On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation
aa r X i v : . [ m a t h . A P ] J un On the determination of the nonlinearity fromlocalized measurements in a reaction-diffusionequation
L. Roques a,* and M. Cristofol b a UR 546 Biostatistique et Processus Spatiaux, INRA, F-84000 Avignon, Franceb Aix-Marseille Universit´e, LATP, F-13397 Marseille, France* Author for correspondence. ([email protected]) Abstract
This paper is devoted to the analysis of some uniqueness propertiesof a classical reaction-diffusion equation of Fisher-KPP type, comingfrom population dynamics in heterogeneous environments. We workin a one-dimensional interval ( a, b ) and we assume a nonlinear term ofthe form u ( µ ( x ) − γu ) where µ belongs to a fixed subset of C ([ a, b ]).We prove that the knowledge of u at t = 0 and of u , u x at a singlepoint x and for small times t ∈ (0 , ε ) is sufficient to completely de-termine the couple ( u ( t, x ) , µ ( x )) provided γ is known. Additionally,if u xx ( t, x ) is also measured for t ∈ (0 , ε ), the triplet ( u ( t, x ) , µ ( x ) , γ )is also completely determined. Those analytical results are completedwith numerical simulations which show that, in practice, measurementsof u and u x at a single point x (and for t ∈ (0 , ε )) are sufficient toobtain a good approximation of the coefficient µ ( x ) . These numericalsimulations also show that the measurement of the derivative u x isessential in order to accurately determine µ ( x ). Keywords : reaction-diffusion · heterogeneous media · uniqueness · inverseproblem Reaction-diffusion models (hereafter RD models), although they sometimesbear on simplistic assumptions such as infinite velocity assumption and com-pletely random motion of animals [1], are not in disagreement with certain1ispersal properties of populations observed in natural as well as experimen-tal ecological systems, at least qualitatively [2, 3, 4, 5]. In fact, since thework of Skellam [6], RD theory has been the main analytical framework tostudy spatial spread of biological organisms, partly because it benefits froma well-developed mathematical theory.The idea of modeling population dynamics with such models has emergedat the beginning of the 20th century, with random walk theories of organ-isms, introduced by Pearson and Blakeman [7]. Then, Fisher [8] and Kol-mogorov, Petrovsky, Piskunov [9] independently used a reaction-diffusionequation as a model for population genetics. The corresponding equation is ∂u∂t − D ∂ u∂x = u ( µ − γu ) , t > , x ∈ ( a, b ) ⊂ R , (1.1)where u = u ( t, x ) is the population density at time t and space position x , D is the diffusion coefficient, and µ and γ respectively correspond to the constant intrinsic growth rate and intraspecific competition coefficients. Inthe 80’s, this model has been extended to heterogeneous environments byShigesada et al. [10]. The corresponding model is of the type: ∂u∂t − ∂∂x (cid:18) D ( x ) ∂u∂x (cid:19) = u ( µ ( x ) − γ ( x ) u ) , t > , x ∈ ( a, b ) . (1.2)The coefficients µ ( x ) and γ ( x ) now depend on the space variable x and cantherefore include some effects of environmental heterogeneity. More recently,this model revealed that the heterogeneous character of the environmentplayed an essential role on species persistence and spreading, in the sensethat for different spatial configurations of the environment, a populationcan survive or become extinct and spread at different speeds, dependingon the habitat spatial structure ([2], [11], [12],[13], [14] ,[15], [16]). Thus,determining the coefficients in model (1.2) is an important question, evenfor areas other than ecology (see [17] and references therein).In this paper, we focus on the case of constant coefficients D and γ : ∂u∂t − D ∂ u∂x = u ( µ ( x ) − γu ) , t > , x ∈ ( a, b ) , (1.3)and we address the question of the uniqueness of couples ( u, µ ( x )) and triples( u, µ ( x ) , γ ) satisfying (1.2), given a localized measurement of u .Uniqueness results of this type have been obtained for reaction-diffusionmodels, through the Lipschtiz stability of the coefficient with respect to thesolution u . Lipschtiz stability is generally obtained by using the method of2arleman estimates [18]. Several publications starting from the paper byIsakov [19] and including the recent overview of the method of Carlemanestimates applied to inverse coefficients problems [20] provide results for thecase of multiple measurements. The particular problem of the uniqueness ofthe couple ( u, µ ( x )) satisfying (1.3) given such multiple measurements hasbeen investigated, together with Lipschtiz stability, in a previous work [21].Placing ourselves in a bounded domain Ω of R N with Dirichlet boundaryconditions, we had to use the following measurements: (i) the density u (0 , x )in Ω at t = 0; (ii) the density u ( t, x ) for ( t, x ) ∈ ( t , t ) × ω , for some times0 < t < t and a subset ω ⊂⊂ Ω; (iii) the density u ( θ, x ) for all x ∈ Ω, atsome time θ ∈ ( t , t ).Although the result of [21] allows to determine µ ( x ) using partial mea-surements of u ( t, x ) , assumption (iii) implies that u has to be known in thewhole set Ω. This last measurement (iii) is a key assumption in several otherpapers on uniqueness and stability of solutions to parabolic equations withrespect to parameters (see Imanuvilov and Yamamoto [22], Yamamoto andZou [23], Belassoued and Yamamoto [24] for scalar equations and Cristofol,Gaitan and Ramoul [25] or Benabdallah, Cristofol, Gaitan and Yamamoto[26] for systems).Here, contrarily to previous results obtained for this type of reaction-diffusion models, there are some regions in ( a, b ) where u is never measured:we only require to know (i’) the density u (0 , x ) in ( a, b ) at t = 0 and (ii’) thedensity u ( t, x ) and its spatial derivative ∂u∂x ( t, x ) for t ∈ (0 , ε ) and somepoint x in ( a, b ) (see Remark 2.5 for a particular example of hypothesis(ii’)). Thus a measurement of type (iii) is no more necessary. Furthermore,we show simultaneous uniqueness of two coefficients µ ( x ) and γ providedthat measurements of the second derivative ∂ u∂x ( t, x ) are available.Our paper is organized as follows: in the next section, we give precisestatements of our hypotheses and results; Section 3 is then dedicated to theproof of the results. Section 4 is devoted to the description of numericalexamples illustrating how the coefficient µ ( x ) can be approached using mea-sures of the type (i’) and (ii’). Those results are further discussed in Section5. 3 Hypotheses and main results
Let ( a, b ) be an interval in R . We consider the problem: ∂u∂t − D ∂ u∂x = u ( µ ( x ) − γu ) , t ≥ , x ∈ ( a, b ) ,α u ( t, a ) − β ∂u∂x ( t, a ) = 0 , t > ,α u ( t, b ) + β ∂u∂x ( t, b ) = 0 , t > ,u (0 , x ) = u i ( x ) , x ∈ ( a, b ) . ( P µ,γ )Our hypotheses on the coefficients are the following. Firstly, we assumethat: µ ∈ M := { ψ ∈ C ,η ([ a, b ]) such that ψ is piecewise analytic on ( a, b ) } , (2.4)for some η ∈ (0 , C ,η corresponds to H¨older continuous func-tions with exponent η (see e.g. [27]). A function ψ ∈ C ,η ([ a, b ]) is calledpiecewise analytic if it exists n > i k ) ≤ k ≤ n such that i = a , i n = b , andfor all x ∈ ( a, b ) , ψ ( x ) = n − X j =1 χ [ i j ,i j +1 ) ( x ) ϕ j ( x ) , for some analytic functions ϕ j , defined on the intervals [ i j , i j +1 ], and where χ [ i j ,i j +1 ) are the characteristic functions of the intervals [ i j , i j +1 ) for j =1 , . . . , n − γ is a positive constant and that the boundarycoefficients satisfy: α , α , β , β ≥ α + β > α + β > . (2.5)We furthermore make the following hypotheses on the initial condition: u i ≥ , u i u i ∈ C ,η ([ a, b ]) , (2.6)for some η in (0 , u i is a C function such that u ′′ i is H¨older contin-uous. In addition to that, we assume the following compatibility conditions: α u i ( a ) − β u ′ i ( a ) = 0 , α u i ( b ) + β u ′ i ( b ) = 0 , δ β u ′′ i ( a ) = 0 , δ β u ′′ i ( b ) = 0 , (2.7)where δ y is verifies: δ = 1 and δ y = 0 if y = 0. We also need to assumethat: measure( { x ∈ ( a, b ) , u i ( x ) = 0 } ) = 0 . (2.8)4nder the assumptions (2.4)-(2.7), for each µ ∈ M and γ >
0, theproblem ( P µ,γ ) has a unique solution u ∈ C ,η ,η/ ([0 , + ∞ ) × [ a, b ]) (i.e. thederivatives up to order two in x and order one in t are H¨older continuous,see [27, 28] for a definition of H¨older continuity). Existence, uniqueness andregularity of the solution u are classical. See e.g. [28, Ch. 1].Let us state our main results: Theorem 2.1.
Let µ, ˜ µ ∈ M, and γ > , and assume that the solutions u and ˜ u to ( P µ,γ ) and ( P ˜ µ,γ ) satisfy, at some x ∈ ( a, b ) , and for some ε > and all t in (0 , ε ) : u ( t, x ) = ˜ u ( t, x ) , (2.9) ∂u∂x ( t, x ) = ∂ ˜ u∂x ( t, x ) . (2.10) Assume furthermore that u i ( x ) = 0 or ∂ u∂x ( t, x ) = ∂ ˜ u∂x ( t, x ) for t ∈ (0 , ε ) . (2.11) Then, we have µ ≡ ˜ µ on [ a, b ] and consequently u ≡ ˜ u in [0 , + ∞ ) × [ a, b ] .If β > (resp. β > ), this statement remains true when x = a (resp. x = b ). Remark 2.2.
This result remains valid if γ = γ ( x ) is a given, positivefunction in C ,η ([ a, b ]) . However, the conclusion of Theorem 2.1 is not true in general withoutthe assumption (2.10):
Proposition 2.3.
Let µ ∈ M and γ > . Assume that α = α and β = β and that u i is symmetric with respect to x = ( a + b ) / . Let ˜ µ := µ ( b − ( x − a )) for x ∈ [ a, b ] . Then, the solutions u and ˜ u to ( P µ,γ ) and ( P ˜ µ,γ ) satisfy u ( t, a + b ) = ˜ u ( t, a + b ) for all t ≥ . Under an additional assumption on the initial condition u i , we are ableto obtain a uniqueness result for triples ( u, µ, γ ): Theorem 2.4.
Let µ, ˜ µ ∈ M, and γ, ˜ γ > . Assume that, at some x ∈ ( a, b ) , u i ( x ) = 0 . Assume furthermore that the solutions u and ˜ u to ( P µ,γ ) and ( P ˜ µ, ˜ γ ) satisfy, for some ε > and for all t in (0 , ε ) : u ( t, x ) = ˜ u ( t, x ) , (2.12) ∂u∂x ( t, x ) = ∂ ˜ u∂x ( t, x ) , (2.13) ∂ u∂x ( t, x ) = ∂ ˜ u∂x ( t, x ) . (2.14)5 hen, we have µ ≡ ˜ µ on [ a, b ] and γ = ˜ γ . Consequently u ≡ ˜ u in [0 , + ∞ ) × [ a, b ] . If β > (resp. β > ), this statement remains true for x = a (resp. x = b ). Remarks 2.5. • A particular example where hypotheses (2.9-2.11) ofTheorem 2.1 (resp. hypotheses (2.12-2.14) of Theorem 2.4) are fulfilledis whenever, for some subset ω of ( a, b ) , u ( t, x ) = ˜ u ( t, x ) for t ∈ (0 , ε ) and all x ∈ ω (resp. x ∈ ω and u ( t, x ) = ˜ u ( t, x ) in (0 , ε ) × ω ).Note that, under this hypothesis, the previous results [21] did not implyuniqueness; indeed, an additional assumption of type (iii) was required(cf. the introduction section). • The uniqueness result of Theorem 2.4 cannot be adapted to the sta-tionary equation associated to ( P µ,γ ) : − p ′′ = p ( µ ( x ) − γp ) (see e.g.[11] for the existence and uniqueness of the stationary state p > ).Indeed, for any τ ∈ (0 , , setting ˜ µ = µ − τ γp and ˜ γ = (1 − τ ) γ ,we obtain − p ′′ = p (˜ µ ( x ) − ˜ γp ) , whereas ˜ µ µ and ˜ γ γ . Thus, ameasurement of p , even on the whole interval [ a, b ] , does not providea unique couple ( µ, γ ) . • The subset M of C ,η ([ a, b ]) made of piecewise analytic functions ismuch larger than the set of analytic functions on [ a, b ] . It indeed con-tains some functions whose regularity is not higher than C ,η , andsome functions which are constant on some subsets of [ a, b ] . Our re-sults hold true if M is replaced by any subset M ′ of C ,η ([ a, b ]) suchthat for any couple of elements in M ′ , the subset of [ a, b ] where thesetwo elements intersect has a finite number of connected components. Let µ, ˜ µ ∈ M, and γ, ˜ γ >
0. Let u be the solution to ( P µ,γ ) and ˜ u thesolution to ( P ˜ µ, ˜ γ ). We set U := u − ˜ u and m := µ − ˜ µ. The function U satisfies: ∂U∂t − D ∂ U∂x = ˜ µU − ˜ γU ( u + ˜ u ) + u ( m − u ( γ − ˜ γ )) , (3.15)for t ≥ x ∈ ( a, b ) , and (cid:26) α U ( t, a ) − β ∂U∂x ( t, a ) = 0 , α U ( t, b ) + β ∂U∂x ( t, b ) = 0 , t > ,U (0 , x ) = 0 , x ∈ ( a, b ) . (3.16)6roof of Theorem 2.1: In that case γ = ˜ γ . Equation (3.15) then reducesto ∂U∂t − D ∂ U∂x = ˜ µU − γU ( u + ˜ u ) + u m. (3.17) Step 1: We prove that m ( x ) = 0 . It follows from hypothesis (2.9) that, for all t ∈ [0 , ε ), U ( t, x ) = 0 andthereby, ∂U∂t ( t, x ) = 0 for all t ∈ [0 , ε ) . If u i ( x ) = 0, then, since U (0 , · ) ≡ t = 0and x = x that u i ( x ) m ( x ) = 0 , and therefore m ( x ) = 0.If u i ( x ) = 0, from (2.11), we have ∂ U∂x ( t, x ) = 0 for all t ∈ [0 , ε ).Applying equation (3.17) at t = ε/ x = x , we get u (cid:16) ε , x (cid:17) m ( x ) = 0 . If x ∈ ( a, b ), the strong parabolic maximum principle (Corollary 5.2) ap-plied to u implies that u ( ε/ , x ) >
0. As a consequence we again get m ( x ) = 0. Lastly, if x = a and β > u again implies that u ( ε/ , x ) >
0. Indeed, assume on the contrary that u ( ε/ , x ) = u ( ε/ , a ) = 0. The boundary condition α u ( t, a ) − β ∂u∂x ( t, a ) =0 implies: β ∂u∂x (cid:16) ε , a (cid:17) = 0 , which is impossible from Hopf’s Lemma (Corollary 5.2 and Theorem 5.1 (b)and (c)). Thus u ( ε/ , x ) > m ( x ) = 0. A similar argumentholds for x = b , whenever β > . Under the assumptions of Theorem 2.1, we therefore always obtain m ( x ) =0 . Step 2: We prove that m ≡ . Let us now set b := sup { x ∈ [ x , b ] s.t. m has a constant sign on [ x , x ] } . By “constant sign” we mean that either m ≥ x , x ] or m ≤ x , x ].Then, four possibilities may arise: 7 (i) m = 0 on [ x , b ] and b < b , • (ii) m ≥ x , b ] and it exists x ∈ ( x , b ) such that m ( x ) > • (iii) m ≤ x , b ] and it exists x ∈ ( x , b ) such that m ( x ) < • (iv) b = b , and m = 0 on [ x , b ].Assume (i). Then, by definition of b , there exists a decreasing sequence y k → b , y k > b , such that | m ( y k ) | > k ≥
0. Assume that itexists k such that | m ( x ) | > x ∈ ( b , y k ). By continuity, m doesnot change sign in ( b , y k ), and therefore in [ x , y k ]. This contradicts thedefinition of b . Thus,for all k, it exists z k ∈ ( b , y k ) such that m ( z k ) = 0 . (3.18)Since µ and ˜ µ belong to M , the function m also belongs to M and is thereforepiecewise analytic on ( a, b ). Thus, the set { x ∈ ( a, b ) s.t. m ( x ) = 0 } has afinite number of connected components. This contradicts (3.18) and rulesout possibility (i).Now assume (ii). By continuity of m , and from hypothesis (2.8) on u i ,we can assume that u i ( x ) >
0. Since m ( x ) > U (0 , · ) ≡
0, it followsfrom (3.17) that ∂U∂t (0 , x ) = u i ( x ) m ( x ) > . Thus, for ε > U ( t, x ) > t ∈ (0 , ε ]. As a consequence, U satisfies: ∂U∂t − D ∂ U∂x − (˜ µ − γu − γ ˜ u ) U ≥ , t ∈ (0 , ε ] , x ∈ ( x , x ) ,U ( t, x ) = 0 and U ( t, x ) > , t ∈ (0 , ε ] ,U (0 , x ) = 0 , x ∈ ( x , x ) . (3.19)Moreover, Lemma 3.1.
We have U ( t, x ) > in (0 , ε ) × ( x , x ) . Proof of Lemma 3.1:
Set W = U e − λt , for some λ > c ( t, x ) := ˜ µ − γu − γ ˜ u − λ ≤ x , x ) . The function W satisfies ∂W∂t − D ∂ W∂x − c ( t, x ) W ≥ , t ∈ (0 , ε ] , x ∈ ( x , x ) . Assume that it exists a point ( t ∗ , x ∗ ) in (0 , ε ) × ( x , x ) such that U ( t ∗ , x ∗ ) <
0. Then, since W ( t, x ) = 0 and W ( t, x ) > t ∈ (0 , ε ), and since8 (0 , x ) = U (0 , x ) = 0 , W admits a minimum m ∗ < , ε ] × ( x , x ).Theorem 5.1 (a) applied to W implies that W ≡ m ∗ < , ε ] × [ x , x ],which is impossible. Thus W ≥ , ε ] × [ x , x ] . Theorem 5.1 (a) and (c)then implies that
W > U ( t, x ) > , ε ) × ( x , x ). (cid:3) Since U ( t, x ) = 0, the Hopf’s lemma (Theorem 5.1 (b) and (c)) alsoimplies that ∂U∂x ( t, x ) > t ∈ (0 , ε ) . This contradicts hypothesis(2.10). Possibility (ii) can therefore be ruled out.Applying the same arguments to − U , possibility (iii) can also be rejected.Finally, only (iv) remains.Setting a := inf { x ∈ [ a, x ] s.t. m has a constant sign on [ x, x ] } , the same argument as above shows that a = a and m = 0 on [ a, x ]. Thus,finally, m ≡ a, b ] and this concludes the proof of Theorem 2.1. (cid:3) Proof of Theorem 2.4: From the assumptions (2.12) and (2.14) of Theo-rem 2.4, equation (3.15) at x = x reduces to u ( t, x ) ( m ( x ) − u ( t, x )( γ − ˜ γ )) = 0 for t ∈ [0 , ε ) . If x ∈ ( a, b ) , the strong parabolic maximum principle (Corollary 5.2) impliesthat u ( t, x ) > t >
0. This remains true if x = a (if β >
0) or x = b (if β > m ( x ) = u ( t, x )( γ − ˜ γ ) for t ∈ (0 , ε ) . (3.20)From the continuity of t u ( t, x ) up to t = 0, we have m ( x ) = u i ( x )( γ − ˜ γ ). Thus, u i ( x ) = 0 implies that m ( x ) = 0 which in turns implies from(3.20), and since u ( t, x ) > t >
0, that ˜ γ = γ . The end of the proof istherefore similar to that of Theorem 2.1. (cid:3) Remark 3.2.
Extension of the arguments used in the previous proof tohigher dimensions is not straightforward. Indeed, placing ourselves in abounded domain Ω of R N , with N ≥ , we may consider the largest region Ω in Ω , containing x and such that m has a constant sign in Ω . Considerin the above proof the possibility (ii) N (instead of (ii)): m ≥ on Ω and itexists x ∈ Ω such that m ( x ) > . Then it exists a subset ω of Ω , suchthat x ∈ ∂ω and u ( t, x ) > on a portion of ∂ω . However, we cannotassert that U ( t, x ) ≥ on ∂ω , and (ii) N can therefore not be ruled out aswe did for (ii). u ( t, b − ( x − a )) is a solution of ( P µ,γ ) . In particular, by unique-ness, we have u ( t, x ) = ˜ u ( t, b − ( x − a )) , for all t ≥ x ∈ [ a, b ] . It follows that u ( t, a + b ) = ˜ u ( t, a + b ) for all t ≥ . (cid:3) The purpose of this section is to verify numerically that the measurements(2.9-2.10) of Theorem 2.1 allow to obtain a good approximation of the co-efficient µ ( x ), when γ is known.Assuming that µ belongs to a finite-dimensional subspace E ⊂ M andmeasuring the distance between the measurements of the solutions of ( P µ,γ )and ( P ˜ µ,γ ) through the function G µ (˜ µ ) = k u ( · , x ) − ˜ u ( · , x ) k L (0 ,ε ) + k ∂u∂x ( · , x ) − ∂ ˜ u∂x ( · , x ) k L (0 ,ε ) , we look for the coefficient µ ( x ) as a minimizer of the function G µ . Indeed, G µ ( µ ) = 0 and, from Theorem 2.1, this is the unique global minimum of G µ in M .Solving ( P µ,γ ) by a numerical method (see Appendix B) gives an ap-proximate solution u h . In our numerical tests, we therefore replace G µ bythe discretized functional b G µ (˜ µ ) := k u h ( · , x ) − ˜ u h ( · , x ) k L (0 ,ε ) + k ∂u h ∂x ( · , x ) − ∂ ˜ u h ∂x ( · , x ) k L (0 ,ε ) . Remark 4.1.
Since ( P µ,γ ) and ( P ˜ µ,γ ) are solved with the same (determin-istic) numerical method, we have b G µ ( µ ) = 0 . Thus µ is a global minimizerof b G µ . However, this minimizer might not be unique. E We fix ( a, b ) = (0 ,
1) and we assume that the function µ belongs to a subspace E ⊂ M defined by: E := ( ˜ µ ∈ C ,η ([0 , | ∃ ( h i ) ≤ i ≤ n ∈ R n +1 , ˜ µ ( x ) = n X i =0 h i · j (( n −
2) ( x − c i )) on [0 , ) , with c i = i − n − and j ( x ) = ( exp (cid:16) x x − (cid:17) , if x ∈ ( − , , .2 Minimization of b G µ in E For the numerical computations, we fixed D = 0 . γ = 1, α = α = 0 and β = β = 1 (Neumann boundary conditions). Besides, we assumed that u i ≡ . ε = 0 . x = 2 / . The integer n was set to 10 in the definitionof E .Numerical computations were carried out for 100 functions µ k in E : µ k = n X i =0 h ki · j [( n −
2) ( x − c i )] , k = 1 . . . , whose components h ki were randomly drawn from a uniform distribution in( − , . Minimizations of the functions b G µ k were performed using MATLAB’s r fminunc solver . This led to 100 functions µ ∗ k in E , each one correspondingto a computed approximation for a minimizer of the function b G µ k . In ournumerical tests, we obtained values of b G µ k ( µ ∗ k ) in (5 · − , − ), with anaverage of 5 · − and a standard deviation of 2 · − . The values k µ k − µ ∗ k k L ([0 , / k µ k k L ([0 , , for k = 1 . . . · − and 0 .
16, with an average value of 0 .
04 and a standarddeviation of 0 . µ in E , together with a function µ ∗ which was obtained by minimizing b G µ . H µ In this section, we illustrate that measurement (2.9) alone cannot be usedfor reconstructing µ . Replacing G µ by: H µ (˜ µ ) = k u ( · , x ) − ˜ u ( · , x ) k L (0 ,ε ) , and setting b H µ (˜ µ ) := k u h ( · , x ) − ˜ u h ( · , x ) k L (0 ,ε ) , we performed the sameanalysis as above, with the same samples µ k ∈ E and the same parameters.The corresponding values of b H µ ( µ ∗ k ) are comparable to those obtainedin Section 4.2. Namely, these values are included in (2 · − , − ) , with MATLAB’s r fminunc medium-scale optimization algorithm uses a Quasi-Newtonmethod with a mixed quadratic and cubic line search procedure. Our stopping criterionwas based on the maximum number of evaluations of the function b G µ , which was set at2 · . (a) µ (plain line) and µ ∗ (dotted line) (b) µ (plain line) and µ ∗ (dotted line) Figure 1: (a) An example of function µ in E , together with a func-tion µ ∗ which was obtained by minimizing b G µ . In this case k µ − µ ∗ k L ([0 , / k µ k L ([0 , = 0 .
03 and b G µ ( µ ∗ ) = 2 · − . (b) The same func-tion µ together with the function µ ∗ obtained by minimizing b H µ . Here, k µ − µ ∗ k L ([0 , / k µ k L ([0 , = 0 .
47 and b H µ ( µ ∗ ) = 2 · − . · − , and standard deviation 3 · − . However, the correspondingvalues of the distance k µ k − µ ∗ k k L ([0 , / k µ k k L ([0 , are far larger than thoseobtained in Section 4.2: these values are comprised between 0 .
08 and 1 . .
56 and a standard deviation of 0 . µ ∈ E as in Fig. 1 (a), we present in Fig. 1 (b) theapproximation µ ∗ obtained by minimizing b H µ . In this case, the distance k µ − µ ∗ k L ([0 , / k µ k L ([0 , is 18 times larger than k µ − µ ∗ k L ([0 , / k µ k L ([0 , . Studying the reaction-diffusion problem ( P µ,γ ) with a nonlinear term of thetype u ( µ ( x ) − γu ) , we have proved in Section 2 that knowing u and itsfirst spatial derivative at a single point x and for small times t ∈ (0 , ε ) issufficient to completely determine the couple ( u ( t, x ) , µ ( x )). Additionally, ifthe second spatial derivative is also measured at x for t ∈ (0 , ε ), the triplet( u ( t, x ) , µ ( x ) , γ ) is also completely determined.These uniqueness results are mainly the consequences of Hopf’s Lemmaand of an hypothesis on the set M of coefficients which µ ( x ) belongs to.This hypothesis implies that two coefficients in M can be equal only over aset having a finite number of connected components.The theoretical results of Section 2 suggest that the coefficients µ ( x ) and γ can be numerically determined using only measurements of the solution u of ( P µ,γ ) and of its spatial derivatives at one point x , and for t ∈ (0 , ε ).Indeed, the numerical computations of Section 4 show that, when γ is known,the coefficient µ ( x ) can be estimated by minimizing a function G µ . Thefunction ˜ u being the solution of ( P ˜ µ,γ ) , we defined G µ (˜ µ ) as the distancebetween ( u, ∂u/∂x )( · , x ) and (˜ u, ∂ ˜ u/∂x )( · , x ) , in the L (0 , ε ) sense.The numerical computations presented in Section 4.2 were carried outon 100 samples of functions µ k chosen in a finite-dimensional subspace of M . In each case, a good approximation µ ∗ k of µ k was obtained. The averagerelative L -error between µ k and µ ∗ k is 30 times smaller than the averagerelative L -error between µ k and the constant function µ k ( x ). Thus, ameasurement of u and of its first spatial derivative at a point x (and for t ∈ (0 , ε )) indirectly gives more information on the global shape of µ than adirect measure of µ at x . These good results, in spite of the computationalerror, indicate L -stability of the coefficient µ with respect to single-pointmeasurements of the solution u of ( P µ,γ ) and of its spatial derivative.Proposition 2.3 shows that the uniqueness result of Theorem 2.1 is nottrue without the assumption (2.10) on the spatial derivatives. This suggests13hat measurement (2.9) alone cannot be used for reconstructing µ. In Sec-tion 4.3, working with the same samples µ k as those discussed above, weobtained approximations µ ∗ k of µ k by minimizing a new function H µ , whichmeasures the distance between u ( · , x ) and ˜ u ( · , x ) . The average relative L -error between µ k and µ k ∗ was 14 times larger than the average relative L -error separating µ k and µ ∗ k . This confirms the usefulness of the spatialderivative measurements for the reconstruction of µ. Acknowledgements
The authors would like to thank two anonymous referees for their valuablecomments on an earlier version of this paper. The first author is supportedby the French “Agence Nationale de la Recherche” within the projects Colon-SGS, PREFERED, and URTICLIM.
Appendix A: maximum principle
The following version of the parabolic maximum principle can be found in[27, Ch. 2] and [29, Ch. 3].
Theorem 5.1.
Let u ∈ C ((0 , T ] × ( x , x )) ∩ C ([0 , T ] × [ x , x ]) , for some T > and x , x ∈ R . Let c ( t, x ) ≤ ∈ C ,η ,η/ ([0 , T ] × [ x , x ]) , for some η ∈ (0 , .Suppose that ∂u∂t − D ∂ u∂x − c ( x ) u ≥ for t ∈ (0 , T ] and x ∈ ( x , x ) .(a) If u attains a minimum m ∗ ≤ at a point ( t ∗ , x ∗ ) ∈ (0 , T ] × ( x , x ) , then u ( t, x ) ≡ m ∗ on [0 , t ∗ ] × [ x , x ] .(b) (Hopf ’s Lemma) If u attains a minimum m ∗ ≤ at a point ( t ∗ , x ) (resp. ( t ∗ , x ) ), with t ∗ > , then either ∂u∂x ( t ∗ , x ) > (resp. ∂u∂x ( t ∗ , x ) < )or u ( t, x ) ≡ m ∗ on [0 , t ∗ ] × [ x , x ] .(c) If u ≥ , the results (a) and (b) remain true without the assumption c ( t, x ) ≤ . An immediate corollary of this theorem is:
Corollary 5.2.
The solution u ( t, x ) of ( P µ,γ ) is strictly positive in (0 , + ∞ ) × ( a, b ) . Proof of Corollary 5.2:
Assume that it exists ( t ∗ , x ∗ ) ∈ (0 , + ∞ ) × ( a, b )such that u ( t ∗ , x ∗ ) <
0. 14et w ( t, x ) = u e − λt , for λ > c ( t, x ) := µ ( x ) − γu − λ ≤ , t ∗ ] × [ a, b ] . The function w satisfies: ∂w∂t − D ∂ w∂x − c ( t, x ) w = 0 . Since w (0 , x ) = u i ( x ) ≥ a, b ) and w ( t ∗ , x ∗ ) < , the function w admitsa minimum m ∗ < , t ∗ ] × [ a, b ] . From Theorem 5.1 (a), and since u i , this minimum is attained at a boundary point: it exits t ′ ∈ (0 , t ∗ ] such that w ( t ′ , a ) = m ∗ < w ( t ′ , b ) = m ∗ <
0. Without loss of generality, we canassume in the sequel that w ( t ′ , a ) = m ∗ < . From Theorem 5.1 (b), weobtain ∂w∂x ( t ′ , a ) > . Using the boundary conditions in problem ( P µ,γ ), wefinally get: α m ∗ = β ∂w∂x ( t ′ , a ) > . Using assumption (2.5), we get a contradiction. Thus u ( t, x ) ≥ , + ∞ ) × ( a, b ) . The conclusion then follows from Theorem 5.1 (c). (cid:3)
Appendix B: numerical solutions of ( P µ,γ ) and ( P ˜ µ,γ ) The equations ( P µ,γ ) and ( P ˜ µ,γ ) were solved using Comsol Multiphysics r time-dependent solver, using second order finite element method (FEM)with 960 elements. This solver uses a method of lines approach incorporat-ing variable order variable stepsize backward differentiation formulas. Non-linearities are treated using a Newton’s method. The interested reader canget more information in Comsol Multiphysics r user’s guide. ReferencesReferences [1] E E Holmes. Are diffusion models too simple? A comparison withtelegraph models of invasion.
American Naturalist , 142:779–795, 1993.[2] N Shigesada and K Kawasaki.
Biological invasions: theory and practice .Oxford Series in Ecology and Evolution, Oxford: Oxford UniversityPress, 1997. 153] P Turchin.
Quantitative analysis of movement: measuring and modelingpopulation redistribution in animals and plants . Sinauer Associates,Sunderland, MA, 1998.[4] J D Murray.
Mathematical Biology . Third Edition. InterdisciplinaryApplied Mathematics 17, Springer-Verlag, New York, 2002.[5] A Okubo and S A Levin.
Diffusion and ecological problems – modernperspectives . Second edition, Springer-Verlag, New York, 2002.[6] J G Skellam. Random dispersal in theoretical populations.
Biometrika ,38:196–218, 1951.[7] K Pearson and J Blakeman.
Mathematical contributions of the theory ofevolution. A mathematical theory of random migration . Drapersi Com-pany Research Mem. Biometrics Series III, Dept. Appl. Meth. Univ.College, London, 1906.[8] R A Fisher. The wave of advance of advantageous genes.
Annals ofEugenics , 7:335–369, 1937.[9] A N Kolmogorov, I G Petrovsky, and N S Piskunov. ´Etude de l’´equationde la diffusion avec croissance de la quantit´e de mati`ere et son appli-cation `a un probl`eme biologique.
Bulletin de l’Universit´e d’ ´Etat deMoscou, S´erie Internationale A , 1:1–26, 1937.[10] N Shigesada, K Kawasaki, and E Teramoto. Traveling periodic-waves in heterogeneous environments.
Theoretical Population Biology ,30(1):143–160, 1986.[11] H Berestycki, F Hamel, and L Roques. Analysis of the periodicallyfragmented environment model: I - Species persistence.
Journal ofMathematical Biology , 51(1):75–113, 2005.[12] S Cantrell, R and C Cosner.
Spatial ecology via reaction-diffusion equa-tions . John Wiley & Sons Ltd, Chichester, UK , 2003.[13] M El Smaily, F Hamel, and L Roques. Homogenization and influence offragmentation in a biological invasion model.
Discrete and ContinuousDynamical Systems, Series A , 25:321–342, 2009.[14] F Hamel, J Fayard, and L Roques. Spreading speeds in slowly oscillatingenvironments.
Bulletin of Mathematical Biology , DOI 10.1007/s11538-009-9486-7, 2010. 1615] L Roques and F Hamel. Mathematical analysis of the optimal habi-tat configurations for species persistence.
Mathematical Biosciences ,210(1):34–59, 2007.[16] L Roques and R S Stoica. Species persistence decreases with habitatfragmentation: an analysis in periodic stochastic environments.
Journalof Mathematical Biology , 55(2):189–205, 2007.[17] J Xin. Front propagation in heterogeneous media.
SIAM Review ,42:161–230, 2000.[18] A L Bukhgeim and M V Klibanov. Global uniqueness of a class of mul-tidimensional inverse problems.
Soviet Mathematics - Doklady , 24:244–247, 1981.[19] V Isakov. An uniqueness in inverse problems for semilinear parabolicequations.
Archive for Rational Mechanics and Analysis , 1993.[20] M V Klibanov and A Timonov.
Carleman estimates for coefficientinverse problems and numerical applications . Inverse And Ill-PosedSeries, VSP, Utrecht, 2004.[21] M Cristofol and L Roques. Biological invasions: Deriving the regions atrisk from partial measurements.
Mathematical Biosciences , 215(2):158–166, 2008.[22] O Y Immanuvilov and M Yamamoto. Lipschitz stability in inverseparabolic problems by the Carleman estimate.
Inverse Problems ,14:1229–1245, 1998.[23] M Yamamoto and J Zou. Simultaneous reconstruction of the initialtemperature and heat radiative coefficient.
Inverse Problems , 17:1181–1202, 2001.[24] M Belassoued and M Yamamoto. Inverse source problem for a transmis-sion problem for a parabolic equation.
Journal of Inverse and Ill-PosedProblems , 14(1):47–56, 2006.[25] M Cristofol, P Gaitan, and H Ramoul. Inverse problems for a twoby two reaction-diffusion system using a carleman estimate with oneobservation.
Inverse Problems , 22:1561–1573, 2006.[26] A Benabdallah, M Cristofol, P Gaitan, and M Yamamoto. Inverseproblem for a parabolic system with two components by measurementsof one component.
Applicable Analysis , 88(5):683–710, 2009.1727] A Friedman.
Partial differential equations of parabolic type . Prentice-Hall, Englewood Cliffs, NJ, 1964.[28] C V Pao.
Nonlinear Parabolic and Elliptic Equations . Plenum Press,New York, 1992.[29] M H Protter and H F Weinberger.