≤ C 5 H θy(0, .) 2 L 2 (Ω 0 (r)) + z 2 L 2 (-T,T ;H 1 (Ω 0 (r)\Ω 1 (ℓ))) , (3.36)
under the same conditions on s, where C 5 is some positive real number depending only on ω, L, T and ε. The last step involves noticing that χ α L 2 (Ω 0 (ℓ)) ≥ q mes(ω 1 ) α L 2 (I ℓ ) , s > 0, (3.37) since χ(x) = 1 for every x ∈ Ω 1 (ℓ) and α(x 3 ) = 0 whenever x 3 ∈ R \ I ℓ , by (1.5) and (3.30)-(3.31). Finally, bearing in mind that H θy(0, .) L 2 (Ω 0 (r)) ≤ C 6 y(0, .) H 2 (Ω 0 (r)) , where the constant C 6 > 0 depends only on ω 0 and ε, we get the result by combining (3.36) and (3.37).

Proof of Theorem 1.2

Set z := µz, where µ denotes the cut off function defined by (3.30). Since z is solution to the system (3.32) where µ is substituted for χ, we have the following Carleman estimate

I(z) ≤ C 0 e -sη (µR ′ (t, x) + [H θ, µ]z) 2 L 2 (M T ) + s e -sη ψ 1/2 ∂ ν z 2 L 2 (Σ 0 ) , s ≥ s 0 ,
arising from Proposition 2.1. Putting Ω(ℓ, r) := Ω(r)\Ω(ℓ) and Σ 0 (r) := (-T, T )×Γ 0 (r) with Γ 0 (r) := Γ 0 ∩ (∂ω × I r ), there is thus a constant C ′ 1 > 0, depending only on, such that we have

I(z) ≤ C ′ 1 m=0,1 e -sη α (m) 2 L 2 (-T,T ;L 2 (Ω(ℓ))) + e -sη ∇ m z L 2 (-T,T ;L 2 (Ω(ℓ,r))) + s e -sη ∂ ν z 2 L 2 (Σ 0 (r)) , (3.38)
for all s ≥ s 0 . The strategy of the proof of Theorem 1.1 applies unchanged provided (3.33) is replaced by (3.38). Indeed, arguing as in the derivation of (3.34) from (3.32), we find out some positive constant

C ′ 2 = C ′ 2 () satisfying e -sη 0 ∂ ϕ q0 α 2 L 2 (Ω(ℓ)) ≤ C ′ 2 e -sη 0 z(0, .) 2 L 2 (M) + e -sη 0 H θy(0, .) 2 L 2 (Ω(r)) + e -sη 0 α 2 L 2 (Ω(ℓ)) ,
(3.39) for all s > 0. Further, by substituting (3.38) for (3.33) in the proof of Lemma 3.1, we find out that

C ′ 3 s 3/2 e -sη 0 z(0, .) 2 L 2 (M) ≤ m=0,1 e -sη 0 α (m) 2 L 2 (-T,T ;L 2 (Ω(ℓ))) + e -sη ∇ (m) z 2 L 2 (-T,T ;L 2 (Ω(ℓ,r))) +s e -sη ∂ ν z 2 L 2 (Σ 0 (r)) , (3.40)
for s ≥ s 0 and some constant C ′ 3 = C ′ 3 () > 0. Last, by mimicking the proof of Lemma 3.2, we find out that e -sη 0 α L 2 (Ω(ℓ)) ≤ T 2 d 3 s e -sη 0 α L 2 (Ω(ℓ)) , s > 0.

(3.41)

Now, upon setting

obs := e -sη ∂ ν z 2 L 2 (Σ 0 (r)) + e -sη 0 z L 2 (-T,T ;H 1 (Ω(ℓ,r))) + e -sη 0 H θy(0, .) 2 L 2 (Ω(r)) , (3.42)
it turns out by putting (3.39), (3.40) and (3.41) together, that

e -sη 0 ∂ ϕ q0 α 2 L 2 (Ω(ℓ)) ≤ C ′ 4 (obs + s -3/2 e -sη 0 α 2 L 2 (Ω(ℓ)) ), (3.43)
provided s is sufficiently large. Here C ′ 4 is some positive constant depending only on . The next step of the proof involves splitting e -sη 0 α 2 L 2 (Ω(ℓ)) into the sum e -sη 0 α 2 L 2 (Ω 0 (ℓ)) + e -sη 0 α 2 L 2 (Ω(ℓ)\Ω 0 (ℓ)) , where the subset Ω 0 (ℓ) = ω 0 × I ℓ is chosen in accordance with the following statement.

Lemma 3.3. Assume (2.22)-(2.23). Then there exist two proper open subsets of ω, ω 1 ω 0 , with positive R 2 -Lebesgue measure, satisfying:

m 1 = sup x∈Ω 1 (ℓ) η 0 (x) < m 0 = inf x∈Ω(ℓ)\Ω 0 (ℓ) η 0 (x).
Proof. For ǫ = 2L(L + d 3 )/d τ , where d τ is the same as in (2.22), we put

ω j = {x τ ∈ ω, |x τ -a τ | > d τ + (j + 1)ǫ}, j = 0, 1,
in such a way that ω 1 ω 0 ω according to (2.22)-(2.23), and

inf xτ ∈ω 1 |x τ -a τ | ≥ sup xτ ∈ω\ω 0 |x τ -a τ | + ǫ. (3.44) Further, as |x τ -a τ | 2 + d 2 3 ≤ ϑ(x) ≤ |x τ -a τ | 2 + (2L + d 3 ) 2 for all x = (x τ , x 3 ) ∈ Ω(ℓ), by (2.19)-(2.20), it follows from (3.44) that ϑ(x) ≥ ϑ(x ′ ) + R(x ′ τ ), x ∈ Ω 1 (ℓ), x ′ = (x ′ τ , x ′ 3 ) ∈ Ω(ℓ) \ Ω 0 (ℓ), (3.45)
where

R(x ′ τ ) = 2ǫ(|x ′ τ -a τ | -d τ ) + ǫ 2 . Since |x ′ τ -a τ | ≥ d τ from (2.22), then R(x ′ τ ) ≥ ǫ 2 and ϑ(x) ≥ ϑ(x ′ ) + ǫ 2 , x ∈ Ω 1 (ℓ), x ′ ∈ Ω(ℓ) \ Ω 0 (ℓ).
This entails inf x∈Ω 1 (ℓ) ϑ(x) > sup x∈Ω(ℓ)\Ω 0 (ℓ) ϑ(x), hence the result by (2.15).

Since e -sη 0 ∂ ϕ q0 α L 2 (Ω(ℓ)) ≥ q e -sη 0 α L 2 (Ω 0 (ℓ)) by (1.5), we deduce from (3.43) that

(q -C ′ 4 s -3/2 ) e -sη 0 α 2 L 2 (Ω 0 (ℓ)) ≤ C ′ 4 (obs + s -3/2 e -sη 0 α 2 L 2 (Ω\Ω 0 (ℓ)) ).
Therefore we have

e -sη 0 α 2 L 2 (Ω 1 (ℓ)) ≤ C ′ 5 (obs + s -3/2 e -sη 0 α 2 L 2 (Ω(ℓ)\Ω 0 (ℓ)) ), (3.46)
for some C ′ 5 = C ′ 5 () > 0 and all s large enough. Here Ω 1 (ℓ) is the same as in Lemma 3.3 so (3.46) yields e -sm 1 (mes(ω 1 )s -3/2 e -s(m 0 -m 1 ) mes(ω)) α 2 L 2 (I ℓ ) ≤ C ′ 5 obs, (3.47)

under the above prescribed conditions on s. Finally, by recalling (3.42) and noticing that obs

≤ C ′ 6 ( ∂ ν z 2 L 2 (Σ 0 (r)) + z L 2 (-T,T ;H 1 (Ω(ℓ,r))) + y(0, .) 2 H 2 (Ω(r))
), where C ′ 6 = C ′ 6 () > 0, the desired result follows directly from this, (3.47) and Lemma 3.3 upon taking s sufficiently large.

4 Appendix : proof of Proposition 2.1

4.1 Preliminaries Put div g X := 1 √ detg n j=1 ∂ j ( detgα j ), X = (α j ) 1≤j≤n ∈ C n ,
in such a way that (2.7) reads ∆ g = div g ∇ g . For X and f in H 1 (M) we have

div g (f X) = ∇ g f, X g + f div g X. (4.48)
For X and f in H 1 (M) we have the following Green formula

M div g (X)f dv g = - M X, ∇ g f g dv g + ∂M X • νf dω g , (4.49)
which yields

M (∆ g w)f dv g = - M ∇ g w, ∇ g f g dv g + ∂M (∂ ν w)f dω g , (4.50)
whenever f ∈ H 1 (M) and w ∈ H 2 (M).

Completion of the proof

In light of (2.16)-( 2.17) we shall first compute each term in the rhs of the following identity Re M T M 1 wM 2 w dv g dt = j,k=1,2,3

I j,k , (4.51)
separately. We start by noticing that

I 1,1 := Re M T ıw ′ (-ısη ′ w) dv g dt = sRe M T w ′ η ′ w dv g dt = -sRe M T w(η ′′ w + η ′ w ′ ) dv g dt = -sRe M T η ′′ |w| 2 dv g dt -I 1,1 ,
where we used the fact that w(±T ) = 0. Hence

I 1,1 = - s 2 M T η ′′ |w| 2 dv g dt. (4.52)
Next, bearing in mind that w(±T ) = 0, we have

I 1,2 := 2sRe M T ı ∇ g η, ∇ g w g w ′ dv g dt = -2sIm M T ∇ g η, ∇ g w g w ′ dv g dt = 2sIm M T ∇ g η ′ , ∇ g w g + ∇ g η, ∇ g w ′ g w dv g dt , (4.53)
with, in addition,

M ∇ g η, ∇ g w ′ g w dv g = - M div g ((∇ g η)w)w ′ dv g = - M ((∆ g η)w+ ∇ g η, ∇ g w g )w ′ dv g ,
by taking into account that w ′ vanishes on ∂M. Therefore, it follows from this and (4.53) that

I 1,2 = 2sIm M T ∇ g η ′ , ∇ g w g -(∆ g η)w ′ w dv g dt -I 1,2 ,
from where we get that

I 1,2 = sγIm M T ψ(∆ g ϑ + γ|∇ g ϑ| 2 g )w ′ -ψ ′ ∇ g ϑ, ∇ g w g w dv g dt . (4.54)
Further we have

I 1,3 := sRe M T ıw ′ (∆ g η)w dv g dt = -sγIm M T ψ(∆ g ϑ + γ|∇ g ϑ| 2 g )ww ′ dv g dt , (4.55) and I 2,1 := Re M T (∆ g w)(-ısη ′ w) dv g dt = sIm M T (∆ g w)η ′ w dv g dt , where M (∆ g w)η ′ w dv g = - M ∇ g η ′ , ∇ g w g w dv g - M η ′ ∇ g w, ∇ g w g dv g ,
since w = 0 on ∂M, so we find that

I 2,1 = -sγIm M T ψ ′ ∇ g ϑ, ∇ g w g w dv g dt .
(4.56)

Now, putting (4.54), (4.55) and (4.56) together we obtain that

I 1,2 + I 1,3 + I 2,1 = -2sγIm M T ψ ′ ∇ g ϑ, ∇ g w g w dv g dt . (4.57)
We turn now to computing I 2,2 := 2sRe M T (∆ g w) ∇ g η, ∇ g w g dv g dt . In light of (4.50) we see that

M (∆ g w) ∇ g η, ∇ g w g dv g = - M ∇ g w, ∇ g ∇ g η, ∇ g w g g dv g + ∂M ∇ g η, ∇ g w g ∂ ν w dω g .
(4.58) Bearing in mind that w = 0 and hence ∇ g w • τ = 0 on ∂M, where τ denotes the unit tangential vector to ∂M, we have

∇ g η, ∇ g w g = (∇η • ν)∂ ν ω, hence ∂M ∇ g η, ∇ g w g ∂ ν w dω g = ∂M (∇η • ν)|∂ ν w| 2 dω g = -γ ∂M ψ(∇ϑ • ν)|∂ ν w| 2 dω g . (4.59) Moreover it holds true that ∇ g w, ∇ g ∇ g η, ∇ g w g g = -γ ∇ g w, ∇ g (ψ ∇ g ϑ, ∇ g w g ) g = -γψ ∇ g w, ∇ g ∇ g ϑ, ∇ g w g g -γ 2 ψ| ∇ g ϑ, ∇ g w g | 2 .
(4.60)

Further, as ∇ g w, ∇ g ∇ g ϑ, ∇ g w g g = ∇ g w( ∇ g ϑ, ∇ g w g ), we have

∇ g w, ∇ g ∇ g ϑ, ∇ g w g g = D ∇gw ∇ g ϑ, ∇ g w g + ∇ g ϑ, D ∇gw ∇ g w g = D 2 ϑ(∇ g w, ∇ g w) + D 2 w(∇ g w, ∇ g ϑ), so (4.58)-(4.60) yields I 2,2 = 2sγ M T ψD 2 ϑ(∇ g w, ∇ g w) dv g dt + 2sγRe M T ψD 2 w(∇ g w, ∇ g ϑ) dv g dt +2sγ 2 M T ψ| ∇ g ϑ, ∇ g w g | 2 dv g dt -2sγ ∂M T ψ(∇ϑ • ν)|∂ ν w| 2 dω g dt.(4.61)
The next step involves noticing that ∇ g η, ∇ g ∇ g w, ∇ g w g g = ∇ g η( ∇ g w, ∇ g w g ) is equal to D ∇gη ∇ g w, ∇ g w g + ∇ g w, D ∇gη ∇ g w g , in such a way that we have From this and the two identities div g (|∇ g η| 2 g ∇ g η) = |∇ g η| 2 g ∆ g η + ∇ g η, ∇ g |∇ g η| 2 g g and ∇ g η, ∇ g |∇ g η| 2 g g = -γ 3 ψ 3 (2γ|∇ g ϑ| 2 g + ∇ g ϑ, ∇ g |∇ g ϑ| 2 g g ) then follows that

∇ g η, ∇ g ∇ g w, ∇ g w g g = D 2 w(∇ g η, ∇ g w) + D 2 w(∇ g η, ∇ g w) = -γψ D 2 w(∇ g ϑ, ∇ g w) + D 2 w(∇ g ϑ, ∇ g w) = -2γψRe D 2 w(∇ g ϑ, ∇ g w) ,
I 3,2 = s 3 γ 3 M T ψ 3 (

M

  ψD 2 w(∇ g w, ∇ g ϑ) dv g = -M ∇ g η, ∇ g ∇ g w, ∇ g w g g dv g = -M (∆ g η) ∇ g w, ∇ g w g dv g + ∂M (∇η • ν)|∂ ν ω| 2 dω g = -γ M ψ(∆ g ϑ + γ|∇ g ϑ| 2 g ) ∇ g w, ∇ g w g dv g + γ ∂M ψ(∇ϑ • ν)|∂ ν ω| 2 dω g ,by recalling that ∇ g w • τ = 0 on ∂M. From this and (4.61) then follows thatI 2,2 = 2sγ

M T |∇ g η| 2 g η ′ |w| 2

 2 dv g dt = 0, (4.66) being easily treated, we turn our attention to the computation ofI 3,2 := 2s 3 Re M T |∇ g η| 2 g ∇ g η, ∇ g w g w dv g dt = M T |∇ g η| 2 g ∇ g η, ∇ g |w| 2 g dv g dt = -M T div g (|∇ g η| 2 g ∇ g η)|w| 2 dv g dt.

  1.1 and 1.2 is based on a global Carleman estimate for the operator H θ. A Carleman inequality for the dynamic Schrödinger operator defined in a compact manifold M ⊂ R n , n ≥ 2, with smooth metric g, is established in Proposition 2.1 in §2. Its proof being quite lengthy it is postponed to Appendix A. The inverse problem of determining the twisting function (actually the second derivative of the twisting function) is addressed in §3, which contains the proofs of Theorems 1.1-1.2.

  M T ψD 2 ϑ(∇ g w, ∇ g w) dv g dtsγ M T ψ(∆ g ϑ + γ|∇ g ϑ| 2 g ) ∇ g w, ∇ g w g dv g dt +2sγ 2 M T ψ| ∇ g ϑ, ∇ g w g | 2 dv g dtsγ ∂M T ψ(∇ϑ • ν)|∂ ν w| 2 dω g dt. (4.62)and then applying (4.49), we find out that ReM T ψ ∇ g (∆ g ϑ + γ|∇ g ϑ| 2 g ), ∇ g w g w dv g dt = -1 2 M T div g (ψ∇ g (∆ g ϑ + γ|∇ g ϑ| 2 g ))|w| 2 dv g dt = -1 2 M T ψ ∆ g (∆ g ϑ + γ|∇ g ϑ| 2 g ) + γ ∇ g ϑ, ∇ g (∆ g ϑ + γ|∇ g ϑ| 2 g ) g |w| 2 dv g dt. (∆ g ϑ + γ|∇ g ϑ| 2 g ) ∇ g w, ∇ g w g dv g dt sγ 2 M T ψ ∆ g (∆ g ϑ + γ|∇ g ϑ| 2 g ) + 2γ ∇ g ϑ, ∇ g (∆ g ϑ + γ|∇ g ϑ| 2 g ) g +γ(∆ g ϑ + γ|∇ g ϑ| 2 g ) 2 |w| 2 dv g dt. (4.65)Further, the case of I 3,1 := s 3 Re

	Along with (4.63)-(4.64), this yields
	I 2,3 = sγ
	M T

ψM T |∇ g η| 2 g w(-ıη ′ w) dv g dt = Im

  3γ|∇ g ϑ| 4 g + ∇ g ϑ, ∇ g |∇ g ϑ| 2 g g + |∇ g ϑ| 2 g ∆ g ϑ) dv g dt. |∇ g η| 2 g (∆ g η)|w| 2 dv g dt = -s 3 γ 3 M T ψ 3 (γ|∇ϑ| 4 g + |∇ϑ| 2 g ∆ g ϑ)|w| 2 dv g dt, (4.68) by straighforward computations. Now, putting (4.51)-(4.52), (4.57), (4.62) and (4.65)-(4.68) together, we have obtained that

	(4.67)
	Finally, we get
	I 3,3 := s 2
	M T

Here the various physical constants are taken equal to 1.

Let us now consider

The strategy now is to re-express each of the two last terms in the rhs of (4.63). To do that we substitute (1/2) ∇ g ϑ, ∇ g |w| 2 g for Re ( ∇ g ϑ, ∇ g w g w) in the last integral and apply (4.49), getting Re

since w = 0 on ∂M. In light of (4.48) this entails


Michel Cristofol

Patricia Gaitan
email: gaitan@cmi.univ-mrs.fr

Hichem Ramoul
email: ramoul2477@yahoo.fr

Inverse problems for a two by two reaction-diffusion system using a Carleman estimate with one observation

Keywords: Inverse problem, Reaction-diffusion system, Carleman estimate AMS classification scheme numbers: 35K05, 35K57, 35R30

For a two by two reaction-diffusion system on a bounded domain we give a simultaneous stability result for one coefficient and for the initial conditions. The key ingredient is a global Carleman-type estimate with a single observation acting on a subdomain.

Introduction

This paper is devoted to the simultaneous identification of one coefficient and the initial conditions in a reaction-diffusion system using the least number of observations as possible.

Let Ω ⊂ R n be a bounded domain of R n with n ≤ 3. We denote by n the outward unit normal to Ω on Γ = ∂Ω assumed to be of class C 1 . Let T > 0 and t 0 ∈ (0, T ). We shall use the following notations Q 0 = Ω × (0, T ), Q = Ω × (t 0 , T ), Σ = Γ × (t 0 , T ) and Σ 0 = Γ × (0, T ). We consider the following reaction-diffusion system which arises for instance in mathematical biology:

         ∂ t u = ∆u + a(x)u + b(x)v in Q 0 , ∂ t v = ∆v + c(x)u + d(x)v
in Q 0 , u(t, x) = g(t, x), v(t, x) = h(t, x) on Σ 0 , u(0, x) = u 0 and v(0, x) = v 0 in Ω,

Throughout this paper, let us consider the following set

Λ(R) = {Φ ∈ L ∞ (Ω); Φ L ∞ (Ω) R},
where R is a given positive constant.

If we assume that (u 0 , v 0 ) belongs to (H 2 (Ω)) 2 and g, h are sufficiently regular (e.g. ∃ ǫ > 0 such that g, h ∈ H 1 (t 0 , T, H 2+ε (∂Ω)) ∩ H 2 (t 0 , T, H ε (∂Ω))), then (1) admits a solution in H 1 (t 0 , T, H 2 (Ω)) (see [START_REF] Lions | Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles[END_REF]). We will later use this regularity result. We also assume that

        
a, b, c, d ∈ Λ(R), There exist r > 0, c 0 > 0 such that u 0 ≥ 0, v 0 ≥ r, c ≥ c 0 , b > 0, c + dr ≥ 0, g ≥ 0 and h ≥ r.

Let ω be a subdomain of Ω. Let (u, v) (resp. ( u, v)) be solution of (1) associated to (a, b, c, d, u 0 , v 0 ) (resp. (a, b, c, d, u 0 , v 0 )) satisfying some regularity and "positivity" properties. We assume that we can measure ∂ t v on ω in the time interval (t 0 , T ) for some t 0 ∈ (0, T ) and ∆u, u and v in Ω at time T ′ ∈ (t 0 , T ). Our main results are

• A stability result for the coefficient b(x) (or a(x)): For u 0 , v 0 in H 2 (Ω) there exists a constant

C = C(Ω, ω, c 0 , t 0 , T, r, R) > 0 such that |b -b| 2 L 2 (Ω) ≤ C|∂ t v -∂ t v| 2 L 2 ((t 0 ,T )×ω) + C|∆u(T ′ , •) -∆ u(T ′ , •)| 2 L 2 (Ω) +C|u(T ′ , •) -u(T ′ , •)| 2 L 2 (Ω) + C|v(T ′ , •) -v(T ′ , •)| 2 L 2 (Ω) .
• A stability estimate for the initial conditions u 0 , v 0 : For u 0 , v 0 , u 0 , v 0 in H 4 (Ω) there exists a constant

C = C(Ω, ω, c 0 , t 0 , T, r, R) > 0 such that |u 0 -u 0 | 2 L 2 (Ω) + |v 0 -v 0 | 2 L 2 (Ω) ≤ C | log E| , where E = |∂ t v -∂ t v| 2 L 2 ((t 0 ,T )×ω) + |u(T ′ , •) -u(T ′ , •)| 2 H 2 (Ω) + |v(T ′ , •) -v(T ′ , •)| 2 H 2 (Ω) .
The key ingredient to these stability results is a global Carleman estimate for a two by two system with one observation. Controllability for such parabolic systems has been studied in [START_REF] Khodja | Null controllability of some systems of parabolic type by one control force[END_REF]. The Carleman estimate obtained in [START_REF] Khodja | Null controllability of some systems of parabolic type by one control force[END_REF] cannot be used to solve the inverse problem of identification of one coefficient and initial conditions because of the weight functions which are different in the left and right hand side of their estimate. We establish a new Carleman estimate with one observation involving the same weight function in the left and right hand side. Concerning the stability of the initial conditions we use an extension of the logarithmic convexity method (see [START_REF] Isakov | Inverse problems for partial differential equations[END_REF]). The simultaneous reconstruction of one coefficient and initial conditions from the measurement of one solution v over (t 0 , T ) × ω and some measurement at fixed time T ′ is an essential aspect of our result. In the perspective of numerical reconstruction, such problems are ill-posed. Stability results are thus of importance. Inverse problems for parabolic equations are well studied (see [START_REF] Cannon | Structural identification of an unknown source term in a heat equation[END_REF], [START_REF] Yu | Lipschitz stability in inverse problems by Carleman estimates[END_REF], [START_REF] Yamamoto | Simultaneous reconstruction of the initial temperature and heat radiative coefficient[END_REF]). A recent book of Klibanov and Timonov [START_REF] Klibanov | Carleman estimates for coefficient inverse problems and numerical applications[END_REF] is devoted to the Carleman estimates applied to inverse coefficient problems. In our knowledge, there is no work about inverse problems for coupled parabolic systems. The used method allows us to give a stability result for the coefficient a(x) adapting assumption 3.1. On the other hand, since we only measure ∂ t v on ω, we cannot obtain such stability results for the coefficients c(x) or d(x) of the second equation of [START_REF] Khodja | Null controllability of some systems of parabolic type by one control force[END_REF]. For the reconstruction of two coefficients the problem is more complicated. We obtain partial results with restrictive assumptions on the coefficients a(x), b(x), c(x) and d(x). In order to avoid such assumptions, we think it is necessary to use other methods such as those used in [START_REF] Yu | An Inverse Problem for the Dynamical Lamé system with two set of boundary data[END_REF]. Our paper is organized as follows. In Section 2, we derive a global Carleman estimate for system (1) with one observation, i.e. the measurement of one solution v over (t 0 , T ) × ω. In Section 3, we prove a stability result for the coefficient b(x) when one of the solutions v is in a particular class of solutions with some regularity and "positivity" properties. In Section 4, we prove a stability result for the initial conditions.

Carleman estimate

We prove here a Carleman-type estimate with a single observation acting on a subdomain ω of Ω in the right-hand side of the estimate. Let us introduce the following notations: let ω ′ ⋐ ω and let β be a C 2 (Ω) function such that

β > 0, in Ω, β = 0 on ∂Ω, min{|∇ β(x)|, x ∈ Ω \ ω ′ } > 0 and ∂ n β < 0 on ∂Ω.
Then, we define β = β + K with K = m β ∞ and m > 1. For λ > 0 and t ∈ (t 0 , T ), we define the following weight functions

ϕ(x, t) = e λβ(x) (t -t 0 )(T -t) , η(x, t) = e 2λK -e λβ(x) (t -t 0 )(T -t) .
If we set ψ = e -sη q, we also introduce the following operators

M 1 ψ = -∆ψ -s 2 λ 2 |∇β| 2 ϕ 2 ψ + s(∂ t η)ψ, M 2 ψ = ∂ t ψ + 2sλϕ∇β.∇ψ + 2sλ 2 ϕ|∇β| 2 ψ.
Then the following result holds (see [START_REF] Fernández-Cara | Global Carleman inequalities for parabolic systems and applications to controllability[END_REF]).

Theorem 2.1 There exist λ 0 = λ 0 (Ω, ω) ≥ 1, s 0 = s 0 (λ 0 , T ) > 1 and a positive constant C 0 = C 0 (Ω, ω, T ) such that, for any λ ≥ λ 0 and any s ≥ s 0 , the next inequality holds:

M 1 (e -sη q) 2 L 2 (Q) + M 2 (e -sη q) 2 L 2 (Q) (2)
+sλ 2 Q e -2sη ϕ|∇q| 2 dx dt + s 3 λ 4 Q e -2sη ϕ 3 |q| 2 dx dt ≤ C 0 s 3 λ 4 T t 0 ω e -2sη ϕ 3 |q| 2 dx dt + Q e -2sη |∂ t q -∆q| 2 dx dt ,
for all q ∈ H 1 (t 0 , T, H 2 (Ω)) with q = 0 on Σ.

From the above theorem we have also the following result (see [START_REF] Fernández-Cara | Global Carleman inequalities for parabolic systems and applications to controllability[END_REF] and [START_REF] Fursikov | Optimal control of distribued systems[END_REF]).

Proposition 2.2 There exist λ 0 = λ 0 (Ω, ω) ≥ 1, s 0 = s 0 (λ 0 , T ) > 1 and a positive constant C 0 = C 0 (Ω, ω, T ) such that, for any λ ≥ λ 0 and any s ≥ s 0 , the next inequality holds:

s -1 Q e -2sη ϕ -1 (|∂ t q| 2 + |∆q| 2 ) dx dt (3) +sλ 2 Q e -2sη ϕ|∇q| 2 dx dt + s 3 λ 4 Q e -2sη ϕ 3 |q| 2 dx dt ≤ C 0 s 3 λ 4 T t 0 ω e -2sη ϕ 3 |q| 2 dx dt + Q e -2sη |∂ t q -∆q| 2 dx dt ,
for all q ∈ H 1 (t 0 , T, H 2 (Ω)) with q = 0 on Σ.

We consider the solutions (u, v) and ( u, v) to the following systems

         ∂ t u = ∆u + au + bv in Q 0 , ∂ t v = ∆v + cu + dv in Q 0 , u(t, x) = g(t, x), v(t, x) = h(t, x) on Σ 0 , u(0, x) = u 0 and v(0, x) = v 0 in Ω, (4)
and

         ∂ t u = ∆ u + a u + b v in Q 0 , ∂ t v = ∆ v + c u + d v in Q 0 , u(t, x) = g(t, x), v(t, x) = h(t, x) on Σ 0 , u(0, x) = u 0 and v(0, x) = v 0 in Ω. (5)
We set

U = u -u, V = v -v, y = ∂ t (u -u), z = ∂ t (v -v) and γ = b -b. Then (y, z) is solution to the following problem              ∂ t y = ∆y + ay + bz + γ∂ t v in Q 0 , ∂ t z = ∆z + cy + dz in Q 0 , y(t, x) = z(t, x) = 0 on Σ 0 , y(0, x) = ∆U(0, x) + aU(0, x) + bV (0, x) + γ v(0, x), in Ω, z(0, x) = ∆V (0, x) + cU(0, x) + dV (0, x) in Ω. (6)
Note that the previous initial conditions are available for all T ′ ∈ (0, T ). We consider the functional

I(q) = s -1 Q e -2sη ϕ -1 (|∂ t q| 2 + |∆q| 2 ) dx dt +sλ 2 Q e -2sη ϕ|∇q| 2 dx dt + s 3 λ 4 Q e -2sη ϕ 3 |q| 2 dx dt.
Then using the Carleman estimate (3), the solution (y, z) of ( 6) satisfies

I(y) + I(z) ≤ C 1 [s 3 λ 4 T t 0 ω e -2sη ϕ 3 |z| 2 dx dt (7) +s 3 λ 4 T t 0 ω e -2sη ϕ 3 |y| 2 dx dt + Q e -2sη (|ay| 2 + |bz| 2 + |γ∂ t v| 2 ) dx dt + Q e -2sη (|cy| 2 + |dz| 2 ) dx dt]
Let ξ be a smooth cut-off function satisfying

     ξ(x) = 1 ∀x ∈ ω ′ , 0 < ξ(x) ≤ 1 ∀x ∈ ω ′′ , ξ(x) = 0 ∀x ∈ R n \ ω ′′ ,
where

ω ′ ⋐ ω ′′ ⋐ ω ⋐ Ω.
We shall to estimate the following three terms

I := s 3 λ 4 T t 0 ω e -2sη ϕ 3 |y| 2 dx dt, J := Q e -2sη |bz| 2 dx dt or Q e -2sη |dz| 2 dx dt, K := Q e -2sη |ay| 2 dx dt or Q e -2sη |cy| 2 dx dt.
For the first term I, we multiply the second equation of ( 6) by s 3 λ 4 e -2sη ξϕ 3 y and we integrate over (t 0 , T ) × ω. We obtain

I ′ := s 3 λ 4 T t 0 ω c e -2sη ξϕ 3 |y| 2 dx dt = s 3 λ 4 T t 0 ω e -2sη ξϕ 3 (∂ t z -∆z -dz)y dx dt = s 3 λ 4 T t 0 ω e -2sη ξϕ 3 (∂ t z)y dx dt -s 3 λ 4 T t 0 ω e -2sη ξϕ 3 (∆z)y dx dt -s 3 λ 4 T t 0 ω de -2sη ξϕ 3 zy dx dt = I 1 + I 2 + I 3 .
By integration by parts with respect to the time variable, the first integral, I 1 , can be written as

I 1 = -s 3 λ 4 T t 0 ω e -2sη ξϕ 3 z(∂ t y) dx dt + 2s 4 λ 4 T t 0 ω e -2sη ξϕ 3 (∂ t η) zy dx dt -3s 3 λ 4 T t 0 ω e -2sη ξϕ 2 (∂ t ϕ) zy dx dt.
We write

I 1 = I 1 1 + I 2 1 with I 1 1 = -s 3 λ 4 T t 0 ω e -2sη ξϕ 3 z(∂ t y) dx dt, I 2 1 = 2s 4 λ 4 T t 0 ω e -2sη ξϕ 3 (∂ t η) zy dx dt -3s 3 λ 4 T t 0 ω
e -2sη ξϕ 2 (∂ t ϕ) zy dx dt.

Using Young inequality, we estimate the two integrals I 1 1 and I 2 1 . We have

|I 1 1 | ≤ s 3 λ 4 C ε s 4 λ 4 T t 0 ω e -2sη ξ 2 ϕ 7 |z| 2 dx dt + εs -4 λ -4 T t 0 ω e -2sη ϕ -1 |∂ t y| 2 dx dt ≤ C ε s 7 λ 8 T t 0 ω e -2sη ϕ 7 |z| 2 dx dt + εs -1 Q e -2sη ϕ -1 |∂ t y| 2 dx dt.
The last term of the previous inequality can be "absorbed" by the terms in I(y) for ε sufficiently small.

|I 2 1 | ≤ Cs 4 λ 4 sλ T t 0 ω e -2sη ξ 2 ϕ(ϕ 2 |∂ t η| 2 + |∂ t ϕ| 2 )|z| 2 dx dt +s -1 λ -1 T t 0 ω e -2sη ϕ 3 |y| 2 dx dt ≤ C s 5 λ 5 T t 0 ω e -2sη ϕ 7 |z| 2 dx dt + s 3 λ 3 Q e -2sη ϕ 3 |y| 2 dx dt .
The last inequality holds through the following estimates

|∂ t ϕ| ≤ C(Ω, ω)T ϕ 2 , |∂ t η| ≤ C(Ω, ω)T ϕ 2 , ϕ ≤ C(Ω, ω)T 4 ϕ 3 .
The last term of the previous inequality can be "absorbed" by the terms in I(y) for s and λ sufficiently large. Finally, we obtain

|I 1 | ≤ Cs 7 λ 8 T t 0 ω e -2sη ϕ 7 |z| 2 dx dt + "absorbed terms" ,
where C is a generic constant which depends on Ω, ω and T .

Integrating by parts the second integral I 2 with respect to the space variable, we obtain

I 2 = -s 3 λ 4 T t 0 ω
∆(e -2sη ξϕ 3 y)z dx dt.

If we denote by P = e -2sη ξϕ 3 , then we have

I 2 = -s 3 λ 4 T t 0 ω (P ∆y + 2∇P ∇y + y∆P )z dx dt.
We compute ∇P and ∆P and we obtain the following estimation for I 2

|I 2 | ≤ s 3 λ 4 εs -4 λ -4 T t 0 ω e -2sη ϕ -1 |∆y| 2 dx dt + C ε s 4 λ 4 T t 0 ω e -2sη ϕ 7 |z| 2 dx dt +εs -2 λ -2 T t 0 ω e -2sη ϕ|∇y| 2 dx dt + C ε s 2 λ 2 T t 0 ω e -2sη ϕ 5 |z| 2 dx dt +ε T t 0 ω e -2sη ϕ 3 |y| 2 dx dt + C ε T t 0 ω e -2sη ϕ 3 |z| 2 dx dt .
Therefore we obtain

|I 2 | ≤ ε s -1 Q e -2sη ϕ -1 |∆y| 2 dx dt + sλ 2 Q e -2sη ϕ|∇y| 2 dx dt +s 3 λ 4 Q e -2sη ϕ 3 |y| 2 dx dt + C ε s 7 λ 8 T t 0 ω e -2sη ϕ 7 |z| 2 dx dt +s 5 λ 6 T t 0 ω e -2sη ϕ 5 |z| 2 dx dt + s 3 λ 4 T t 0 ω e -2sη ϕ 3 |z| 2 dx dt .
The first three integrals of the r.h.s. of the previous inequality can be "absorbed" by the terms in I(y) for ε sufficiently small. Finally, we have

|I 2 | ≤ Cs 7 λ 8 T t 0 ω
e -2sη ϕ 7 |z| 2 dx dt + "absorbed terms" .

For the last integral I 3 , we have

|I 3 | ≤ Cs 3 λ 4 C ε T t 0 ω e -2sη ϕ 3 |z| 2 dx dt + ε Q e -2sη ϕ 3 |y| 2 dx dt
Finally, if we assume that there exists c 0 > 0 such that c ≥ c 0 in ω, we have thus obtained for λ and s sufficiently large and ε sufficiently small the following estimate:

|I| ≤ 1 c 0 |I ′ | ≤ C c 0 s 7 λ 8 T t 0 ω e -2sη ϕ 7 |z| 2 dx dt.
For the integrals J and K, since a, b, c, d ∈ Λ(R) and using the estimate

1 ≤ C(Ω, ω)T 6 ϕ 3 /4, we have |J| ≤ C Q e -2sη ϕ 3 |z| 2 dx dt, |K| ≤ C Q e -2sη ϕ 3 |y| 2 dx dt,
and these terms can be "absorbed" by the terms I(y) and I(z) for λ and s sufficiently large. If we now come back to inequality (7), using the estimates for I, J and K, and choosing λ and s sufficiently large and ε sufficiently small, we can thus write

I(y) + I(z) ≤ C 1 [s 3 λ 4 T t 0 ω e -2sη ϕ 3 |z| 2 dx dt + s 7 λ 8 T t 0 ω e -2sη ϕ 7 |z| 2 dx dt + Q e -2sη |γ∂ t v| 2 dx dt].
Observing that

ϕ 3 ≤ C(Ω, ω)T 8 ϕ 7 ,
We have thus obtained the fundamental result Theorem 2.3 We assume a, b, c, d ∈ Λ(R) and that exists c 0 > 0 such that c ≥ c 0 in ω. Then there exist

λ 1 = λ 1 (Ω, ω) ≥ 1, s 1 = s 1 (λ 1 , T ) > 1 and a positive constant C 1 = C 1 (Ω, ω, c 0 , R, T
) such that, for any λ ≥ λ 1 and any s ≥ s 1 , the following inequality holds:

I(y)+I(z) ≤ C 1 s 7 λ 8 T t 0 ω e -2sη ϕ 7 |z| 2 dx dt + Q e -2sη |γ∂ t v| 2 dx dt ,( 8
)
for any solution (y, z) of (6).

Uniqueness and stability estimate with one observation

In this section, we establish, a stability inequality and deduce a uniqueness result for the coefficient b. This inequality [START_REF] Yamamoto | Simultaneous reconstruction of the initial temperature and heat radiative coefficient[END_REF] estimates the difference between the coefficients b and b with an upper bound given by some Sobolev norms of the difference between the solutions v, and v of ( 4) and [START_REF] Fernández-Cara | Global Carleman inequalities for parabolic systems and applications to controllability[END_REF]. Recall that

U = u -u, V = v -v, y = ∂ t (u -u), z = ∂ t (v -v), γ = b -b and              ∂ t y = ∆y + ay + bz + γ∂ t v in Q 0 , ∂ t z = ∆z + cy + dz in Q 0 , y(t, x) = z(t, x) = 0
on Σ 0 , y(0, x) = ∆U(0, x) + aU(0, x) + bV (0, x) + γ v(0, x), in Ω, z(0, x) = ∆V (0, x) + cU(0, x) + dV (0, x)

in Ω.

The Carleman estimate (8) proved in the previous section will be the key ingredient in the proof of such a stability estimate.

Let T ′ = 1 2 (T + t 0 ) the point for which Φ(t) = 1 (t-t 0 )(T -t) has its minimum value. For ( u, v) solutions of (5), we make the following assumption: Assumption 3.1 There exist r > 0, c 0 > 0 such that b ≥ 0, c ≥ c 0 , c + dr ≥ 0, u 0 ≥ 0, v 0 ≥ r, g ≥ 0 and h ≥ r.

Such assumption allows us to state that the solution v is such that | v(x, T ′ )| ≥ r > 0 in Ω (see [START_REF] Smoller | Shock waves and reaction-diffusion equations[END_REF], theorem 14.7 p.200). Furthermore if we assume that u 0 , v 0 in H 2 (Ω),the solutions of ( 5) belong to H 1 (t 0 , T, H 2 (Ω)). Then using classical Sobolev imbedding (see [START_REF] Brezis | Analyse fonctionnelle[END_REF]), we can write for n ≤ 3, that ∂ t v belongs to L 2 (t 0 , T, L ∞ (Ω)) and we assume that

|∂ t v| L 2 (t 0 ,T ) ∈ Λ(R).
We set ψ = e -sη y. With the operator

M 2 ψ = ∂ t ψ + 2sλϕ∇β.∇ψ + 2sλ 2 ϕ|∇β| 2 ψ, (9)
we introduce, following [START_REF] Baudouin | Uniqueness and stability in an inverse problem for the Schrödinger equation[END_REF],

I = ℜ T ′ t 0 Ω M 2 ψ ψ dxdt
We have the following estimates.

Proof:

Observe that

|I| ≤ s -3/2 λ -2 T ′ t 0 Ω |M 2 ψ| 2 dx dt 1/2 s 3 λ 4 T ′ t 0 Ω e -2sη |y| 2 dx dt 1/2
, thus using Young inequality and the estimate 1 ≤ C ′ T 6 ϕ 3 , we obtain

|I| ≤ Cs -3/2 λ -2 |M 2 ψ| 2 L 2 (Q) + s 3 λ 4 T t 0 Ω e -2sη ϕ 3 |y| 2 dxdt ,
which yields the result from Carleman estimate [START_REF] Yu | Lipschitz stability in inverse problems by Carleman estimates[END_REF].

Lemma 3.3 Let λ ≥ λ 1 , s ≥ s 1 and let a, b, c, d ∈ Λ(R)
. Furthermore, we assume that u 0 , v 0 in H 2 (Ω) and the assumption 3.1 is satisfied. Then there exists a constant

C = C(Ω, ω, T ) such that Ω e -2sη(T ′ ,x) |γ v(T ′ , x)| 2 dx (10) ≤ Cs -3/2 λ -2 s 7 λ 8 T t 0 ω e -2sη ϕ 7 |z| 2 dx dt + T t 0 Ω e -2sη |γ| 2 |∂ t v| 2 dx dt +C Ω e -2sη(T ′ ,x) |∆ U(T ′ , x) + aU(T ′ , x) + bV (T ′ , x)| 2 dx.

Proof:

We evaluate integral I using ( 9)

I = 1 2 T ′ t 0 Ω ∂ t |ψ| 2 dxdt + sλ T ′ t 0 Ω ϕ∇β • ∇|ψ| 2 dxdt + 2sλ 2 T ′ t 0 Ω ϕ|∇β| 2 |ψ| 2 dxdt = 1 2 T ′ t 0 Ω ∂ t |ψ| 2 dxdt -sλ T ′ t 0 Ω ∇ • (ϕ∇β)|ψ| 2 dxdt + 2sλ 2 T ′ t 0 Ω ϕ|∇β| 2 |ψ| 2 dxdt,
by integration by parts. With an integration by parts w.r.t. t in the first integral, we then obtain

1 2 Ω |ψ(T ′ , .)| 2 dx = I -sλ 2 T ′ t 0 Ω ϕ|∇β| 2 |ψ| 2 dxdt + sλ T ′ t 0 Ω ϕ(∆β)|ψ| 2 dxdt
since ψ(t 0 ) = 0 and ∇ϕ = λϕ∇β. Then, we have

Ω (e -2sη)(T ′ ,x) |y(T ′ , x)| 2 dx ≤ 2|I|+Csλ(λ+1) T ′ t 0 Ω e -2sη(t,x) ϕ|y| 2 dxdt.( 11
)
Using ϕ ≤ T 4 4 ϕ 3 the last term in ( 11) is overestimated by the left hand side of ( 8) and this last one is absorbed by the l.h.s. of the inequality obtained in lemma 3.2. If we now observe that

y(T ′ , x) = ∆U(T ′ , x) + aU(T ′ , x) + bV (T ′ , x) + γ v(T ′ , x),
we have

|y(T ′ , x)| 2 ≥ 1 2 |γ v(T ′ , x)| 2 -|∆U(T ′ , x) + aU(T ′ , x) + bV (T ′ , x)| 2 .
The regularity of the solutions of (5) allows us to write that for n ≤ 3,

∂ t v is an element of L 2 (t 0 , T, L ∞ (Ω)). So, from | v(x, T ′ )| ≥ r > 0, we have ∃ k ∈ L 2 (t 0 , T ), |∂ t v(x, t)| ≤ k(t)| v(x, T ′ )|, ∀x ∈ Ω, t ∈ (t 0 , T ).
Hence [START_REF] Lions | Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles[END_REF] can be written

Ω (e -2sη )(T ′ , x)|γ| 2 | v(x, T ′ )| 2 dx ≤ Cs -3/2 λ -2 T t 0 Ω e -2sη |γ| 2 |k(t)| 2 | v(x, T ′ )| 2 dx dt +Cs 11/2 λ 6 T t 0 ω e -2sη ϕ 7 |z| 2 dx dt +C Ω (e -2sη)(T ′ ,x) (|∆U(T ′ , x)| 2 + |U(T ′ , x)| 2 + |V (T ′ , x)| 2 ) dx. Since k ∈ L 2 (t 0 , T ) implies that T t 0 |k(t)| 2 dt ≤ k 0 < +∞.
For λ large enough, the term (1 -Cs -3/2 λ -2 k 0 ) can be made positive:

1 -Cs -3/2 λ -2 k 0 ≥ C 2 > 0.
Using the fact that e -2sη(t,x) ≤ e -2sη(T ′ ,x) ∀x ∈ Ω, ∀t ∈ (t 0 , T ), we deduce that

r 2 (1 -Cs -3/2 λ -2 k 0 ) Ω (e -2sη )(T ′ , x)|γ| 2 dx ≤ Cs 11/2 λ 6 T t 0 ω e -2sη ϕ 7 |z| 2 dx dt +C Ω (e -2sη)(T ′ ,x) (|∆U(T ′ , x)| 2 + |U(T ′ , x)| 2 + |V (T ′ , x)| 2 ) dx,
where we have also used that | v(x, T ′ )| ≥ r > 0 in Ω. Then, by virtue of the properties satisfied by ϕ and η, we finally obtain

|γ| 2 L 2 (Ω) ≤ C r 2 C 2 s 11/2 λ 6 T t 0 ω |z| 2 dx dt (12)
+ C r 2 C 2 Ω (|∆U(T ′ , x)| 2 + |U(T ′ , x)| 2 + |V (T ′ , x)| 2 ) dx.
With [START_REF] Smoller | Shock waves and reaction-diffusion equations[END_REF], recalling that Furthermore, we assume that u 0 , v 0 in H 2 (Ω) and the assumption 3.1 is satisfied. Let (u, v), ( u, v) be solutions to ( 4)- [START_REF] Fernández-Cara | Global Carleman inequalities for parabolic systems and applications to controllability[END_REF]. Then there exists a constant C

U = u -u, V = v -v, y = ∂ t (u -u) and z = ∂ t (v -v),
C = C(Ω, ω, c 0 , t 0 , T, r, R) > 0 such that |b -b| 2 L 2 (Ω) ≤ C|∂ t v -∂ t v| 2 L 2 ((t 0 ,T )×ω) + C|∆u(T ′ , •) -∆ u(T ′ , •)| 2 L 2 (Ω) (13) +C|u(T ′ , •) -u(T ′ , •)| 2 L 2 (Ω) + C|v(T ′ , •) -v(T ′ , •)| 2 L 2 (Ω) .
Remark 3.5 If we assume that u(T ′ , •) = u(T ′ , •) and v(T ′ , •) = v(T ′ , •) (such an additional assumption is sometimes made, e.g. in [START_REF] Yu | Lipschitz stability in inverse problems by Carleman estimates[END_REF]), then the stability estimate becomes

|b -b| 2 L 2 (Ω) ≤ C|∂ t v -∂ t v| 2 L 2 ((t 0 ,T )×ω) .
With Theorem 3.4 we have the following uniqueness result Corollary 3.6 Under the same assumptions as in theorem 3.4 and if

(∂ t v -∂ t v)(t, x) = 0 in (t 0 , T ) × ω, ∆u(T ′ , x) -∆ u(T ′ , x) = 0 in Ω, v(T ′ , x) -v(T ′ , x) = 0 in Ω, Then b = b.

A uniqueness and stability estimate for the initial conditions

In this section, we use the same method as in [START_REF] Yamamoto | Simultaneous reconstruction of the initial temperature and heat radiative coefficient[END_REF] to state a stability estimate for the initial conditions u 0 , v 0 . The idea is to prove logarithmic-convexity inequality. The following method has been used to obtain continuous dependence inequalities in initial value problems. If (y, z) is solution of (6), we introduce (y 1 , z 1 ) and (y 2 , z 2 ) that satisfy

             ∂ t y 1 = ∆y 1 + ay 1 + bz 1 + γ∂ t v in Q 0 , ∂ t z 1 = ∆z 1 + cy 1 + dz 1 in Q 0 , y 1 (t, x) = z 1 (t, x) = 0 on Σ 0 , y 1 (0, x) = 0, in Ω, z 1 (0, x) = 0 in Ω, (14)
and

             ∂ t y 2 = ∆y 2 + ay 2 + bz 2 in Q 0 , ∂ t z 2 = ∆z 2 + cy 2 + dz 2 in Q 0 , y 2 (t, x) = z 2 (t, x) = 0 on Σ 0 , y 2 (0, x) = ∆U(0, x) + aU(0, x) + bV (0, x) + γ v(0, x) in Ω, z 2 (0, x) = ∆V (0, x) + cU(0, x) + dV (0, x)
in Ω.

(15)

Then, we have

y = y 1 + y 2 and z = z 1 + z 2 . (16)
In a first step, we give an L 2 estimate for (y 1 , z 1 ) Lemma 4.1 Let a, b, c, d, |∂ t v| L 2 (t 0 ,T ) ∈ Λ(R). Then there exists a constant

C = C(t 0 , T ′ , R) > 0, such that |y 1 (t)| 2 L 2 (Ω) + |z 1 (t)| 2 L 2 (Ω) ≤ C|γ| 2 L 2 (Ω) , t 0 ≤ t ≤ T ′ . (17)
Proof:

We multiply the first (resp. the second) equation of ( 14) by y 1 (resp. by z 1 ). Then, after integrations by parts with respect to the space variable, we obtain

1 2 ∂ t Ω (|y 1 | 2 + |z 1 | 2 ) dx = - Ω (|∇y 1 | 2 + |∇z 1 | 2 ) dx + Ω ay 2 1 dx + Ω dz 2 1 dx + Ω (b + c)y 1 z 1 dx + Ω γ(∂ t v)y 1 dx.
We use Cauchy-Schwarz and Young inequalities and we integrate over (t 0 , t) for t 0 ≤ t ≤ T ′ , and we obtain

|y 1 (t)| 2 L 2 (Ω) + |z 1 (t)| 2 L 2 (Ω) ≤ C 2 |γ| 2 L 2 (Ω) + C 1 t t 0 (|y 1 (s)| 2 L 2 (Ω) + |z 1 (s)| 2 L 2 (Ω) ) ds.
The result follows by a Gronwall inequality.

In a second step, we use a logarithmic-convexity inequality for (y 2 , z 2 ) Lemma 4.2 Let a, b, c, d ∈ Λ(R) and u 0 , v 0 , u 0 , v 0 in H 4 (Ω). Then there exist constants M > 0, C = C(R) > 0 and

C 1 = C 1 (t 0 , T ′ , R) > 0 such that |y 2 (t)| 2 L 2 (Ω) + |z 2 (t)| 2 L 2 (Ω) ≤ C 1 M 1-µ(t) (|y 2 (T ′ )| 2 L 2 (Ω) + |z 2 (T ′ )| 2 L 2 (Ω) ) µ(t) ,( 18
)
for t 0 ≤ t ≤ T ′ , where µ(t) = (e -Ct 0 -e -Ct ) (e -Ct 0 -e -CT ′ ) .

Proof:

The proof of this lemma is just an application of Theorem 3.1.3 in [START_REF] Isakov | Inverse problems for partial differential equations[END_REF]. In fact, system (15) can be written in the following form

∂ t W + AW = BW, in Q 0 , W (t, x) = 0 on Σ 0 , W (0, x) = W 0 (x) in Ω, where W = y 2 z 2 , A = -∆ 0 0 -∆ , B = a b c d .
The operator A is symetric and the solution W satisfies

||∂ t W + AW || L 2 (Ω) ≤ α||W || L 2 (Ω) ,
where α = ||B|| L ∞ (Ω) < +∞ since a, b, c and d are in Λ(R). If we assume that u 0 , v 0 , u 0 , v 0 are in H 4 (Ω), the hypothesis of Theorem 3.1.3 in [START_REF] Isakov | Inverse problems for partial differential equations[END_REF] are satisfied, thus we have

||W (t)|| L 2 (Ω) ≤ C 1 ||W (t 0 )|| 1-µ(t) L 2 (Ω) ||W (T ′ )|| µ(t) L 2 (Ω) ,
with µ(t) = (e -Ct 0 -e -Ct ) (e -Ct 0 -e -CT ′ ) .

Since W ∈ C(t 0 , T, L 2 (Ω)), we have ||W (t 0 )|| L 2 (Ω) ≤ M Furthermore, we assume that u 0 , v 0 , u 0 , v 0 in H 4 (Ω) and Assumption 3.1 is satisfied. Let (u, v), ( u, v) be solutions to (4)- [START_REF] Fernández-Cara | Global Carleman inequalities for parabolic systems and applications to controllability[END_REF]. We set

E = |∂ t v -∂ t v| 2 L 2 ((t 0 ,T )×ω) + |u(T ′ , •) -u(T ′ , •)| 2 H 2 (Ω) + |v(T ′ , •) -v(T ′ , •)| 2 H 2 (Ω) .
Then there exists a constant C = C(Ω, ω, c 0 , t 0 , T, r, R) > 0 such that ). If we use [START_REF] Yamamoto | Simultaneous reconstruction of the initial temperature and heat radiative coefficient[END_REF], the last estimate yields

|u 0 -u 0 | 2 L 2 (Ω) + |v 0 -v 0 | 2 L 2 (Ω) ≤ C |
|u 0 -u 0 | 2 L 2 (Ω) + |v 0 -v 0 | 2 L 2 (Ω) = |U(t 0 , •)| 2 L 2 (Ω) + |V (t 0 , •)| 2 L 2 (Ω) = | - T ′ t 0 y(s, •) ds + U(T ′ , •)| 2 L 2 (Ω) + | - T ′ t 0 z(s, •) ds + V (T ′ , •)| 2 L 2 (Ω) ≤ M 3 T ′ t 0 E µ(s) ds + C 2 E ≤ C 3 E -1 log E + C 4 E ≤ C | log E| .

Lemma 3 . 2 s 7 λ 8 T t 0 ω e -2sη ϕ 7

 3287 Let λ ≥ λ 1 and s ≥ s 1 and let a, b, c, d ∈ Λ(R). We assume that assumption 3.1 is satisfied then there exists a constant C = C(Ω, ω, T ) such that|I| ≤ Cs -3/2 λ -2 |z| 2 dx dt + T t 0 Ω e -2sη |γ| 2 |∂ t v| 2 dx dt .

  we have thus obtained the following stability result. Theorem 3.4 Let ω be a subdomain of an open set Ω of R n , let a, b, c, d ∈ Λ(R).

1 2 , 4 . 3

 243 and the result follows. The two previous lemmas allow us to prove the following Theorem Let ω be a subdomain of an open set Ω of R n , let a, b, c, d ∈ Λ(R).

  Proof:Since v ∈ H 1 (t 0 , T, H 2 (Ω)), we have v ∈ L ∞ (Ω). In view of (16), inequalities (17), (18) imply|y(t, •)| 2 L 2 (Ω) ≤ 2(|y 1 (t, •)| 2 L 2 (Ω) + |y 2 (t, •)| 2 L 2 (Ω) ) ≤ C 1 |γ| 2 L 2 (Ω) + M 1-µ(t) (|y 2 (T ′ , •)| 2 L 2 (Ω) + |z 2 (T ′ , •)| 2 L 2 (Ω) ) µ(t). Now, with (16), we write y 2 = y -y 1 and z 2 = z -z 1 . Inequality (17) gives us an estimation of |y1 (T ′ , •)| 2 L 2 (Ω) and |z 1 (T ′ , •)| 2 L 2 (Ω) in terms of |γ| 2 L 2 (Ω). Then the definition of y and z in[START_REF] Fursikov | Optimal control of distribued systems[END_REF] gives us an estimation of|y(T ′ , •)| 2 L 2 (Ω) and |z(T ′ , •)| 2 L 2 (Ω) in terms of |U(T ′ , •)| 2 H 2 (Ω) and |V (T ′ , •)| 2 H 2 (Ω). Finally, we obtain|y(t, •)| 2 L 2 (Ω) ≤ C 1 |γ| 2 L 2 (Ω) + M 2 (|γ| 2 L 2 (Ω) + |U(T ′ , •)| 2 H 2 (Ω) + |V (T ′ , •)| 2 H 2 (Ω) ) µ(t) ,(a similar estimate is obtained for |z(t, •)| 2

log E| , for 0 < E < 1

Acknowledgments

The authors wish to thank A. Benabdallah, Y. Dermenjian, M. Henry, J. Le Rousseau and O. Poisson for numerous usefull discussions on the subject of this paper.