Identification of the collision kernel in the linear Boltzmann equation by a finite number of measurements on the boundary

Identification of the collision kernel in the linear Boltzmann equation by a finite number of measurements on the boundary

Rolci Cipolatti

Departamento de Métodos Matemáticos, Instituto de Matemática Universidade Federal do Rio de Janeiro, Cx. Postal 68530, 21945-970 Rio de Janeiro, RJ, Brasil E-mail: [e-mail]

ABSTRACT

In this paper we consider the inverse problem of recovering the collision kernel for the time dependent linear Boltzmann equation via a finite number of boundary measurements. We prove that this kernel can be uniquely determined by at most k measurements, provided that it belongs to a finite k-dimensional vector space.

Mathematical subject classification: 35R30, 83D75.

Key words: inverse problem, linear Boltzmann equation, albedo operator, boundary measurements.

1 Introduction

In this paper we consider an inverse problem for the linear Boltzmann equation

where T > 0, W is a smooth bounded convex domain of ^N, N > 2, denotes the unit sphere of ^N, q Î L^¥(W) and K_k is the integral operator with kernel k(x,w',w) defined by

In applications, the equation (1.1) describes the dynamics of a monokinetic flow of particles in a body W under the assumption that the interaction between them is negligible (which allows us to discard nonlinear terms). For instance, in the case of a low-density flux of neutrons (see [7], [10]), q > 0 is the total extinction coefficient and the collision kernel k is given by

k(x,w',w) = c (x)h (x,w',w)

where c corresponds to the within-group scattering probability and h describes the anisotropy of the scattering process. In this model, q(x)u(t,w,x) describes the loss of particles at x in the direction w at time t due to absorption or scattering and q(x)K_k[u](t,w,x) represents the production of particles at x in the direction w from those coming from directions w'.

Our focus here is the inverse problem of recovery the coefficients in (1.1) via boundary measurements. More precisely, we are interested to recover q and k by giving the incoming flux of particles on the boundary and measuring the outgoing one. Since these operations are described mathematically by the albedo operator

_q,k: L¹(0,T,L¹(S^–;dx)) ® L¹ (0,T,L¹(S⁺;dx))

(the spaces will be precised below), a general mathematical question concerning this inverse problem is to know if the knowledge of

_q,k uniquely determines q,k, i.e., if the map (q,k)

_q,k is invertible.

Taking into account the applications, we have to precise this question. A first one is to know if the knowledge of

_q,k [f] for all f determines (q,k) (infinitely many measurements ); a second one is to know if the knowledge of _q,k [f_j], for j = 1,2,...,k, determines (q,k) (finite number of measurements ).

There is a wide bibliography devoted to the first problem. We specially mention the general results obtained by Choulli and Stefanov [4]: they show that q and k are uniquely determined by the albedo operator (see also [9]). We also mention the stability results obtained by Cipolatti, Motta and Roberty (see [5] and the references therein).

There is also a lot of papers concerning the stationary case (see for instance those by V.G. Romanov [11], [12], P. Stefanov and G. Uhlmann [13], Tamasan [14], J.N. Wang [15], and also the references therein).

In this work we focus on the second question, concerning the recovery by a finite number of measurements. This may be interesting from the numerical point of view (finite element methods, for instance). Assuming that k(t,w',w) = c(x)h(w',w), we prove that c can be uniquely determined by at most k measurements, provided that c belongs to a finite k-dimensional vector space of C(). More precisely:

Theorem 1.1.Let W Ì ^Nbe a bounded convex domain of class C¹, T > (W) and = span{r₁,r₂,...,r_k}, where {r₁,r₂,...,r_k} is a linearly independent subset of C(). We assume that c Î and k(x,w',w) = c(x)h(w',w), where h Î C(×) satisfies h(w,w) ¹ 0 for every w Î . Then, there exist f₁,...,f_k Î C₀((0,T)×S^-) and ₁,...,_k Î that determine k uniquely.

The proof of Theorem 1.1 is based on the construction of highly oscillatory solutions (à la Calderón [1]) introduced in [5] and some arguments already used by the author in [6]. In fact, we consider solutions of the form

u_j(t,w,x) = c_s(_j,w)f_j(x-tw)+R_l,s(t,w,x),

where c_s converges (as s ® 1) to , the spherical atomic measure concentrated on _j and R_l,s vanishes as l ® ¥. Therefore, by choosing _j and f_j conveniently, we obtain the result.

We organize the paper as follows: in Section 2 we recall the standard functional framework in which the Cauchy problem for (1.1) is well posed in the sense of the semigroup theory and the albedo operator is defined; in Section 3, we introduce the highly oscillatory functions that will be used, in Section 4, to prove Theorem 1.1.

2 Notation and functional framework

In this section we introduce the notation and we recall some well known results on the Transport Operator and the semigroup it generates in the Neutronic Function Spaces (see [5] and the references therein for the proofs).

Let W Ì

^N (N > 2) be a convex and bounded domain of class C¹ and the unit sphere of ^N. We denote by Q: = ×W and S its boundary, i.e., S = ׶W. For p Î [1,+¥) we consider the space L^p(Q) with the usual norm

where dw denotes the surface measure on associated to the Lebesgue measure in ^N-1.

For each u Î L^p(Q) we define A₀u by

(A₀u)(w,x): = w· Ñ_xu(w,x) = (w,x), w= (w₁,...,w_N)

where the derivatives are taken in the sense of distributions in W.

One checks easily that setting

_p: = {u Î L^p(Q) ; A₀u Î L^p(Q)}, the operator (A₀,_p) is a closed densely defined operator and _p with the graph norm is a Banach space.

For every s Î ¶W, we denote n(s) the unit outward normal at s Î ¶W and we consider the sets (respectively, the incoming and outgoing boundaries)

S⁺: = {(w,s) Î × ¶W; w · n(s) >0}.

In order to well define the albedo operator as a trace operator on the outgoing boundary, we consider L^p(S^±;dx), where dx: = |w · n(s)|dsdw, and we introduce the spaces

which are Banach spaces if equipped with the norms

The next two lemmas concern the continuity and surjectivity of the trace operators (see [2], [3] and [5]):

Lemma 2.2. Let 1< p < +¥. Then there exists C > 0 (depending only on p) such that

Moreover, if p > 1 and 1/p+1/p' = 1, we have the Gauss identity

for all u Î and v Î .

As an immediate consequence of Lemma 2.1, we can introduce the space

an we have that = = _p with equivalent norms.

Lemma 2.2. The trace operators g_± are surjective from

onto L^p(;dx). More precisely, for each f Î L^p(;dx), there exists h Î such that g_±(h) = f and

where C > 0 is independent of f.

We consider the operator A: D(A)® L^p(Q), defined by (Au)(w,x): = w · Ñu(w,x), with D(A): = {u Î _p ; g_–(u) = 0}.

Theorem 2.3. The operator A is m-accretive in L^p(Q), for p Î [1,+¥).

Corollary 2.4. Let f Î L^p(Q), p Î [1,+¥) and assume that u Î D(A) is a solution of u+Au = f. If f > 0 a.e. in Q, then u > 0 a.e. in Q. In particular, it follows that

It follows from Theorem 2.3 and Corollary 2.4 that the operator A generates a positive semigroup {U₀(t)}_{t> 0} of contractions acting on L^p(Q).

Let q Î L^¥(W) and k:W××® be a real measurable function satisfying

Associated to these functions, we define the following continous operators:

1) B Î (L^p(Q),L^p(Q)) defined by B[u](w,x): = q(x)u(w,x),

2) K_k[u](w,x): = k(x,w',w)u(w',x) dw'.

It follows from (2.4) that K_kÎ (L^p(Q),L^p(Q)) "p Î [1,+¥) and (see [7])

The operator A+B-K_k: D(A)® L^p(Q) generates a c₀-semigroup {U(t)}_{t> 0} on L^p(Q) satisfying

We consider the initial-boundary value problem for the linear Boltzmann equation

where q Î L^¥(W), K_k[u] is defined by (1.2) with k satisfying (2.4).

By the previous results, it follows that, for f Î L^p(0,T;L^p(S^-,dx)), p Î [1,+¥), there exists a unique solution u Î C([0,T];_p)ÇC¹([0,T];L^p(Q)) of (2.7). This solution u allows us to define the albedo operator

_q,k:L^P(0,T,L^P(S^-,dx)) ® L^P(0,T,L^P(S⁺,dx))

_q,k:[f](t,w,s):= u (t,w,s), (w,s) Î S⁺.

As a consequence of Lemmas 2.1 and 2.2,

_q,k is a linear and bounded operator.

We also consider the following backward-boundary value problem, called the adjoint problem of (2.7):

where f* Î L^p'(0,T;L^p'(S⁺,dx)), p' Î [1,+¥),

with the corresponding albedo operator

The operators

_q,k and satisfy the following property:

Lemma 2.5. Let

f Î L^P(0,T;L^P(S^-;dx)) and f* Î L^P'(0,T;L^P(S⁺;dx))

where p,p' Î (1,+¥) are such that 1/p+1/p' = 1. Then, we have

Proof. It is a direct consequence of Lemma 2.1. Let u(t,w,x) the solution of (2.7) with boundary condition f and u*(t,w,x) the solution of (2.8) with boundary f*. We obtain the result by using (2.3), once the equation in (2.7) is multiplied by u* and integrated over (0,T)×Q.

As a direct consequence of Lemma 2.5, we have:

Lemma 2.6. Let T > 0, q₁,q₂Î L^¥(W) and k₁,k₂ satisfying (2.4). Assume that u₁ is the solution of (2.7) with coefficients q₁,k₁ and satisfying the boundary condition f Î L^p(0,T;L^p(S^-,dx)), p Î (1,+¥) and that is the solution of (2.8), with q₂,k₂ and boundary condition f^*Î L^p'(0,T;L^p'(S⁺,dx)), 1/p+1/p' = 1. Then we have

3 Highly oscillatory solutions

In this section we prove some technical results related to special solutions of (2.7) and (2.8) that will be useful in the proof of Theorem 1.1. We denote by the zero extension of q in the exterior of W.

Proposition 3.1. Let T > 0, q₁,q₂Î L^¥(W), and k₁, k₂ satisfying (2.4). We consider y₁,y₂Î C() such that

Then, there exists C₀ > 0 such that, for each l > 0, there exist R_1,lÎ C([0,T];₂) and Î C([0,T];₂) satisfying

for which the functions u₁,defined by

are solutions of (2.7) with (q, k) = (q₁, k₁) and (2.8) with (q, k) = (q₂, k₂) respectively. Moreover, if k_jÎ L^¥(W;L²(×)), then we have

Proof. Let u be the function

By direct calculations, we easily verify that

where

From (2..6), there exists R_1,lÎ C¹([0,T];L²(Q))ÇC([0,T];D(A)) a unique solution of

and it follows from (3.1) that the function u defined by (3.5) satisfies (2.7) with boundary condition

Multiplying both sides of the equation in (3.7) by the complex conjugate of R, integrating it over Q and taking its real part, we get, from Lemma 2.1,

It follows from the Cauchy-Schwarz inequality and (2.5) that

where C₁: = max{M₁,M₂}. Therefore, we obtain

where C₂: = 3+2C₁. Since R(0) = 0, we get, by integrating this last inequality on [0,t],

The first inequality in (3.2) follows easily because

and, as the same arguments hold for and , we also obtain the second inequality.

We assume now k Î L^¥(W;L²(×)). For each x Î ^N, the map w' exp (ilw'·x) converges weakly to zero in L²() when l ® +¥ and the integral operator with kernel k(x,·,·) is compact in L²(). So, we obtain from (3.6),

Moreover, ||Z_1,l(t,·,x) || < C, where C > 0 is a constant that does not depend on l. The Lebesgue's Dominated Convergence Theorem implies that

From (3.9) and (3.8) we obtain (3.4), and our proof is complete.

Corollary 3.2. Under the hypothesis of Proposition 3.1, if q₁,q₂Î C() and k₁,k₂Î L^¥(W;C(×)), we have, for every w Î ,

Proof. By multiplying both sides of the equation in (3.7) by the complex conjugate of R(t,w,x), integrating it over W, taking its real part and applying the Hölder inequality, we get

Since

we obtain

From (3.8), (3.10) and (3.11) we have

Now, integrating this last inequality on time, we get

From Proposition 3.1 we know that ® 0 as l ® + ¥. On the other hand, as the map w' e^iw'·x converges weakly to zero in L²(), we have from (3.6), for almost x Î W,

and the conclusion follows from the Lebesgue's Theorem.

Lemma 3.3. We assume that q Î L^¥(W) and k satisfies (2.4). Let be the solution of

where Z Î H¹(0,T;L²(Q)) such that Z(T) = 0. Then we have

where C₀is a constant independent of l.

Proof. Multiplying both sides of the equation in (3.12) by the complex conjugate of , integrating it over Q and taking its real part, we get

Since

we have

where C₂: = (3+2max{M₁,M₂})||q||_¥. Integrating this last inequality on [t,T] and taking into account that (T) = 0, we obtain

and the inequality in (3.13) follows easily.

We consider now

Then, it is easy to check that w_l satisfies

Multiplying both sides of the equation in (3.16) by the complex conjugate of w_l, integrating it over Q, taking its real part and applying the Cauchy-Schwarz inequality, we get as before,

As = -¶_t w_l, it follows from (3.14) and (3.17) that the set {w_l} is bounded in C¹([0,T];L²(Q)) and, in particular, is relatively compact in C([0,T];L²(Q)).

On the other hand, by integrating by parts the second integral in (3.15), it is easy to check that there exists C > 0 (depending only on T) such that

Hence, by (3.17), it follows that ® 0 as l ® ¥. Since the partial derivative in t, ¶_t:C([0,T];L²(Q)) ®H^-1(0,T;L²(Q)), is a continuous operator, there exists a constant C₃ > 0 such that

and we have the conclusion.

4 Recovery by a finite number of boundary measurements

In this section we assume that {r₁,r₂,...,r_k} is a given linearly independent set of functions of C() and we denote : = span{r₁,r₂,...,r_k}. For each Î we consider [r_i] the X-ray transform of r_i in the direction , i.e.,

and, for each e > 0, W_e: = {x Î ^N\ ; dist (x,W) < e}.

The following Lemma, which the proof is given in [6], will be essential for the proof of Theorem 1.1:

Lemma 4.1. For all e > 0, there exist _jÎ and f_jÎ (W_e), j = 1,...,k, such that the matrix A = (a_ij), with entries defined by

is invertible.

In order to prove Theorem 1.1, we define, for 0 < r < 1, the function c_r: × ® as c_r(,w): = P(r,w), where P is the Poisson kernel for B₁(0), i.e.,

From the well known properties of P (see [8]), we have

where the above limit is taken in the topology of L^p(), p Î [1,+¥) and uniformly on if y Î C(). We are now in position to prove our main result.

Proof of Theorem 1.1. Let e: = (T-diam (W))/2. We assume that q₁ = q₂ = q and k_i(x,w',w) = c_i(x)h(w',w), where c₁,c₂Î . For Î , we define y₁(w,x) = c_s(,w)f(x) and y₂(w,x) = c_r(,w)f(x), where 0 < r,s < 1 and f Î (W_e). Then y₁ and y₂ satisfy the condition (3.1) and we may consider the solutions u₁ and defined by (3.3), i.e.,

where l > 0 will be chosen a posteriori. We shall write

in such a way that

Substituting u₁ and in the indentity given in Lemma 2.6, we have

where

In the above formulas, we are denoting

_i = _ci, i = 1,2 and

In particular, it follows from the definition of the Albedo Operator and (4.3),

By denoting h(x) = (x)(₁(x)-₂(x)) and by considering the special form of u₁ and , we may write J(l,r,s) as J = J₁+J₂+J₃+J₄, where

Taking the limit as r ® 1^- in the above expressions, we get from the definition of c_r, J_i(l,r,s)®J_i(l,s), where

and is the unique solution of

Moreover, from (4.5) and (4.2), it follows that L(l,r,s)® L(l,s), where

where

_i[f_l,s] denotes the zero extension of _i[f_l,s] on ¶W. Therefore, by taking the limit as r ® 1^- in (4.4), we have

j₁(l,s) + j₂(l,s)+ j₃(l,s) + j₄(l,s) = L(l,s).

Now, it is time to take the limit as s ® 1^-. For the first two terms of the right hand side of the above identity, we get (for i = 1,2) J_i(l,s)® J_i(l), where

On the other hand, the dependence on s in the other terms is given by R_1,l,s and R_2,l,s, which are the solution of (j = 1,2)

where

It is an immediate consequence of (4.2) and the Lebesgue's Theorem that, as s ® 1, Z_j,l,s® Z_j,l in C([0,T];L²(Q)), where

Hence,

where S_j,l is the solution of

and

Therefore, J_i(l,s)® J_i(l), (i = 3,4) and L(l,s)® L(l), where

and we obtain

where

Since f Î (W_e), it follows from the choice of e that the function (t,w,x)f(x-t) belongs to (0,T;L²(Q)) (as a constant function on w). Hence, we have

On the other hand, from the weak convergence to zero in L²(0,T;L²(Q)) of S_1,l, it follows that

Hence, we have from (4.15)–(4.17) and Lemma 3.3,

where C(l)® 0 as l ®+¥.

Since (supp f+s)ÇW = Æ for all |s| > T, we have

and we get

We are now in position to conclude the proof. First of all, we consider in the above inequality the directions

₁,...,_k and the functions f₁,...,f_k given by Lemma 4.1, in such a way that we can write

for some constant C₀ > 0. If we denote by

it follows from (4.2) that, as s ® 1^-, u_i,j ® , where

and is the spherical atomic measure concentrated on _j.

It is clear from (4.13) that (t,w,s) = (t,w,s), for s Î and j = 1,...,k. Moreover, – = S_1,l-S_2,l. Therefore, if (t,_j,s) = (t,_j,s) on , for j = 1,...,k, it follows that

and the conclusion follows easily if we choose l > 0 large enough.

Received: 01/IX/06. Accepted: 01/X/06.

#691/06.

• [1] A.P. Calderón, On an inverse boundary value problem. Seminars on Numerical Analysis and Application to Continuum Physics, SBM (Rio de Janeiro 1980), 65-73.
• [2] M. Cessenat, Théorčmes de trace L^p pour des espaces de fonctions de la neutronique. C.R. Acad. Sci. Paris, Série I, 299 (1984), 831-834.
• [3] M. Cessenat, Théorčmes de trace pour des espaces de fonctions de la neutronique. C.R. Acad. Sci. Paris, Série I, 300 (1985), 89-92.
• [4] M. Choulli and P. Stefanov, Inverse scattering and inverse boundary value problem for the linear Boltzmann equation. Comm. Part. Diff. Equations, 21(5-6) (1996), 763-785.
• [5] R. Cipolatti, C.M. Motta and N.C. Roberty, Stability Estimates for an Inverse Problem for the Linear Boltzmann Equation. Revista Matemática Complutense, 19(1) (2006), 113-132.
• [6] R. Cipolatti and Ivo F. Lopez, Determination of coefficients for a dissipative wave equation via boundary measurements. J. Math. Anal. Appl., 306 (2005), 317-329.
• [7] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, 6 (1993), Springer-Verlag.
• [8] G.B. Folland, Introduction to Partial Differential Equations. Mathematical Notes, 17 (1976), Princeton University Press.
• [9] M. Mokhtar-Kharroubi, Mathematical Topics in Neutron Transport Theory - New Aspects; Series on Advances in Mathematics for Applied Sciences, 46 (1997), World Scientific.
• [10] M. Reed and B. Simon, Methods of Modern Physics, 3 (1993), Springer-Verlag.
• [11] V.G. Romanov, Estimation of stability in the problem of determining the attenuation coefficient and the scattering indicatrix for the transport equation. Sibirsk. Mat. Zh., 37(2) (1996), 361-377, iii; translation in Siberian Math. J., 37(2) (1996), 308-324.
• [12] V.G. Romanov, Stability estimate in the three-domensional inverse problem for the transport equation. J. Inverse Ill-Posed Probl., 5(5) (1997), 463-475.
• [13] P. Stefanov and G. Uhlmann, Optical tomography in two dimensions. Methods and Applications of Analysis, 10 (2002), 1-9.
• [14] A. Tamasan, An inverse boundary value problem in two dimensional transport. Inverse Problems, 18 (2002), 209-219.
• [15] J.N. Wang, Stability estimates of an inverse problem for the stationary transport equation. Ann. Inst. Henri Poincaré, 70(5) (1999), 473-495.

Publication Dates

• Publication in this collection
  19 Mar 2007
• Date of issue
  2006

History

• Received
  01 Sept 2006
• Accepted
  01 Oct 2006