Uniqueness from pointwise observations in a multi-parameter inverse problem

In this paper, we prove a uniqueness result in the inverse problem of determining several non-constant coefficients of one-dimensional reaction-diffusion equations. Such reaction-diffusion equations include the classical model of Kolmogorov, Petrovsky and Piskunov as well as more sophisticated models from biology. When the reaction term contains an unknown polynomial part of degree $N,$ with non-constant coefficients $\mu_k(x),$ our result gives a sufficient condition for the uniqueness of the determination of this polynomial part. This sufficient condition only involves pointwise measurements of the solution $u$ of the reaction-diffusion equation and of its spatial derivative $\partial u / \partial x$ at a single point $x_0,$ during a time interval $(0,\epsilon).$ In addition to this uniqueness result, we give several counter-examples to uniqueness, which emphasize the optimality of our assumptions. Finally, in the particular cases N=2 and $N=3,$ we show that such pointwise measurements can allow an efficient numerical determination of the unknown polynomial reaction term.


Introduction
Reaction-diffusion equations arise as models in many fields of mathematical biology [20]. From morphogenesis [33] to population genetics [11,17] and spatial ecology [27,29,32], these partial differential equations benefit from a well-developed mathematical theory.
In the context of spatial ecology, single-species reaction-diffusion models generally deal with polynomial reaction terms. In a one-dimensional case, and if the environment is supposed to be homogeneous they take the form: ∂u ∂t − D ∂ 2 u ∂x 2 = P (u), (1.1) where u = u(t, x) is the population density at time t and space position x and D > 0 is the diffusion coefficient. The function P, which stands for the growth of the population, is a polynomial of order N ≥ 1. The Fisher-Kolmogorov, Petrovsky, Piskunov (F-KPP) equation is the archetype of such models. In this model, we have P (u) = µ u − γu 2 . The constant parameters µ and γ respectively correspond to the intrinsic growth rate and intraspecific competition coefficients. In this model, the lower the population density u, the higher the per capita growth rate P (u)/u. More complex models can involve polynomial nonlinearities of higher order. Examples are those taking account of an Allee effect. This effect occurs when the per capita growth rate P (u)/u reaches its maximum at a strictly positive population density and is known in many species [1,9,34]. A typical example of reaction term involving an Allee effect is [15,18,25]: with r > 0 and ρ ∈ (0, 1). The parameter ρ corresponds in that case to the "Allee threshold" below which the growth rate becomes negative.
In the previous examples, the reaction terms were assumed to be independent of the space variable. However, real world is far from begin homogeneous. In order to take the heterogeneities into account, models have been adapted and constant coefficients have been replaced by space or time dependant functions. In his pioneering work, Skellam [29] (and later, Shigesada, Kawasaki and Teramoto [28]) mentioned the following extension of the F-KPP model to heterogeneous environments: Here, the values of µ(x) and γ(x) depend on the position x. For instance, regions of the space associated with high values of µ(x) correspond to favorable regions, whereas those associated with low or negative values of µ(x) correspond to unfavorable regions. As emphasized by recent works, the precise spatial arrangement of these regions plays a crucial role in this model, since it controls persistence and spreading of the population [4,6,10,22,24,26,27]. Models involving an Allee effect can be extended as well to heterogeneous environments, as in [13,25], where the effects of spatial heterogeneities are discussed for models of the type: We also refer to [19] for an analysis of propagation phenomena related to a reaction-diffusion model with an Allee effect in infinite cylinders having undulating boundaries. In this paper, we focus on reaction-diffusion models with more general heterogeneous nonlinearities: Since the behavior of such models depends on the precise form of the coefficients, their empirical use requires an accurate knowledge of the coefficients. Unfortunately, in applications, the coefficients cannot be directly measured since they generally result from intertwined effects of several factors. Thus, the coefficients are generally measured through the density u(t, x) [30]. From a theoretical viewpoint, if u(t, x) is measured at any time t ≥ 0 and at all points x in the considered domain, all the coefficients in the model can generally be determined. However, in most cases, u(t, x) can only be measured in some -possibly small -subregions of the domain (a, b) [35]. For reaction-diffusion models as well as for many other types of models, the determination of the coefficients in the whole domain (a, b) bears on inference methods which consist in comparing the solution of the model with hypothetical values of coefficientsμ k , with the measurements on the subregions [31]. The underlying assumption behind this inference process is that there is a one-toone and onto relationship between the value of the solutions of the model over the subregions and the space of coefficients. This assumption is of course not true in general.
In this paper, we obtain uniqueness results for the coefficients µ k (x), k = 1, . . . , N , based on localized measurements of the solution u(t, x) of (1.4). The major differences with previous works dealing with comparable uniqueness results are (1) the size of the regions where u(t, x) has to be known in order to prove uniqueness, (2) the number of parameters we are able to determine, and (3) the general type of nonlinearity we deal with.
Uniqueness of the parameters, given some values of the solution, corresponds to an inverse coefficient problem, which is generally dealt with -for such reaction-diffusion equations -using the method of Carleman estimates [5,16]. This method provides Lipschitz stability, in addition to the uniqueness of the coefficients. However, this method requires, among other measurements, the knowledge of the density u(θ, x) at some time θ and for all x in the domain (a, b) (see [2,3,7,14,36]). The uniqueness of the couple (u, µ(x)) satisfying the equation (1.2) given such measurements has been investigated in a previous work [8], in any space dimension.
In a recent work, Roques and Cristofol [23] have proved the uniqueness of the coefficient µ(x) in (1.2) when γ(x) is known under the weaker assumption that the density u(t, x 0 ) and its spatial derivative ∂u ∂x (t, x 0 ) are known at a point x 0 in (a, b) for all t ∈ (0, ε) and that the initial density u(0, x) is known over (a, b). This result shows that the coefficient µ(x) is uniquely determined in the whole domain (a, b) by the value of the solution u(t, x) and of its spatial derivative at a single point x 0 . The present work extends this result to the case of several coefficients µ k (x), k = 1, . . . , N : given any point x 0 in (a, b) we establish a uniqueness result for the N −uple (µ 1 (x), . . . , µ N (x)) given measurements of the N solutions u(t, x) of (1.4) and of their first spatial derivatives in (0, ε) × {x 0 }, starting with N nonintersecting initial conditions.

Hypotheses and main result
Let (a, b) be a bounded interval in R. We consider, for some T > 0, the problem: for some N ∈ N * , and for -unknown -functions µ k which belong to the following space M: for some η ∈ (0, 1]. The space C 0,η corresponds to Hölder continuous functions with exponent η (see e.g. [12]). A function ψ ∈ C 0,η ([a, b]) is called piecewise analytic if their exist n > 0 and an increasing sequence (κ j ) 1≤j≤n such that κ 1 = a, κ n = b, and for some analytic functions ϕ j , defined on the intervals [κ j , κ j+1 ], and where χ [κ j ,κ j+1 ) are the characteristic functions of the intervals [κ j , κ j+1 ) for j = 1, . . . , n − 1. In particular, if ψ ∈ M, The assumptions on the function g are: We also assume that the diffusion coefficient D is positive and that the boundary coefficients satisfy: We furthermore make the following hypotheses on the initial condition: that is u 0 is a C 2 function such that (u 0 ) ′′ is Hölder continuous. In addition to that, we assume the following compatibility conditions: Under the assumptions (2.1)-(2.5), for each sequence (µ k ) 1≤k≤N ∈ M N , there exists a time the derivatives up to order two in x and order one in t are Hölder continuous). In the sequel, even if it means decreasing T u 0 (µ k ) in some cases and dropping the indices (µ k ) and u 0 , we only deal with values of t smaller than T so that the problem (P u 0 (µ k ) ) is well posed. Existence, uniqueness and regularity of the solution u are classical (see e.g. [21]).
Our main result is a uniqueness result for the sequence of coefficients (µ k ) 1≤k≤N associated with observations of the solution and of its spatial derivative at a single point x 0 in [a, b]. Consider N ordered initial conditions u 0 i and, for each sequence (µ k ) 1≤k≤N , let u i be the solution of (P u 0 i (µ k ) ). Our result shows that for any ε > 0 the function is one-to-one. In other words, we have the following theorem: Let (u 0 i ) 1≤i≤N be N positive functions fulfilling (2.4) and (2.5) and such that u 0 We assume that u i andũ i satisfy at some x 0 ∈ [a, b], and for some ε ∈ (0, T ]: The main result in [23] was a particular case of Theorem 2.1. A similar conclusion was indeed proved in the case N = 1 and for g(x, u) = −γ u 2 . In such case, the determination of one coefficient µ 1 (x) only requires the knowledge of the initial condition u 0 and of (u(t, x 0 ), ∂u/∂x(t, x 0 )) for t ∈ (0, ε). When N ≥ 2, the above theorem requires more than the knowledge of the initial condition for the determination of the coefficients: we need a control on the initial condition, which enables to obtain N measurements of the solution of (P u 0 (µ k ) ), starting from N different initial conditions. A natural question is whether the result of Theorem 2.1 remains true when the number of measurements is smaller than N. In Section 4, we prove that the answer is negative in general.

Proof of Theorem 2.1
For the sake of clarity, we begin with proving Theorem 2.1 in the particular case N = 2 (the proof in the case N = 1 would be similar to that of [23], which was concerned with g(x, u) = −γ u 2 ). We then deal with the general case of problems (P Let (µ 1 ,μ 1 ) and (µ 2 ,μ 2 ) be two pairs of coefficients in M. Let u 0 1 (x), u 0 2 (x) be two functions verifying (2.4) and (2.5) and such that u 0 . Let u 1 andũ 1 be respectively the solutions of (P u 0 1 µ 1 ,µ 2 ) and (P u 0 1 µ 1 ,μ 2 ) and u 2 andũ 2 be the solutions of (P u 0 2 µ 1 ,µ 2 ) and (P u 0 2 µ 1 ,μ 2 ). We set, for i = 1, 2, The functions U i satisfy: and the boundary and initial conditions: Let us first assume that x 0 < b, and set: and Let us assume on the contrary that x 1 < b.
Step 2: We prove that x 1 = b.
We set for all i, k ∈ {1, · · · , N }, The functions U i satisfy: Moreover, the functions U i satisfy the following boundary and initial conditions: ∈ (a, b). (3.10)

Let us set
Let us assume by contradiction that x 1 < b. As in the case N = 2, we prove, as a first step, that there exist θ ∈ (0, T ), To do so, observe that, from the definitions of x 1 and M, there exists δ > 0 such that x 1 + δ < b and all functions m k are analytic on [x 1 , x 1 + δ] and not all identically zero. Therefore, the integer is well-defined. Furthermore, the function h can then be written as Consequently, h(x, u j (t, x)) = 0 for all (t, The remaining part of the proof of Theorem 2.1 in the general case N ≥ 1 is then similar to Steps 2 and 3 of the proof in the particular case N = 2. Namely, we eventually get a contradiction with the assumption that

Non-uniqueness results
This section deals with non-uniqueness results for the coefficients (µ k ) in (P u 0 (µ k ) ) under assumptions weaker than those of Theorem 2.1. These results emphasize the optimality of the assumptions of Theorem 2.1.

1-Number of measurements smaller than number of unknown coefficients.
We give a counter-example to the uniqueness result of Theorem 2.1 in the case where the number of measurements is smaller than the number of unknown coefficients N .
Assume that the coefficients µ 1 , . . . , µ N are constant, not all zero, and such that the polynomial µ k u k admits exactly N − 1 positive and distinct roots z 1 , . . . , z N −1 . Assume furthermore that α 1 = α 2 = 0 (Neumann boundary conditions) and that g ≡ 0. Then for each i = 1, . . . , N − 1, z i is a (stationary) solution of (P z i (µ k ) ). Consider a similar problem with the coefficientsμ k = τ µ k for τ = 1 and k = 1, . . . , N. Then, again, for each i = 1, . . . , N − 1, z i is a solution of (P z i (μ k ) ). In particular, assumption (2.6) is fulfilled at any point This shows that the determination of N coefficients (µ k ) 1≤k≤N requires in general N observations of the solution of (P u 0 (µ k ) ), starting from N different initial conditions.

2-Lack of measurement of the spatial derivatives.
We show that if hypothesis (2.6) in Theorem 2.1 is replaced with the weaker assumption: x 0 ), for all t ∈ (0, ε) and all i ∈ {1, · · · , N }, (4.1) then the conclusion of the theorem is false in general. Let (µ k ) 1≤k≤N ∈ M N and assume that α 1 = α 2 = 0 (Neumann boundary conditions). Let (u 0 i ) 1≤i≤N satisfy the assumptions of Theorem 2.1 and assume furthermore that the functions u 0 i and g(·, u) are symmetric with respect to x = (a + b)/2, i.e.
The above result is an adaptation of Proposition 2.3 in [23] to the general case N ≥ 1.

3-Time-dependent coefficients.
We show here that the result of Theorem 2.1 is not true in general when the coefficients (µ k ) are allowed to depend on the variable t.
Case N = 2 : 25 couples of functions (µ 1 , µ 2 ) have been randomly sampled in E 2 : for k = 1 and k = 2, the components h k j , in the expression were randomly drawn from a uniform distribution in (−5, 5).

Discussion
We have obtained a uniqueness result in the inverse problem of determining several non-constant coefficients of reaction-diffusion equations. With a reaction term containing an unknown polynomial part of the form N k=1 µ k (x) u k , our result provides a sufficient condition for the uniqueness of the determination of this nonlinear polynomial part.
This sufficient condition, which is detailed in Theorem 2.1, involves pointwise measurements of the solution u(t, x 0 ) and of its spatial derivative ∂u/∂x(t, x 0 ) at a single point x 0 , during a time interval (0, ε), and starting with N nonintersecting initial conditions.
The results of Section 4 show that most conditions of Theorem 2.1 are in fact necessary. In particular, the first counter-example of Section 4 shows that, for the result of Theorem 2.1 to hold in general, the number of measurements of the couple (u, ∂u/∂x)(t, x 0 ) needs to be at least equal to the degree (N ) of the unknown polynomial term.
From a practical point of view, such measurements can be obtained if one has a control on the initial condition. Nevertheless, since our result does not provide a stability inequality, the possibility to do a numerical reconstruction of the unknown coefficients µ k , on the basis of pointwise measurements, was uncertain. In Section 5, we have shown in the cases N = 2 and N = 3 -which include the classical models (1.2) and (1.3) -that such measurements can indeed lead to good numerical approximations of the unknown coefficients, at least if they are assumed to belong to a known finite-dimensional space.