Stability estimates in stationary inverse transport

We study the stability of the reconstruction of the scattering and absorption coefficients in a stationary linear transport equation from knowledge of the full albedo operator in dimension $n\geq3$. The albedo operator is defined as the mapping from the incoming boundary conditions to the outgoing transport solution at the boundary of a compact and convex domain. The uniqueness of the reconstruction was proved in [M. Choulli-P. Stefanov, 1996 and 1999] and partial stability estimates were obtained in [J.-N. Wang, 1999] for spatially independent scattering coefficients. We generalize these results and prove an $L^1$-stability estimate for spatially dependent scattering coefficients.


Introduction
Let the spatial domain X ⊂ R n , n ≥ 2, be a convex, open bounded subset with C 1 boundary ∂X, and let the velocity domain V be S n−1 or an open subset of R n which satisfies inf v∈V |v| > 0. Let Γ ± = {(x, v) ∈ ∂X ×V ; ±n(x)v > 0} where n(x) denotes the outward normal vector to ∂X at x ∈ ∂X. The set Γ − is the set of incoming boundary condition while Γ + is the set where we measure the outgoing solution to the following stationary linear Boltzmann transport equation in X × V : Here, f (x, v) models the density of particles at position x ∈ X with velocity v ∈ V . The albedo operator A is then defined by where f (x, v) is the solution to (1.1). The inverse transport problem consists of reconstructing the absorption coefficient σ(x, v) and the scattering coefficient k(x, v ′ , v) from knowledge of A. Stability estimates aim at controlling the variations in the reconstructed coefficients σ(x, v) and k(x, v ′ , v) from variations in A in suitable metrics. The forward transport equation has been analyzed in e.g. [5,6,8]. The inverse transport problem has been addressed in e.g. [3,4,9,10] with stability estimates obtained in [9,13]. For the two-dimensional case, in which proofs of uniqueness of the scattering coefficient are available only when it is sufficiently small or independent of the spatial variable, we refer the reader to e.g. [1,11,12].
To obtain our stability estimates, we follow a methodology based on the decomposition of the albedo operator into singular components [3,4] and the use of appropriate functions on Γ ± with decreasing support [13]. In dimensions n ≥ 3 the contribution due to single scattering is more singular than the contribution due to higher orders of scattering. As a consequence, the single scattering in a direction v ′ generated by a delta function f − = δ x 0 (x)δ(v − v 0 ) is a one-dimensional curve on ∂X. In order to obtain general stability estimates for the scattering coefficient, one way to proceed is to construct test functions whose support converges to that specific curve. It turns out that it is simpler to work in a geometry in which this curve becomes a straight line.
We now briefly introduce that geometry and refer the reader to section 2 below for a formal presentation. Let R be a positive real constant such that X is included in the ball B(R) of radius R centered at x = 0. On (B(R)\X) × V , the absorption and scattering coefficients vanish and we may solve the equation v∇ x f = 0. This allows us to map back the incoming conditions f − on Γ − as incoming conditions, which we shall still denote by f − , on F − and map forward the outgoing solution f |Γ + to an outgoing solution f + on F + , where we have defined F ± := {(x ± Rv, v) ∈ R n × V for (x, v) ∈ R n × V s.t. vx = 0, |x| < R}.
(1. 3) In other words, F ± is the union for each v ∈ V of the spatial points on a disc of radius R in a plane orthogonal to v and tangent to the sphere of radius R. The incoming boundary condition is thus now defined on F − while measurements occur on F + and we may define the albedo operator still called A as an operator mapping f − defined on F − to the outgoing solution f + on F + . We may now verify that the single scattering in a direction v ′ generated by a delta function Fig. 1. Note also that the geometry we consider here may be more practical than the geometry based on Γ ± . Indeed, we assume that the incoming conditions are generated on a plane for each direction of incidence, and, more importantly, that our measurements are acquired on a plane for each outgoing direction. This is how the collimators used in Computerized Tomography [7] are currently set up.
Under appropriate assumptions on the coefficients, we aim to show that A is a well posed operator from L 1 (F − ) to L 1 (F + ). We shall then obtain a stability esti- Figure 1: Geometry of the ballistic and single scattering components in dimension n = 3. The source term is non-zero in the vicinity (in . The ballistic part is non-zero in the vicinity (in F + ) of x ′ 0 + Rv ′ 0 in P 2 = Rv ′ 0 + Π v ′ 0 (R). The thick "line" represents the support of the single scattering contribution in the vicinity (in F + ) of the segment {x ′ 0 − sv ′ 0 + t(s)v; s ∈ (−R, R)} ⊂ P 3 = Rv + Π v (R). See text for the notation. mate for the reconstruction σ(x) (or σ(x, |v|)) and k(x, v, v ′ ) with respect to the norm L(L 1 (F − ), L 1 (F + )) of A.
The rest of the paper is structured as follows. Because our geometry is not standard, we present a detailed analysis of the linear transport equation and of the singular decomposition of the albedo operator in section 2. Most of the material in that section is similar to that in [4]. One of the main physical constraints in the existence of solutions to (1.1) is that the system be "subcritical", in the sense that the "production" of particles by the scattering term involving the scattering coefficient k(x, v, v ′ ) has to be compensated by the absorption of particles and the leakage of particles at the domain's boundary. Although this may be seen implicitly in [4], we state explicitly that the decomposition of the albedo operator used in the stability estimates holds as soon as the forward transport problem is well-posed in a reasonable way.
The stability results are stated in section 3. Under additional continuity assumptions on the absorption and scattering coefficients, we obtain that (i) the exponential of line integrals of the absorption coefficient and (ii) the scattering coefficient multiplied by the exponential of the integral of the absorption coefficient on a broken line are both stably determined by A in L(L 1 (F − ), L 1 (F + )); see Theorem 3.2. Under additional regularity hypotheses on the absorption coefficient, we obtain a stability result for the absorption coefficient in some Sobolev space H s and for the scattering coefficient in the L 1 norm. The stability results in the geometry of (1.1) are presented in section 4. The proof of the stability results and the construction of the appropriate test functions are presented in section 5. Several proofs on the decomposition of the albedo operator and the uniqueness of the transport equation have been postponed to sections 6 and 7, respectively.

Transport equation and albedo operator
We now state our main results on the stationary linear transport equation and the corresponding albedo operator.
Let R be a positive real constant and let n ∈ N, n ≥ 2. Let V be S n−1 or an open subset of R n which satisfies v 0 : and let F be the set When V = S n−1 , then F is an open subset of T S n−1 := {(x, v) ∈ R n ×S n−1 | vx = 0}, the tangent space to the unit sphere. When V is an open subset of R n (which satisfies v 0 = inf v∈V |v| > 0) then F is an open subset of the 2n − 1 dimensional manifold {(x, v) ∈ R n × V | vx = 0}. We also define F ± by and recall that F − is the set of incoming conditions for the transport equation while F + is the set in which measurements are performed. We consider the space L 1 (O) with the usual norm We also consider the space L 1 (F ) defined as the completed Banach space of the vector space of compactly supported continuous functions on F for the norm and similarly the spaces L 1 (F ± ) defined as the completed Banach space of the vector space of compactly supported continuous functions on F ± for the norm We assume that: Under these conditions, we consider the stationary linear Boltzmann transport equation Throughout the paper, for m ∈ N and for any subset U of R m we denote by χ U the characteristic function defined by We now analyze the well-posedness of (2.10). The following change of variables is useful.
Then for a.e. v ∈ V using the change of variables We introduce the following notation:

12)
and the Banach spaces We consider the space L(F ± ) defined as the completed Banach space of the vector space of compactly supported continuous functions on F ± for the norm Note that W ⊆W and L 1 (F ± ) ⊆ L(F ± ). The spacesW and L(F ± ) are used only to define the unbounded operators T and T 1 below. We obtain the following trace result. 14) for f ∈ W, where C = max((2R) −1 , 1) and Proof. Let f be a C 1 function in R n × V with compact support. Then from (2.11), it follows that Upon integrating the latter equality, we obtain Combining (2.16) et (2.17), we obtain (2.14). The proof of (2.15) is similar.
For a continuous function f − on F − , we define the following extension of f − in O: Proof. Let f − be a compactly supported continous function on F − . From (2.11) and (2.18) it follows that One can check that Jf − satisfies T 0 Jf − = −A 1 Jf − in the distributional sense. Therefore using also (2.20) we obtain which proves the lemma.

Existence theory for the albedo operator
We consider the following unbounded operators: The operator T 1 : D(T 1 ) → L 1 (O) is close, one-to-one, onto, and its inverse T 1 −1 is given for all f ∈ L 1 (O) by Lemma 2.4. The following statements hold: i. The bounded operator |v|T 1 −1 in L 1 (O) has norm less or equal to 2R and the bounded operator ii. Under the hypothesis iii. Assume either condition (2.23) or Lemma 2.4 is proved in section 7. We denote by K the bounded operator in for all f ∈ L 1 (O, |v|dxdv). The operator K also defines a bounded operator in L 1 (O). This allows us to recast the stationary linear Boltzmann transport equation as the following integral equation: The existence theory for the above integral equation is addressed in the following result.  ii. the albedo operator A : Proposition 2.6 is proved in Section 7.

Singular decomposition of the albedo operator
We assume that condition (2.28) is satisfied. Let us consider the operator R : for ψ ∈ L 1 (O, |v|dxdv). Using the equality (in the distributional sense) T 0 Kf = −A 1 f − A 2 f for f ∈ L 1 (O, |v|dxdv) and the boundedness of the operators A 1 and A 2 from L 1 (O, |v|dxdv) to L 1 (O, dxdv) and using (2.14), we obtain that R is a well defined and bounded operator from L 1 (O, |v|dxdv) to L 1 (F + ). We shall use the following lemma for the kernel distribution of R.
Lemma 2.7. We have the following decomposition: Lemma 2.7 is proved in Section 6. The last inequality shows that the kernel of the second scattering operator R is more regular than is indicated in (2.31). When V is bounded, then we can choose ε ′ = 0 in (2.32), in which case we obtain that |v ′ | −1 β ∈ L ∞ (O, L p (F + )) for 1 < p < n n−1 . This regularity is sufficient (while that described in (2.31) is not) to show that multiple scattering contributions do not interfere with our stability estimates. Taking account of Lemma 2.7, we have the following decomposition for the albedo operator.
Lemma 2.8. Under condition (2.28), the following equality in the distributional sense is valid for a.e. (x, v) ∈ F + and (x ′ , v ′ ) ∈ F , and where β is given by (2.30).
Lemma 2.8 is proved in Section 6. The above decomposition is similar to that obtained in [3,4] except that the multiple scattering contribution is written here in terms of the distribution kernel of R rather than that of R(I + K) −1 J.

Stability estimates
In this section, we give stability estimates for the reconstruction of the absorption and scattering coefficient from the albedo operator following the approach in [13].
We assume that conditions (2.9) and (2.28) are satisfied and that there exists a convex open subset X of R n with C 1 boundary ∂X such thatX ⊂ B(0, R) := {x ∈ R n | |x| < R} and the function 0 ≤ σ |X×V is continous and bounded in X × V, the function 0 ≤ k |X×V ×V is continous and bounded in Let (σ,k) be a pair of absorption and scattering coefficients that also satisfy (2.9), (2.28), and (3.1). We denote by a superscript˜any object (such as the albedo operator A or the distribution kernelsα i , i = 1, 2) associated to (σ,k). Let models the incoming condition and is fixed in the analysis that follows. For The support of f ε is represented in Fig. 1.
Let δ > 0 and let φ be any compactly supported continuous function on Then using (2.33) and (3.2) we obtain for ε > 0 that where and where In addition using the estimate φ ∞ ≤ 1, item ii of Proposition 2.6 and the definition of f ε , we obtain (3.10)

16)
for any compactly supported continuous function φ on F + , which satisfies φ ∞ ≤ 1 and Lemma 3.1 is proved in Section 5.
Taking account of Lemma 3.1 and (3.10), and choosing an appropriate sequence of functions "φ", we obtain the main result of the paper: Theorem 3.2. Assume that n ≥ 3 and that (σ, k) and (σ,k) satisfy conditions (2.9), (2.28), and (3.1). Then the following estimates are valid: where E andẼ are defined by (3.2), and where (t( Theorem 3.2 is proved in Section 5.

Stability results under additional regularity assumptions
The second inequality in Theorem 3.2 provides an L 1 stability result for k(x, v ′ , v) provided that σ(x, v) is known. The first inequality in Theorem 3.2 shows that the Radon transform of σ(x, v) is stably determined by the albedo operator. Because the inverse Radon transform is an unbounded operation, additional constraints, including regularity constraints, on σ are necessary to obtain a stable reconstruction. We assume here and that the absorption coefficient σ does not depend on the velocity variable, i.e. σ(x, v) = σ(x), x ∈ R n ; see also remark 3.5 below. Then let (2.9) and (2.28), and for somer > 0 and M > 0. Using Theorem 3.2 for any ( is not empty, we obtain the following theorem. and where θ = 2(r−r) n+1+2r , 0 < r <r, and C 2 = C 2 (R, X, v 0 , V 0 , M, r,r); in addition,

Stability in Γ ±
We now come back to the original geometry in (1.1) and present a similar stability result (Theorem 4.3 below) to Theorem 3.4. The case of a scattering coefficient k(x, v ′ , v) = k(v ′ , v) that does not depend of the space variable x was studied in [13]. We now introduce the notation we need to state our stability result.
Recall that X ⊂ R n , n ≥ 2, is an open bounded subset with C 1 boundary ∂X, and that V is S n−1 or an open subset of R n which satisfies v 0 := inf v∈V |v| > 0, and that the linear stationary Boltzmann transport equation in X × V takes the form For (x, v) ∈ Γ ∓ , we put τ (x, v) = τ ± (x, v). We consider the measure dξ(x, v) = |n(x)v|dµ(x)dv on Γ ± . We still use the notation T 0 , T 1 , T , A 1 , and A 2 as in (2.12) and introduce the following Banach space We recall the following trace formula (see Theorem 2.1 of [4]) Estimate (4.3) is the analog of the estimate (2.14) in the previous measurement setting.
For a continuous function f − on Γ − , we define J f − as the extension of f − in X × V given by : Note that J has the following trace property (see Proposition 2.1 of [4]):

Existence theory for the albedo operator
We denote by K the bounded operator of L 1 (X × V, τ −1 dxdv) defined by for all f ∈ L 1 (X ×V, τ −1 dxdv). We transform the stationary linear Boltzmann transport equation (4.1) into the following integral equation We have the following proposition, which is the analog of Proposition 2.6.   i. Assume that X is also convex. Let f ∈ L 1 (F ± ) be such that suppf ⊆ {(x, v) ∈ F ± | x + tv ∈ X for some t ∈ R}, where F ± is defined by (2.5) and R > diam(X). Then we obtain that: where γ ± (x, v) = x−(xv)v±Rv for any (x, v) ∈ Γ ± . Therefore, considering results on existence of the albedo operator A obtained in [4] and our assumptions (2.9), equality (4.8) leads us to define the albedo operator A from L 1 (F − ) to L 1 (F + ).
ii. The condition (4.7) is satisfied under either of the following constraints: (4.10) iii. Assume that the bounded operator I + K in L 1 (X × V ) admits a bounded inverse in L 1 (X × V ). (4.11) Then we can define the albedo operator from L 1 (Γ − , dξ) to L 1 (Γ + , dξ) where dξ = min(τ, λ)dξ and where λ is a positive constant. To prove this latter statement, we need trace results for the functions f ∈W : iv. Under (4.9) and the condition τ σ < ∞, the existence of the albedo operator A : v. Under the condition σ − σ p ≥ ν > 0, the existence of the albedo operator A : Finally under (4.7), we also obtain a decomposition of the albedo operator A similar to that of A given in Lemma 2.8.

Proof of the stability results
We now prove Lemma 3.1 and Theorems 3.1 and 3.2.
Proof of Lemma 3.1. Using the fact that X is a convex subset of R n with C 1 boundary and using (3.1), we obtain that at any point (t,x,v) such thatx + ηv ∈ X for some real η.
We also use the following interpolation inequality: for − 1 2 ≤ s ≤ n 2 +r. As σ ∈ M, it follows that for t 1 < t 2 ∈ R and for some c ∈ [t 1 , t 2 ] which depends on t 1 and t 2 ). (In fact, the estimate (5.53) is valid for any (  15) for v ∈ V and t ′ ∈ (−R, R).

Decomposition of the albedo operator
We now prove Lemmas 2.7 and 2.8.
Now we prove (2.32). Assume k ∈ L ∞ (R n × V × V ) and let 1 < p < 1 + 1 n−1 , p ′ −1 + p −1 = 1, be fixed for the rest of the proof of Lemma 2.7. We use (6.10). Using Hölder estimate, the change of variables "y = x − tv" (dy = dxdt) and the spherical coordinates, we obtain Assume first that V = S n−1 and let φ be a continuous function on F + . Then using (6.5), σ ≥ 0, Hölder estimate and (6.10) (with δ = 1 2 ), we obtain Now assume that V is an open subset of R n which satisfies inf v∈V |v| > 0. Let ε ′ > 0 an δ > 0 be positive real numbers. Let φ be a compactly supported and continuous function on F + such that suppφ ⊆ {(x, v) ∈ F + | |v| < δ −1 }. We use the following lemma, whose proof is postponed to the end of this section. (6.13) belongs to L ∞ (O, L 1 (F + )), where E 0 is defined by (6.2).
Let M ε ′ > v 0 be defined by . (6.14) From (6.8), it follows that From the definitions of K and J, we obtain Lemma 2.8 follows from (6.21)-(6.23) and Lemma 2.7.

Proof of existence of the albedo operator
In this section, we prove Lemma 2.4 and Propositions 2.5 and 2.6.
Proof of Lemma 2.4. Using the definition of T 1 −1 , the estimate σ ≥ 0 and (2.11), we have Using the definition of A 2 and (2.11), we have for f ∈ L 1 (O). We also used (2.9) and the change of variables for f ∈ L 1 (R n ) and v, v ′ ∈ V . Using the definition of T 1 −1 and A 2 and Lemma 2.1, (2.23) and (7.1), we obtain for f ∈ L 1 (O). Item iii follows from items i and ii (under (2.24), we also use that Proof of Proposition 2.5. We first prove item i. Assume (2.27). For all f ∈ D(T), From (2.27) it follows that T admits a bounded inverse in L 1 (O) given by T −1 := T 1 −1 (I + A 2 T 1 −1 ) −1 . Using the latter equality, we obtain 3) The proof that (I − T −1 A 2 )(I + K) = I is similar. We now prove that (2.26) implies (2.27). For f ∈ D(T), As K = T 1 −1 A 2 , we have (I + K)(D(T)) ⊆ D(T). Let g ∈ D(T), and let f = (I + K) −1 T 1 −1 g ∈ L 1 (O). Then f = −KT 1 −1 g + g = −T 1 −1 A 2 T 1 −1 g + g ∈ D(T) (we recall that g ∈ D(T)). Equality (7.2) still holds. Using (7.2), (7.5) and the fact that T 1 : D(T) → L 1 (O) is one-to-one and onto L 1 (O), we obtain (2.27). Item i is thus proved. Item ii follows from item iii of Lemma 2.4 and item i. We shall prove item iii. Note that (see (7.3)) (I + K)(I − T −1 A 2 ) = I = (I − T −1 A 2 )(I + K) in L(L 1 (O)). (7.6) Note also that L 1 (O, |v|dxdv) ⊆ L 1 (O) and recall that K is a bounded operator in L 1 (O, |v|dxdv). Therefore, we only have to prove that T −1 A 2 defines a bounded operator in L 1 (O, |v|dxdv). Note that From item i, (7.7) and item i of Lemma 2.4, it follows that T −1 A 2 defines a bounded operator in L 1 (O, |v|dxdv). Thus item iii is proved.