A Fourier approach to the inverse source problem in an absorbing and anisotropic scattering medium
AA FOURIER APPROACH TO THE INVERSE SOURCE PROBLEM IN ANABSORBING AND ANISOTROPIC SCATTERING MEDIUM
HIROSHI FUJIWARA, KAMRAN SADIQ, AND ALEXANDRU TAMASANA
BSTRACT . We revisit the inverse source problem in a two dimensional absorbing and scatteringmedium and present a non-iterative reconstruction method using measurements of the radiating fluxat the boundary. The attenuation and scattering coefficients are known and the unknown source isisotropic. The approach is based on the Cauchy problem for a Beltrami-like equation for the sequencevalued maps, and extends the original ideas of A. Bukhgeim from the non-scattering to scattering me-dia. We demonstrate the feasibility of the method in a numerical experiment in which the scatteringis modeled by the two dimensional Henyey-Greenstein kernel with parameters meaningful in OpticalTomography.
1. I
NTRODUCTION
This work concerns a Fourier approach to the inverse source problem for radiative transport ina strictly convex domain Ω in the Euclidean plane. The attenuation and scattering coefficients areknown real valued functions. Generated by an unknown source f , in the steady state case, thedensity of particles u p z, θ q at z traveling in the direction θ solve the stationary transport equation θ ¨ ∇ u p z, θ q ` a p z q u p z, θ q “ ż S k p z, θ ¨ θ q u p z, θ q d θ ` f p z q , p z, θ q P Ω ˆ S , (1)where S denotes the unit sphere.Let Γ ˘ : “ tp z, θ q P Γ ˆ S : ˘ ν p z q ¨ θ ą u be the incoming (-), respectively outgoing(+), unit tangent sub-bundles of the boundary; where ν p z q is the outer unit normal at z P Γ . The(forward) boundary value problem for (1) assumes a given incoming flux u on Γ ´ , In here weassume that there is no incoming radiation from outside the domain, u | Γ ´ “ . The boundaryvalue problem is know to be well-posed under various admissibility and subcritical assumptions,e.g, in [12, 10, 11, 1, 26], with the most general result for a generic pair of coefficients obtainedby Stefanov and Uhlmann [46]. In here we assume that the forward problem is well-posed, andthat the outgoing radiation u | Γ ` is measured, and thus the trace u | Γ ˆ S is known. Without loss ofgenerality Ω is the unit disc.In here we show how to recover f from knowledge of u on the torus Γ ˆ S and provide an error,and stability estimates.When a “ k “ , this is the classical X -ray tomography problem of Radon [37], where f is tobe recovered from its integrals along lines, see also [32, 18, 25]. For a ‰ but k “ , this is theproblem of inversion of the Attenuated Radon transform in two dimensions, solved successfully byArbuzov, Bukhgeim and Kazantsev [2], and Novikov [34]; see [33, 8, 5] for later approaches. Date : July 18, 2019.2010
Mathematics Subject Classification.
Primary 35J56, 30E20; Secondary 45E05.
Key words and phrases.
Attenuated X -ray transform, Attenuated Radon transform, scattering, A -analytic maps,Hilbert transform, Bukhgeim-Beltrami equation, optical tomography, optical molecular imaging. a r X i v : . [ m a t h . A P ] J u l HIROSHI FUJIWARA, KAMRAN SADIQ, AND ALEXANDRU TAMASAN
The inverse source problem in an absorbing and scattering media, a, k ‰ , has also been con-sidered (e.g., [24, 44]) in the Euclidean setting, and in [42] in the Riemannian setting. The mostgeneral result ( k may vary with two independent directions) on the stable determination of thesource was obtained by Stefanov and Uhlmann [46]. The reconstruction of the source based on[46] is yet to be realized. When the anisotropic part of scattering is sufficiently small, a convergentiterative method for source reconstruction was proposed in [7]. Based on a perturbation argumentto the non-scattering case in [34], the method does not extend to strongly anisotropic scattering.In addition, it requires solving one forward problem (a computationally extensive effort) at eachiteration.The main motivation of this work is to provide a source reconstruction method that applies to theanisotropic scattering media, with non-small anisotropy. In here we propose such a non-iterativemethod. Our approach extends the original ideas in [2] from the non-scattering to the scatteringmedia.Throughout we assume that a P C ,s p Ω q , and k and its angular derivative are periodic in theangular variable, k P C ,sper pr´ , s ; Lip p Ω qq , s ą { , and that the forward problem is well posed.It is known from [46] that for pairs of coefficients p a, k q in an open and dense sets of C ˆ C , andfor any f P L p Ω q , there is a unique solution u P L p Ω ˆ S q to the forward boundary value problem.However, our approach requires a smooth solution u P H p Ω ˆ S q . As a direct consequence of[46, Proposition 3.4] the regularity of u is dictated by its ballistic term. In particular, if f P H p Ω q ,then u P H p Ω ˆ S q . With the exception of the numerical examples in Section 7, we assume thatthe unknown source f P H p Ω q , and thus the unknown solution(2) u P H p Ω ˆ S q . In the numerical experiment we use a discontinuous source, whose successful quantitative recon-struction indicates robustness of the method.Let u p z, θ q “ ř u n p z q e inθ be the formal Fourier series representation of u in the angularvariable θ “ p cos θ, sin θ q . Since u is real valued, u ´ n “ u n and the angular dependence iscompletely determined by the sequence of its nonpositive Fourier modes Ω Q z ÞÑ u p z q : “ x u p z q , u ´ p z q , u ´ p z q , ... y . (3)Let k n p z q “ π ş π ´ π k p z, cos θ q e ´ inθ dθ , n P Z , be the Fourier coefficients of the scattering kernel.Since k p z, cos θ q is both real valued and even in θ , k n p z q are real valued and k n p z q “ k ´ n p z q .Throughout this paper the Cauchy-Riemann operators B “ pB x ` i B y q{ and B “ pB x ´ i B y q{ refer to derivatives in the spatial domain. By using the advection operator θ ¨ ∇ “ e ´ iθ B ` e iθ B , andidentifying the Fourier coefficients of the same order, the equation (1) reduces to the system: B u p z q ` B u ´ p z q ` a p z q u p z q “ k p z q u p z q ` f p z q , (4)and B u n p z q ` B u n ´ p z q ` a p z q u n ´ p z q “ k n ´ p z q u n ´ p z q , n ‰ . (5)In particular, the sequence valued map (3) solves the Beltrami-like equation B u p z q ` L B u p z q ` a p z q L u p z q “ LK u p z q , z P Ω , (6)where L u p z q “ L p u p z q , u ´ p z q , u ´ p z q , ... q : “ p u ´ p z q , u ´ p z q , ... q denotes the left translation,and K u p z q : “ p k p z q u p z q , k ´ p z q u ´ p z q , k ´ p z q u ´ p z q , ... q (7)is a Fourier multiplier operator determined by the scattering kernel. FOURIER APPROACH TO THE INVERSE SOURCE PROBLEM 3
Our data u | Γ ˆ S yields the trace of a solution of (6) on the boundary, g “ u | Γ “ x g , g ´ , g ´ , ... y . (8)Bukhgeim’s original theory in [9] concerns solutions of (6) for a “ and K “ . Solutions of B u ` L B u “ , (9)(called L -analytic) satisfy a Cauchy-like integral formula, which recovers u in Ω from its trace u | Γ . In the explicit form in [15], for each n ě , u ´ n p ζ q “ πi ż Γ u ´ n p z q z ´ ζ dz ` πi ż Γ " dzz ´ ζ ´ dzz ´ ζ * ÿ j “ u ´ n ´ j p z q ˆ z ´ ζz ´ ζ ˙ j , ζ P Ω . (10)In Section 2 we review the absorbing, non-scattering case. While we follow the treatment in[38], it is in this section that the new analytical framework and notation is introduced. Section 3describe the reconstruction method for scattering kernels of polynomial dependence in the angularvariable. Except for the numerical section in the end, the remaining of the paper analyzes the errormade by the polynomial approximation of the scattering kernel. In Section 4 we exhibit the gainin smoothness due to the scattering, in particular, the { -gain in (61) below has been known (withdifferent proofs) see [26], and in a more general case than considered here in [46].The key ingredient in our analysis is an a priori gradient estimate for solutions of the inhomoge-neous Bukhgeim-Beltrami equations, see Theorem 4.2 in Section 4. Our starting point is an energyidentity, an idea originated in the work of Mukhometov [30], and an equivalent of Pestov’s identity[42, 49] for the Bukhgeim-Beltrami equation. The proof of the gradient estimate in Theorem 4.2uses essentially derivative gain in smoothness due to scattering. As a consequence, in Theorem6.1 we establish an error estimate, which yields a stability result for scattering with polynomialangular dependence (see Corollary 6.1). Furthermore, in a weakly anisotropic scattering mediumthe method is convergent (see Theorem 6.2), thus recovering the result in [7].The feasibility of the proposed method is implemented in two numerical experiments in Section7. Among the several models for the scattering kernel used in Optical Tomography [3], we workwith the two dimensional version of the Henyey-Greenstein kernel for its simplicity. In this kernelwe chose the anisotropy parameter to be { ( half way between the ballistic and isotropic regime),and a mean free path of { units of length e.g. (meaningful value for fluorescent light scattering inthe fog). 2. A BRIEF REVIEW OF THE ABSORBING NON - SCATTERING MEDIUM
In the case for a ‰ and K “ , the Beltrami equation (6) can be reduced to (9) via an integratingfactor. While this idea originates in [2], in here (as in [38]) we use the special integrating factorproposed by Finch in [15], which enjoys the crucial property of having vanishing negative Fouriermodes. This special integrating factor is e ´ h , where h p z, θ q : “ Da p z, θ q ´ p I ´ iH q Ra p z ¨ θ K , θ K q , (11)and θ K is orthogonal to θ , Da p z, θ q “ ż a p z ` t θ q dt is the divergent beam transform of theattenuation a , Ra p s, θ K q “ ż a ` s θ K ` t θ ˘ dt is the Radon transform of the attenuation a , andthe classical Hilbert transform Hh p s q “ π ż h p t q s ´ t dt is taken in the first variable and evaluated HIROSHI FUJIWARA, KAMRAN SADIQ, AND ALEXANDRU TAMASAN at s “ z ¨ θ K . The function h appeared first in the work of Natterer [32]; see also [8] for elegantarguments that show how h extends analytically (in the angular variable on the unit circle S ) insidethe unit disc. We recall some properties of h from [40, Lemma 4.1], while establishing notations. Lemma 2.1. [40, Lemma 4.1]
Assume Ω Ă R is C ,s , s ą { , convex domain. For p “ , , let a P C p,s p Ω q , s ą { , and h defined in (11) . Then h P C p,s p Ω ˆ S q and the following hold(i) h satisfies θ ¨ ∇ h p z, θ q “ ´ a p z q , p z, θ q P Ω ˆ S . (12) (ii) h has vanishing negative Fourier modes yielding the expansions e ´ h p z,θ q : “ ÿ k “ α k p z q e ikϕ , e h p z,θ q : “ ÿ k “ β k p z q e ikϕ , p z, θ q P Ω ˆ S , (13) with (iii) z ÞÑ α p z q : “ x α p z q , α p z q , α p z q , α p z q , ..., y P C p,s p Ω; l q X C p Ω; l q , (14) z ÞÑ β p z q : “ x β p z q , β p z q , β p z q , β p z q , ..., y P C p,s p Ω; l q X C p Ω; l q . (15) (iv) For any z P Ω B β p z q “ , B β p z q “ ´ a p z q β p z q , (16) B β k ` p z q ` B β k p z q ` a p z q β k ` p z q “ , k ě . (17) (v) For any z P Ω B α p z q “ , B α p z q “ a p z q α p z q , (18) B α k ` p z q ` B α k p z q ´ a p z q α k ` p z q “ , k ě . (19) (vi) The Fourier modes α k , β k , k ě satisfy α β “ , k ÿ m “ α m β k ´ m “ , k ě . (20)The Fourier coefficients of e ˘ h define the integrating operators e ˘ G u component-wise for each m ď by p e ´ G u q m “ p α ˚ u q m “ ÿ k “ α k u m ´ k , and p e G u q m “ p β ˚ u q m “ ÿ k “ β k u m ´ k , (21)where α k and β k are the Fourier modes of e ´ h and e h in (13), and α , β as in (14), respectively, (15).Note that e ˘ G can also be written in terms of left translation operator as e ´ G u “ ÿ k “ α k L k u , and e G u “ ÿ k “ β k L k u , (22)where L k “ L ˝ ¨ ¨ ¨ ˝ L l jh n k is the k -th composition of left translation operator. It is important to notethat the operators e ˘ G commute with the left translation, r e ˘ G , L s “ . FOURIER APPROACH TO THE INVERSE SOURCE PROBLEM 5
Different from [40], in this work we carry out the analysis in the Sobolev spaces l ,p p N ; H q p Ω qq with the respective norm }¨} p,q given by l ,p p N ; H q p Ω qq : “ u : } u } p,q : “ ÿ j “ p ` j q p } u ´ j } H q p Ω q ă 8 + . (23)In Proposition 2.1 below we revisit the mapping properties of e ˘ G relative to these new spaces.Throughout, in the notation of the norms, the first index p P t , , u refers to the smooth-ness in the angular variable (expressed as decay in the Fourier coefficient), while the second in-dex q P t , u shows the smoothness in the spatial variable. The most often occurring is thespace l p N ; L p Ω qq , when p “ q “ . To simplify notation, in this case we drop the double zerosubindexes, } u } : “ } u } , “ ÿ j “ } u ´ j } L p Ω q . The traces on the boundary Γ of functions in l ,p p N ; H p Ω qq are in l ,p p N ; H p Γ qq , endowedwith the norm } g } p, : “ ÿ j “ p ` j q p } g ´ j } H { p Γ q . (24)Since Γ is the unit circle, the H { p Γ q -norm can be defined in the Fourier domain as follows. Foreach integer j ě , if we consider the Fourier expansion of the trace u ´ j | Γ , u ´ j | Γ p e iβ q “ ÿ k “´8 u ´ j,k e ikβ , for e iβ P Γ, then } u ´ j } H { p Γ q “ ÿ k “´8 p ` k q | u ´ j,k | . (25)In view of (25), if g P l , p N ; H p Γ qq , then } g } , “ ÿ j “ ÿ n “´8 p ` j q p ` n q | g ´ j,n | . (26)In the estimates we need the following variant of the Poincar´e inequality obtained by component-wise summation: If u P l p N ; H p Ω qq , then } u } ď µ ´ }B u } ` } u | Γ } , ¯ , (27)where µ is a constant depending only on Ω ; for the unit disc µ “ .Consider the Banach space l , p Ω q : “ α : “ x α , α , α , ..., y : } α } l , p Ω q : “ sup z P Ω ÿ j “ j | α j p z q | ă 8 + . (28) HIROSHI FUJIWARA, KAMRAN SADIQ, AND ALEXANDRU TAMASAN
Proposition 2.1.
Let a P C ,s p Ω q , s ą { . Then α , B α , β , B β P l , p Ω q , and for any p P t , , u , q P t , u , the operators e ˘ G : l ,p p N ; H q p Ω qq Ñ l ,p p N ; H q p Ω qq (29) are bounded, and satisfy the following estimates ›› e ´ G u ›› ď } α } l , p Ω q } u } , (30) ›› e ´ G u ›› , ď } α } l , p Ω q } u } , , (31) ›› e ´ G u ›› , ď ´ } α } l , p Ω q ` }B α } l , p Ω q ¯ } u } , , (32) ›› e ´ G u ›› , ď ´ } α } l , p Ω q ` }B α } l , p Ω q ¯ } u } , . (33) The same estimate works for e G u with α replaced by β . The proof of the Proposition 2.1 can be found in the Appendix.We remark that cases p p “ “ q q , p p “ , q “ q , and p p “ , q “ q in Proposition 2.1 holdfor a P C ,s p Ω q , s ą { , and these are the only properties needed for the lemma below. Lemma 2.2.
Let a P C ,s p Ω q , s ą { , and e ˘ G as defined in (21) .(i) If u P l p N ; H p Ω qq solves B u ` L B u ` aL u “ , then v “ e ´ G u P l p N ; H p Ω qq solves B v ` L B v “ .(ii) Conversely, if v P l p N ; H p Ω qq solves B v ` L B v “ , then u “ e G v P l p N ; H p Ω qq solves B u ` L B u ` aL u “ .Proof. (i) Let v “ e ´ G u “ ÿ k “ α k L k u . Since u P l p N ; H p Ω qq , then from Proposition 2.1, v P l p N ; H p Ω qq . Then v solves B v ` L B v “ ÿ k “ B α k L k u ` ÿ k “ α k L k B u ` ÿ k “ B α k L k ` u ` ÿ k “ α k L k ` B u “ B α u ` B α L u ` ÿ k “ ` B α k ` ` B α k ˘ L k ` u ` ÿ k “ α k L k ` B u ` L B u ˘ “ B α u ` B α L u ` ÿ k “ ` B α k ` ` B α k ˘ L k ` u ` ÿ k “ α k L k p´ aL u q“ B α u ` ` B α ´ aα ˘ L u ` ÿ k “ ` B α k ` ` B α k ´ aα k ` ˘ L k ` u “ , where in the last equality we have used (18) and (19).An analogue calculation using the properties in Lemma 2.1 (iv) shows the converse. (cid:3) FOURIER APPROACH TO THE INVERSE SOURCE PROBLEM 7
3. S
OURCE RECONSTRUCTION FOR SCATTERING OF POLYNOMIAL TYPE
This section contains the basic idea of reconstruction in the special case of scattering kernel ofpolynomial type, k p z, cos θ q “ M ÿ n “ k ´ n p z q cos p nθ q , (34)for some fixed integer M ě . Recall that since k p z, cos θ q is both real valued and even in θ , k n p z q are real valued and k n p z q “ k ´ n p z q , ď n ď M . We stress here that no smallness assumption on k , k ´ , k ´ , ¨ ¨ ¨ , k ´ M is assumed. Let u p M q be the solution of (1) with k as in (34) and u p M q denotethe sequence valued map Ω Q z ÞÑ u p M q p z q : “ x u p M q p z q , u p M q´ p z q , u p M q´ p z q , ¨ ¨ ¨ , u p M q´ M p z q , u p M q´ M ´ p z q , ¨ ¨ ¨ y . (35)Let also K p M q denote the corresponding Fourier multiplier operator K p M q u p M q p z q “ p k p z q u p M q p z q , k ´ p z q u p M q´ p z q , ¨ ¨ ¨ , k ´ M p z q u p M q´ M p z q , , , ¨ ¨ ¨ q . (36)The transport equation (1) reduces to the system B u p M q´ p z q ` B u p M q´ p z q ` a p z q u p M q p z q “ k p z q u p M q p z q ` f p M q p z q , (37) B u p M q´ n p z q ` B u p M q´ n ´ p z q ` a p z q u p M q´ n ´ p z q “ k ´ n ´ p z q u p M q´ n ´ p z q , ď n ď M ´ , (38) B u p M q´ n p z q ` B u p M q´ n ´ p z q ` a p z q u p M q´ n ´ p z q “ , n ě M. (39)In sequence valued notation, the system (38) and (39) rewrites: B u p M q ` L B u p M q ` a p z q L u p M q “ LK p M q u p M q , (40)where K p M q as in (36).Since f P H p Ω q , the solution u P H p Ω ˆ S q , and consequently u p M q P l , p N ; H p Ω qq . Wenote that in our method we only use u p M q P l , p N ; H p Ω qq , indicating that it may apply to roughersources.Let the transformation v p M q “ e ´ G L M u p M q , then by Proposition 2.1, v p M q P l , p N ; H p Ω qq ,and v p M q is L -analytic:(41) B v p M q ` L B v p M q “ . The trace of the boundary v p M q | Γ is determined by the trace of u p M q | Γ “ g “ p g , g ´ , g ´ , ... q P l , p N ; H { p Γ qq , by v p M q | Γ “ e ´ G L M u p M q | Γ “ e ´ G L M g . (42)By Proposition 2.1, v p M q | Γ P l , p N ; H { p Γ qq .The Bukhgeim-Cauchy integral formula (10) extends v p M q from Γ to Ω as L -analytic map.From the uniqueness of an L -analytic map with a given trace, we recovered for n ě ,v p M q´ n p z q “ πi ż Γ v p M q´ n p ζ q ζ ´ z dζ ` πi ż Γ " dζζ ´ z ´ dζζ ´ z * ÿ j “ v p M q´ n ´ j p ζ q ˆ ζ ´ zζ ´ z ˙ j , z P Ω . (43)Thus v p M q “ x v p M q , v p M q´ , v p M q´ , ... y is recovered in l , p N ; H p Ω qq . HIROSHI FUJIWARA, KAMRAN SADIQ, AND ALEXANDRU TAMASAN
We recover L M u p M q “ x u p M q´ M , u p M q´ M ´ , u p M q´ M ´ , ... y in Ω by using the convolution formula (21) u p M q´ n ´ M p z q “ ÿ k “ β k p z q v p M q´ n ´ k p z q , z P Ω , n ě , (44)where β k ’s as in (13). In particular we recovered u p M q´ M ´ , u p M q´ M P H p Ω q .By applying B to (38), the mode u p M q´ M ` is then the solution to the Dirichlet problem for thePoisson equation (cid:52) u p M q´ M ` “ ´ B u p M q´ M ´ ´ B ” p a ´ k ´ M q u p M q´ M ı , (45a) u p M q´ M ` | Γ “ g ´ M ` , (45b)where the right hand side of (45a) is known.Since by construction u p M q´ M , u p M q´ M ´ P H p Ω q , we have ››› B u p M q´ M ´ ` B ´ p a ´ k ´ M q u p M q´ M ¯››› H ´ p Ω q ď ››› B u p M q´ M ´ ››› L p Ω q ` ›››´ p a ´ k ´ M q u p M q´ M ¯››› L p Ω q ď ››› B u p M q´ M ´ ››› L p Ω q ` } a ´ k ´ M } L p Ω q ››› u p M q´ M ››› L p Ω q ď ››› u p M q´ M ´ ››› H p Ω q ` } a ´ k ´ M } L p Ω q ››› u p M q´ M ››› H p Ω q . Since g ´ M ` P H { p Γ q , the solution u p M q´ M ` P H p Ω q and ››› u p M q´ M ` ››› H p Ω q ď C ˆ››› u p M q´ M ´ ››› H p Ω q ` ››› u p M q´ M ››› H p Ω q ` } g ´ M ` } H { p Γ q ˙ , (46)where the constant C depends only on Ω and max " , max ď j ď M } a ´ k ´ j } L p Ω q * . Successively allthe other modes u p M q´ M ` j for j “ , ¨ ¨ ¨ , M are computed by solving the corresponding Dirichletproblem for the Poisson equation. To account for the successive accumulation of error we note thefollowing result which can be proven by induction. Lemma 3.1.
Let t a n u and t b n u be sequences of nonnegative numbers, such that a n ` ď c p a n ` ` a n ` b n ` q , n ě , where c ą is a constant, then a n ` ď p ` c q n ` ˜ a ` a ` n ÿ k “ b k ` ¸ , n ě . We applying Lemma 3.1 to (46) and estimate ››› u p M q´ ››› H p Ω q ď p ` C q M ´ ˜››› u p M q´ M ´ ››› H p Ω q ` ››› u p M q´ M ››› H p Ω q ` M ´ ÿ j “ } g ´ M ` j } H { p Γ q ¸ , (47)and ››› u p M q ››› H p Ω q ď p ` C q M ˜››› u p M q´ M ´ ››› H p Ω q ` ››› u p M q´ M ››› H p Ω q ` M ÿ j “ } g ´ M ` j } H { p Γ q ¸ . (48) FOURIER APPROACH TO THE INVERSE SOURCE PROBLEM 9
The source f p M q is computed by f p M q p z q “ R e ´ B u p M q´ p z q ¯ ` p a p z q ´ k p z qq u p M q p z q , (49)and we estimate ›› f p M q ›› L p Ω q ď ››› u p M q´ ››› H p Ω q ` C ››› u p M q ››› H p Ω q ď p ` C q M ` ˜››› u p M q´ M ´ ››› H p Ω q ` ››› u p M q´ M ››› H p Ω q ` M ÿ j “ } g ´ M ` j } H { p Γ q ¸ . (50)This method is implemented in the numerical experiments in Section 7. Next we analyze theerror introduced by truncation.4. G RADIENT ESTIMATES OF SOLUTIONS TO NONHOMOGENEOUS B UKHGEIM -B ELTRAMIEQUATION
When applying the reconstruction method to the data arising from a general scattering kernel k p z, cos θ q “ ÿ n “ k ´ n p z q cos p nθ q an error is made due to the truncation in the Fourier series of k .This error is controlled by the gradient of the solution to the Cauchy problem for the inhomogeneousBukhgeim-Beltrami equation B v ` L B v “ B v ` f , (51)for some specific f and operator coefficient B . Estimates of the gradient for solutions of (51) maybe of separate interest, reason for which we treat them here independently of the transport problem.We start with an energy identity (see [49] for f “ ), `a la Mukhometov [30] or Pestov [42]. Theorem 4.1. (Energy identity) Let f P l , p N ; L p Ω qq and let B be a bounded operator such that B : l p N ; L p Ω qq Ñ l , p N ; L p Ω qq and B : l p N ; H p Ω qq Ñ l , p N ; H p Ω qq .If v P l , p N ; H p Ω qq is a solution to (51) , then ż Ω }B v } l dx “ ´ ż Ω ÿ j “ R e x L j B v , L j ´ B v y dx ` ż Ω ÿ j “ ›› L j B v ›› l dx (52) ´ ż Ω ÿ j “ R e x L j B v , L j ´ f y dx ` ż Ω ÿ j “ R e x L j B v , L j f y dx ` ż Ω ÿ j “ ›› L j f ›› l dx ` i ż Γ ÿ j “ x L j v , B s L j v y ds. Proof.
Using the Green’s identity ż Ω }B v } l dx “ ż Ω ›› B v ›› l dx ` i ż Γ x v , B s v y ds, where B s is thetangential derivative at the boundary, it follows that ż Ω }B v } l dx “ ż Ω ›› L B v ´ B v ´ f ›› l dx ` i ż Γ x v , B s v y ds “ ż Ω ›› L B v ›› l dx ´ ż Ω R e x L B v , B v y dx ` ż Ω } B v } l dx ´ ż Ω R e x L B v , f y dx ` ż Ω R e x B v , f y dx ` ż Ω } f } l dx ` i ż Γ x v , B s v y ds. For each n P N , ż Ω ›› L n B v ›› l dx “ ż Ω ›› L n ` B v ›› l dx ´ ż Ω R e x L n ` B v , L n B v y dx ` ż Ω ›› L n B v ›› l dx ´ ż Ω R e x L n ` B v , L n f y dx ` ż Ω R e x L n B v , L n f y dx ` ż Ω ›› L n f ›› l dx ` i ż Γ x L n v , B s L n v y ds. By summing in n , and using lim n Ñ8 ż Ω ›› L n B v ›› l dx “ , we conclude the theorem. (cid:3) We note the general identity [38, Lemma 2.1], for a sequence of nonnegative numbers:
Lemma 4.1.
Let t c n u be a sequence of nonnegative numbers. Then p i q ÿ m “ ÿ n “ c m ` n “ ÿ j “ p ` j q c j , p ii q ÿ m “ ÿ n “ c m ` n “ ÿ j “ ˆ ` Z j ^˙ c j , whenever one of the sides in (i) and (ii) is finite.Proof. (i) By changing the index j “ m ` n , for m ě , ( j ´ n ě , and n ď j ), we get ÿ m “ ÿ n “ c m ` n “ ÿ j “ j ÿ n “ c j “ ÿ j “ c j j ÿ n “ “ ÿ j “ p ` j q c j . (ii) Similarly, by changing the index j “ m ` n , for m ě , ˆ j ´ n ě , and n ď Z j ^˙ , weget ÿ m “ ÿ n “ c m ` n “ ÿ j “ t j u ÿ n “ c j “ ÿ j “ ˆ ` Z j ^˙ c j . (cid:3) Theorem 4.2 (Gradient estimate) . Let f P l , p N ; L p Ω qq , and B be a smoothing operator such that B : l p N ; L p Ω qq Ñ l , p N ; L p Ω qq and B : l p N ; H p Ω qq Ñ l , p N ; H p Ω qq are bounded, and let C B ą be such that } B v } , ď C B } v } , @ v P l p N ; L p Ω qq , (53a) } B v } , ď C B } v } , , @ v P l p N ; H p Ω qq . (53b) FOURIER APPROACH TO THE INVERSE SOURCE PROBLEM 11
Assume that B is such that (cid:15) : “ a µC B ă ? ´ , (54) where µ is the factor in Poincar´e inequality (27) .If v P l , p N ; H p Ω qq is a solution to the inhomogeneous Bukgheim-Beltrami equation (51) , then ď }B v } ď b ` ? b ` ac a , (55) where a “ ´ (cid:15) ´ (cid:15) ą , b “ (cid:15) } v | Γ } , ` ? } f } , , c “ (cid:15) } v | Γ } , ` π } v | Γ } , ` } f } , . Proof.
We estimate each term on the right hand side of the energy identity (52). For brevity if wedenote B v “ x b , b ´ , b ´ , ¨ ¨ ¨ y , then } B v } , “ ÿ j “ p ` j q ż Ω | b ´ j | .I. We estimate the first term in (52): ˇˇˇˇˇż Ω ÿ j “ x L j B v , L j ´ B v y dx ˇˇˇˇˇ “ ˇˇˇˇˇż Ω ÿ j “ ÿ k “ B v ´ j ´ k b ´ j ` ´ k ˇˇˇˇˇ ď ż Ω ÿ j “ |B v ´ j || jb ´ j ` | ď ˜ż Ω ÿ j “ |B v ´ j | ¸ { ˜ż Ω ÿ j “ p ` j q | b ´ j | ¸ { ď ? }B v } ˜ż Ω ÿ j “ p ` j q| b ´ j | ¸ { ď ? }B v } } B v } , ď ? C B }B v } } v } ď ? C B }B v } ? µ ´ }B v } ` } v | Γ } , ¯ “ a µC B }B v } ` a µC B }B v } } v | Γ } , , where in the first inequality we use Lemma 4.1 part (i), in the second inequality we useCauchy-Schwarz inequality, in the third inequality we use p ` x q ď p ` x q , in the fifthinequality we use } B v } , ď C B } v } , and in the next to last inequality we use the Poincar´einequality (27).II. We estimate the second term in (52): ż Ω ÿ j “ |x L j B v , L j ´ f y| “ ż Ω ÿ j “ ÿ k “ |B v ´ j ´ k f ´ j ` ´ k | ď ż Ω ÿ j “ |B v ´ j || jf ´ j ` |ď ˜ż Ω ÿ j “ |B v ´ j | ¸ { ˜ż Ω ÿ j “ p ` j q | f ´ j | ¸ { ď ? }B v } } f } , , where in the first inequality we use Lemma 4.1 part (i), and then Cauchy-Schwarz.III. We estimate the third term in (52): ż Ω ÿ j “ ›› | L j B v ›› l “ ż Ω ÿ j “ ÿ k “ | b ´ j ´ k | ď ż Ω ÿ j “ p ` j q| b ´ j | ď ż Ω ÿ j “ p ` j q| b ´ j | ď } B v } , ď C B } v } ď µC B }B v } ` µC B } v | Γ } , , where in the first inequality we use Lemma 4.1 part (i), in the next to the last inequality weuse } B v } , ď C B } v } , while in the last we use the Poincar´e inequality (27).IV. We estimate the fourth term in (52): ż Ω ÿ j “ ›› L j f ›› l “ ż Ω ÿ j “ ÿ k “ | f ´ j ´ k | ď ż Ω ÿ j “ p ` j q| f ´ j | ď ż Ω ÿ j “ p ` j q| f ´ j | “ } f } , , where in the first inequality we have used Lemma 4.1 part (i).V. We estimate the next to the last term in (52): ż Ω ÿ j “ |x L j B v , L j f y| ď ż Ω ÿ j “ ›› L j B v ›› l ›› L j f ›› l ď ż Ω ÿ j “ ›› L j B v ›› l ` ż Ω ÿ j “ ›› L j f ›› l ď µC B }B v } ` µC B } v | Γ } , ` } f } , where in the first inequality we have used Cauchy-Schwarz, in the second inequality we haveused the fact xy ď p x ` y q , and in the last inequality we have used estimates from (III) and(IV).VI. To estimate the last term in (52), we use the fact that Γ is the unit circle and consider theFourier expansion of the traces of the modes v ´ j | Γ . For e iβ P Γ , v ´ j p e iβ q “ ÿ k “´8 v ´ j,k e ikβ , B β v ´ j p e iβ q “ ÿ m “´8 p im q v ´ j,m e imβ . Using the parametrization, the last term in (52) becomes ˇˇˇˇˇż Γ ÿ j “ x L j v , B s L j v y ds ˇˇˇˇˇ “ ˇˇˇˇˇż Γ ÿ j “ ÿ k “ v ´ j ´ k B s v ´ j ´ k ds ˇˇˇˇˇ “ ˇˇˇˇˇż Γ ÿ j “ ˆ ` Z j ^˙ v ´ j B s v ´ j ds ˇˇˇˇˇ “ ˇˇˇˇˇż π ÿ j “ ˆ ` Z j ^˙ ÿ k “´8 v ´ j,k e ikβ ÿ m “´8 p´ im q v ´ j,m e ´ imβ dβ ˇˇˇˇˇ “ ˇˇˇˇˇ ÿ j “ ˆ ` Z j ^˙ ÿ k “´8 v ´ j,k ÿ m “´8 p´ im q v ´ j,m ż π e i p k ´ m q β dβ ˇˇˇˇˇ ď π ÿ j “ p ` j q ÿ k “´8 | k | | v ´ j,k v ´ j,k |ď π ÿ j “ ÿ k “´8 p ` j q p ` k q | v ´ j,k | “ π } v | Γ } , , (56) where in the second equality we have used Lemma 4.1 part (ii), in the first inequality we haveused the fact ` ` X j \˘ ď p ` j q { , and in the last equality we have used the definition ofthe norm (26).Using the above estimates (I)-(VI) for the expressions in (52), we have proved for τ “ ˆż Ω }B v } l ˙ { ,that aτ ´ bτ ´ c ď . Assumption on (cid:15) as in (54) yield a ą and we have the estimate (55). (cid:3) For the case when B “ we obtain the immediate corollary. FOURIER APPROACH TO THE INVERSE SOURCE PROBLEM 13
Corollary 4.1.
Let f P l , p N ; L p Ω qq . If v P l , p N ; H p Ω qq solves B v ` L B v “ f , (57) then }B v } ď } f } , ` π } v | Γ } , . (58) Proof.
This is the case (cid:15) “ and a “ in (55). We also use ˆ b ` ? b ` c ˙ ď b ` c. (cid:3)
5. S
MOOTHING DUE TO SCATTERING
In this section we explicit the smoothing properties of the Fourier multiplier operator K in(7) as determined by the appropriate decay of the Fourier coefficients of the scattering kernel k n p z q “ π ş π ´ π k p z, cos θ q e ´ inθ dθ , n P Z . The gain of { smoothness in the angular variable(see (61) below) has been shown before by different methods in [26], and in a more general casethan considered here in [46]. Lemma 5.1. (Smoothing due to scattering) Let M ě be a positive integer and K be the Fouriermultiplier in (7) . Assume that k is such that its Fourier coefficients starting from index M onwardsatisfy γ : “ sup j ě M p ` j q p max (cid:32) } k ´ j } , } ∇ x k ´ j } ( ă 8 , for p ą { . (59) (i) If v P l p N ; L p Ω qq , then ÿ n “ ›› L n ` M K v ›› ď γ p M ` q p ´ ›› L M v ›› . (60) (ii) K : l p N ; H p Ω qq Ñ l , p N ; H p Ω qq is bounded. More precisely, ›› L M K v ›› , ď ? γ p M ` q p ´ { ›› L M v ›› , , @ v P l p N ; H p Ω qq . (61) (iii) Moreover, if (59) holds for p ě , then K : l p N ; L p Ω qq Ñ l , p N ; L p Ω qq and K : l p N ; H p Ω qq Ñ l , p N ; H p Ω qq are bounded, and ›› L M K v ›› , ď γ p M ` q p ´ ›› L M v ›› , @ v P l p N ; L p Ω qq , (62) ›› L M K v ›› , ď ? γ p M ` q p ´ ›› L M v ›› , , @ v P l p N ; H p Ω qq . (63) Proof. (i) Let v P l p N ; L p Ω qq . Then ÿ n “ ›› L n ` M K v ›› “ ÿ n “ ÿ m “ ż Ω | k ´ n ´ m ´ M v ´ n ´ m ´ M | “ ÿ j “ p ` j q ż Ω | k ´ j ´ M v ´ j ´ M | ď γ ÿ j “ ` j p ` j ` M q p ż Ω | v ´ j ´ M | ď γ ÿ j “ p ` j ` M q p ´ ż Ω | v ´ j ´ M | ď γ p M ` q p ´ ÿ j “ ż Ω | v ´ j ´ M | ď γ p M ` q p ´ ›› L M v ›› , where in the second equality we have used Lemma 4.1 part (i), and in the first inequality we haveused (59).To prove (ii), let v P l p N ; H p Ω qq . Then ›› L M K v ›› , “ ÿ j “ p ` j q ˆż Ω | k ´ j ´ M v ´ j ´ M | ` ż Ω |Bp k ´ j ´ M v ´ j ´ M q| ˙ ď ÿ j “ p ` j q ˆż Ω | k ´ j ´ M v ´ j ´ M | ` ż Ω | k ´ j ´ M | |Bp v ´ j ´ M q| ` ż Ω | v ´ j ´ M | |B k ´ j ´ M | ˙ . (64)We estimate the first term in (64), ÿ j “ p ` j q ż Ω | k ´ j ´ M v ´ j ´ M | ď γ ÿ j “ p ` j q p ` j ` M q p ż Ω | v ´ j ´ M | ď γ ÿ j “ p ` j ` M q p ´ ż Ω | v ´ j ´ M | ď γ p M ` q p ´ ÿ j “ ż Ω | v ´ j ´ M | ď γ p M ` q p ´ ›› L M v ›› , (65)where in the first inequality we have used decay of k j ’s.Similarly, following the same proof as of the first term (65), the term ÿ j “ p ` j q ż Ω | k ´ j ´ M | |Bp v ´ j ´ M q| ď γ p M ` q p ´ ›› L M B v ›› , and the last term ÿ j “ p ` j q ż Ω | v ´ j ´ M | |Bp k ´ j ´ M q| ď γ p M ` q p ´ ›› L M v ›› . Thus the expression in (64) becomes ›› L M K v ›› , ď γ p M ` q p ´ ›› L M v ›› ` γ p M ` q p ´ ›› L M B v ›› ď γ p M ` q p ´ ´›› L M v ›› ` ›› L M B v ›› ¯ “ γ p M ` q p ´ ›› L M v ›› , . To prove (iii), assume that k is such that (59) holds for p ě . Let v P l p N ; L p Ω qq , then ›› L M K v ›› , “ ÿ j “ p ` j q ż Ω | k ´ j ´ M v ´ j ´ M | ď γ ÿ j “ p ` j qp ` j ` M q p ż Ω | v ´ j ´ M | ď γ ÿ j “ p ` j ` M q p ´ ż Ω | v ´ j ´ M | ď γ p M ` q p ´ ÿ j “ ż Ω | v ´ j ´ M | ď γ p M ` q p ´ ›› L M v ›› , where in the first inequality we have used decay of k j ’s. FOURIER APPROACH TO THE INVERSE SOURCE PROBLEM 15
To prove the last part, let v P l p N ; H p Ω qq . Then ›› L M K v ›› , “ ÿ j “ p ` j q ˆż Ω | k ´ j ´ M v ´ j ´ M | ` ż Ω |Bp k ´ j ´ M v ´ j ´ M q| ˙ . Following the similar proof of (ii), ` with p ` j q instead of p ` j q { ˘ , we have ›› L M K v ›› , ď γ p M ` q p ´ ´›› L M v ›› ` ›› L M B v ›› ¯ “ γ p M ` q p ´ ›› L M v ›› , . (cid:3)
6. E
RROR ESTIMATES
Recall the sequence valued maps u “ x u , u ´ , u ´ , ... y and u p M q “ x u p M q , u p M q´ , u p M q´ , ... y solu-tions of B u ` L B u ` aL u “ LK u , respectively of B u p M q ` L B u p M q ` aL u p M q “ LK p M q u p M q ,where K is the multiplier operator in (7), and K p M q is its truncated version in (36) correspondingto the M -th order polynomial approximation of k .The error we make in the source reconstruction is controlled by the sequence valued map q p M q : “ e ´ G p u ´ u p M q q , (66)which solves B q p M q ` L B q p M q “ e ´ G LK u ´ e ´ G LK p M q u p M q . (67)Recall that the integrating operators e ˘ G commute with the left translation, r e ˘ G , L s “ . Fromthe equation (36) is easy to see that the translated sequence L M q p M q “ e ´ G L M p u ´ u p M q q (68)solves(69) B L M q p M q ` L B L M q p M q “ e ´ G L M ` K u . For a P C ,s p Ω q , s ą { , let α , β be as in Proposition 2.1. For brevity, let us define σ “ max t} α } l , p Ω q , }B α } l , p Ω q , } β } l , p Ω q , }B β } l , p Ω q u . (70)Under sufficient regularity on k , its Fourier coefficients satisfy (59) with p “ . Lemma 6.1. If k P C ,sper pr´ , s ; Lip p Ω qq , s ą { , then γ M “ sup j ě M ` p ` j q max (cid:32) } k ´ j } , } ∇ x k ´ j } ( ă 8 , for any M ě . (71) Proof.
The proof follows directly by Bernstein’s lemma [19, Chap. I, Theorem 6.3]. (cid:3)
As a consequence we obtain the following a priori error estimate for the source reconstructed bythe method in Section 3.
Theorem 6.1. (Error estimate) Let M ě be a positive integer, a P C ,s p Ω q , k P C ,sper pr´ , s ; Lip p Ω qq ,for s ą { , be known, and f P H p Ω q and u P H p Ω ˆ S q be unknown functions satisfying (1) .Let u be the unknown sequence of nonpositive Fourier modes of u as in (3) , and f p M q in (49) be thereconstructed source using the noisy boundary data r g | Γ P l , p N ; H { p Γ qq . Let δ g “ u | Γ ´ r g bethe error in the data. The error δf “ f ´ f p M q in the reconstructed source is estimated by } δf } L p Ω q ď p ` C q M ` ´ c ›› L M ` u ›› ` c ›› L M δ g | Γ ›› , ` c } δ g | Γ } ¯ , (72) where c “ p ` µ q σ γ M , c “ π p ` µ q σ , c “ max t µσ , u , with σ in (70) , γ M in (71) ,and C depends only on Ω and max " , max ď j ď M } a ´ k ´ j } L p Ω q * .Proof. Since u P H p Ω ˆ S q , u P l , p N ; H p Ω qq , in particular u P l , p N ; H p Ω qq .From linearity of the problem, the error δf also satisfy (50): } δf } L p Ω q ď p ` C q M ` ˜ } δu ´ M ´ } H p Ω q ` } δu ´ M } H p Ω q ` M ÿ j “ } δg ´ M ` j } H { p Γ q ¸ , (73)where δg m for ´ M ` ď m ď , is the m -th Fourier coefficient of δ g .Recall that the δu ´ M and δu ´ M ´ are the first two components of e G L M q p M q . Using the gradientestimate in (58) we have ›› L M B q p M q ›› ď } e ´ G L M ` K u } , ` π } L M q p M q | Γ } , ď } α } l , p Ω q ›› L M ` K u ›› , ` π ›› L M q p M q | Γ ›› , ď σ γ M ›› L M ` u ›› ` π ›› L M q p M q | Γ ›› , , (74)where the first inequality uses (31) in Proposition 2.1, and the second inequality uses (62) with p “ . By Poincare and using (74), } δu ´ M ´ } H p Ω q ` } δu ´ M } H p Ω q ď ›› e G L M q p M q ›› , ď σ ›› L M q p M q ›› , ď p ` µ q σ ›› L M B q p M q ›› ` µσ ›› L M q p M q | Γ ›› ď p ` µ q σ γ M ›› L M ` u ›› ` π p ` µ q σ ›› L M q p M q | Γ ›› , ` µσ ›› L M q p M q | Γ ›› ď p ` µ q σ γ M ›› L M ` u ›› ` π p ` µ q σ ›› L M δ g | Γ ›› , ` µσ ›› L M δ g | Γ ›› . (75)From (75) and (73), the expression in (73) becomes } δf } L p Ω q ď p ` C q M ` ˜ c ›› L M ` u ›› ` c ›› L M δ g | Γ ›› , ` µσ ›› L M δ g | Γ ›› ` M ÿ j “ } δg ´ M ` j } H { p Γ q ¸ ď p ` C q M ` ´ c ›› L M ` u ›› ` c ›› L M δ g | Γ ›› , ` c } δ g | Γ } ¯ , where constant c “ p ` µ q σ γ M , c “ π p ` µ q σ , c “ max t µσ , u , and C depends onlyon Ω and max " , max ď j ď M } a ´ k ´ j } L p Ω q * . Thus (72) follows. (cid:3) For the case when k is a polynomial in the angular variable, k ´ j “ for j ě M ` , we obtainthe following stability result. Corollary 6.1. (Stability) Assume the hypotheses in Theorem 6.1 with k being a polynomial ofdegree M , i.e., k ´ j “ for j ě M ` . Then we have the following stability estimate } δf } L p Ω q ď p ` C q M ` ´ c ›› L M δ g | Γ ›› , ` c } δ g | Γ } ¯ , (76) where c “ π p ` µ q σ , c “ max t µσ , u , with σ in (70) , γ M in (71) , and C depends only on Ω and max " , max ď j ď M } a ´ k ´ j } L p Ω q * . FOURIER APPROACH TO THE INVERSE SOURCE PROBLEM 17
Proof.
This is the case γ M “ in (72). So c “ and result (76) follows. (cid:3) We stress that if k is not of polynomial type, then p ` C q M ` ›› L M ` u ›› can not be madearbitrarily small. However, if the anisotropic part of scattering is small enough (same case as in[7]), then we can prove a convergence result.We have the following convergence result for the weakly anistropic scattering media. Theorem 6.2.
Let a P C ,s p Ω q , k P C ,sper pr´ , s ; Lip p Ω qq , for s ą { , and f P H p Ω q . Assumethat γ “ sup j ě p ` j q p max (cid:32) } k ´ j } , } ∇ x k ´ j } ( ă ? ´ ? µσ , (77) where µ as in Poincar´e inequality (27) and σ as in (70) .For each arbitrarily fixed M , let f p M q in (49) be the reconstructed source using the data r g “ p g , g ´ , ¨ ¨ ¨ , g ´ M ` , g ´ M , Č g ´ M ´ , Č g ´ M ´ , ¨ ¨ ¨ q , (78) where g ´ j , for ď j ď M , are assumed exact data and Ă g ´ j , for j ą M , are allowed to be noisy.Then ›› f p M q ´ f ›› L p Ω q Ñ , as M Ñ 8 . (79) Proof.
Since f P H p Ω q , the solution u P H p Ω ˆ S q , and consequently u P l , p N ; H p Ω qq , inparticular u P l , p N ; H p Ω qq .The error we make in the source reconstruction is controlled by the sequence valued map q M “ e ´ G p u ´ u M q P l , p N ; H p Ω qq , which solves B q p M q ` L B q p M q “ B M q M ` A M u , (80)where B M and A M are operators given by B M “ e ´ G LK p M q e G , and A M “ e ´ G L p K ´ K p M q q , (81)with e ˘ G as in (21), K as in (7) and K p M q is the truncated Fourier multiplier as in (36).The trace q M | Γ “ e ´ G ` u ´ u p M q ˘ | Γ “ e ´ G δ g | Γ (82)is determined by the error in the data δ g | Γ “ ` u ´ u p M q ˘ | Γ “ p , , ¨ ¨ ¨ , , g ´ M ´ ´ Č g ´ M ´ , g ´ M ´ ´ Č g ´ M ´ , ¨ ¨ ¨ q . (83) For c “ max (cid:32) }p a ´ k q} , ( , the norm ›› f ´ f p M q ›› L p Ω q ď }p a ´ k q} ››› u ´ u p M q ››› L p Ω q ` ››› B u ´ ´ B u p M q´ ››› L p Ω q ď c ˆ››› u ´ u p M q ››› H p Ω q ` ››› u ´ ´ u p M q´ ››› H p Ω q ˙ ď c ›› e G q p M q ›› , ď c σ ´›› q p M q ›› ` ›› B q p M q ›› ¯ ď c σ p ` µ q ›› B q p M q ›› ` c σ µ ›› q p M q | Γ ›› , , ď c σ p ` µ q ›› B q p M q ›› ` c σ µ ›› L M ` p g ´ r g q ›› , , (84)where in the fourth and last inequality we use Proposition 2.1, in the next to last inequality we usethe Poincar´e inequality (27) and in the last inequality we have used (82).To estimate the gradient term ›› B q p M q ›› in (84), we employ Theorem 4.1 with v “ q M , B “ B M ,and f “ A M u in there as follows. For u P l , p N ; H p Ω qq , we have ›› A M u ›› , ď σ ›› L p K ´ K p M q q u ›› , “ σ ›› L M ` K u ›› , ď γσ ›› L M ` u ›› , where the first inequality uses Proposition 2.1, and the last inequality uses Lemma 5.1 (iii) for p “ . Since q M P l , p N ; H p Ω qq , we have ›› B M q M ›› , ď σ ›› LK p M q e G q M ›› , ď γσ ›› e G q M ›› ď γσ ›› q M ›› , and ›› B M q M ›› , ď σ ›› LK p M q e G q M ›› , ď ? γσ ›› e G q M ›› , ď ? γσ ›› q M ›› , , where the first inequality uses Proposition 2.1, and the last inequality uses Lemma 5.1 (iii) for p “ . We can apply Theorem 4.1 with C B “ ? γσ and (cid:15) “ ? µC B ă ? ´ by (77), to obtain ď ›› B q p M q ›› ď b M ` a b M ` ac M a , where a “ ´ (cid:15) ´ (cid:15) ą ,b M “ (cid:15) ›› q M | Γ ›› , ` ? γσ ›› L M ` u ›› ,c M “ (cid:15) ›› q M | Γ ›› , ` π ›› q M | Γ ›› , ` γ σ ›› L M ` u ›› . Since the data is assumed exact for the first M modes, p g ´ r g q | Γ in (83) has the first M modes equalto zero. Thus ›› q M | Γ ›› , ď σ ›› L M ` p g ´ r g q | Γ ›› , Ñ , ›› q M | Γ ›› , ď σ ›› L M ` p g ´ r g q | Γ ›› , Ñ , as M Ñ 8 . Since u is fixed, the expression ›› L M ` u ›› Ñ , as M Ñ 8 . Therefore, both b M and c M tends to zero and lim M Ñ8 ›› B q p M q ›› Ñ . Using (84), the norm lim M Ñ8 ›› f ´ f p M q ›› L p Ω q Ñ . (cid:3) FOURIER APPROACH TO THE INVERSE SOURCE PROBLEM 19
7. N
UMERICAL E XPERIMENTS
In this section we demonstrate the numerical feasibility of the method on two numerical exam-ples. We focus on the numerical results, and leave their realization details for a separate discus-sion [17]. Although the theoretical results assume a source of square integrable gradient, we shallsee below that the numerical reconstruction works even in the case of a discontinuous source.The numerical experiments consider θ ¨ ∇ u p z, θ q ` a p z q u p z, θ q “ ż S k p z, θ ¨ θ q u p z, θ q d θ ` f p z q , p z, θ q P Ω ˆ S , with the two dimensional Henyey-Greenstein (Poisson) model of scattering kernel k p z, θ ¨ θ q “ µ s p z q π ´ g ´ g θ ¨ θ ` g , (85)and the attenuation coefficient a p z q “ µ s p z q ` µ a p z q , where µ s is the scattering coefficient, and µ a is the absorption coefficient. The parameter g in (85)models a degree of anisotropy, with g “ for the ballistic regime and g “ for the isotropicscattering regime. In our numerical experiments we work with g “ { and µ s p z q “ , the lattervalue shows that, on average, the particle scatters within { units of length along the path.Let R “ p´ . , . q ˆ p´ . , . q be the rectangular region, and B “ tp x, y q ; p x ´ . q ` y ă . u , and B “ p x, y q : p x ` . q ` ˆ y ´ ? ˙ ă . + , be the circular region inside the unit disc Ω as shown in Figure 1. Ω R B B F IGURE
1. Locations of inclusions in numerical examples. The solution u p z, θ q isrelatively strongly absorbed inside the dotted balls, while the source f p z q is positivein the gray regions.We consider two examples for the attenuation coefficient a p z q for two different functions µ a p z q ,while µ s “ is the same. In the first example, we work with a C smooth absorption, whereas in thesecond example we consider attenuation to be discontinuous. In the first example, the attenuation a is C -smooth via some scaled and translated quartic polynomial p| z | ` q p| z | ´ q , ď | z | ď ,in the (cid:15) -neighborhoods of B B and B B : (86) µ a p z q “ $’&’% , in (cid:32) z : dist p z, B B q ě (cid:15) ( X B ;1 , in (cid:32) z : dist p z, B B q ě (cid:15) ( X B ;0 . , in (cid:32) z : dist p z, B q ě (cid:15) ( X (cid:32) z : dist p z, B q ě (cid:15) ( with (cid:15) “ . .In both examples the data is generated with the same source(87) f p z q “ $’&’% , in R ;1 , in B ;0 , otherwise . This is the function we reconstruct by implementing the method in Section 3.We generate the boundary measurement by the numerical computation of the forward problemby piecewise constant approximation following the method in [16]. In the forward problem thetriangular mesh has , , triangles, while the velocity direction is split into equi-spacedintervals.To obtain the data, we disregard the value of the solution inside and only keep the boundaryvalues. The boundary data, u p z, θ q on B Ω ˆ S , is represented in Figure 2: For each z P B Ω z=(1,0) × u ( z , θ ) θ F IGURE
2. Boundary measurement u p z, θ q| B Ω ˆ S obtained by numerical computa-tion of the corresponding forward problem. The red curves are (cid:32) z ` u p z, θ q θ : θ P S ( for z P B Ω indicated by the cross symbols ( ˆ ) (left). The right figure is amagnification of that at z “ p , q .(indicated by a cross), we represent the graph t ` u p z, θ q , θ ˘ : θ P S u as the (red) closed curve inthe polar coordinates (cid:32) z ` u p z, θ q θ : θ P S ( . Note that, since u | Γ ´ “ (there is no incomingradiation), the red curve lies outside the domain Ω . The nearly tangential behavior at the boundaryis the numerical evidence that the regime is far from ballistic, while the irregular shapes show thatit is far from isotropic. FOURIER APPROACH TO THE INVERSE SOURCE PROBLEM 21
The reconstruction starts with solving (39) with M “ via the Bukhgeim-Cauchy integral for-mula (10), where the infinite series is replaced by a finite sums of up to terms.In the reconstruction procedure, all the steps, including the integral transforms D r a s , R r a s and H “ R r a s ‰ , have been processed numerically [17]. In solving the Poisson equation (45a) and (45b),the standard P finite element method (constant and piecewise linear elements) is employed.The triangulation used in the reconstruction is different from that in the forward problem. Inparticular the reconstruction mesh consists of , triangles (much less than the , , trian-gles used in the forward problem), and is generated without any information of the location of thesubsets R , B , and B .In the first numerical experiment the attenuation is given by (86). The reconstructed source f p z q is shown in Figure 3 on the left. On the right we show the cross section of f p z q along the dotteddiameter y “ ´? x passing through the origin and the center of B .The reconstructed f p z q shows a quantitative agreement with the exact source in (87). Similarto the attenuated X-ray tomography, the artifacts appear due to the co-normal singularities in thesource, but also in the attenuation. This is because the most singular is the ballistic part (attenuatedX-ray transform of the source). -1 -0.5 0 0.5 1-1-0.5 0 0.5 1 x y -0.5 0 0.5 1 1.5 2 2.5 -0.5 -0.25 0 0.25 0.5-0.5 0 0.5 1 1.5 2 2.5 xf(z) F IGURE C -smooth attenuation case: Reconstructed source (on the left) and itssection on the dotted line (on the right). The arrows on the right show the pointswhere the dotted line meets B B and B R .In the second example, we apply the proposed algorithm to the case of a discontinuous attenua-tion a “ µ s ` µ a with µ a p z q “ $’&’% , in B ;1 , in B ;0 . , otherwise . While different, the boundary data in the second experiment is graphically indistinguishable fromthe data in Figure 2 in the smooth case.The numerically reconstructed source, shown in Figure 4, agrees well with the exact source (87),implying that the proposed algorithm has a potential in reconstruction even if the attenuation admitsthe discontinuity. -1 -0.5 0 0.5 1-1-0.5 0 0.5 1 x y -0.5 0 0.5 1 1.5 2 2.5 -0.5 -0.25 0 0.25 0.5-0.5 0 0.5 1 1.5 2 2.5 xf(z) F IGURE
4. Discontinuous attenuation case: Reconstructed source (on the left) andits section on the dotted line (on the right).The arrows on the right show the pointswhere the dotted line meets B B and B R . -0.4 -0.2 0 0.2 0.4 0.6-0.5 0 0.5 1 1.5 2 2.5 ∂ B f(z) in Rxf(z) -0.4 -0.2 0 0.2 0.4 0.6-0.5 0 0.5 1 1.5 2 2.5 ∂ B f(z) in Rxf(z) F IGURE
5. The projection on the x -axis of the reconstructed source f in a neighbor-hood of the rectangle R for the C -smooth attenuation (on the left) vs. discontinuousattenuation (on the right). The arrows indicate B B .Figure 5 shows the projection on the x -axis of the reconstructed f p z q for z P r´ . , . s ˆr´ . , . s . The arrows show the location of B B in R , where the attenuation has a jump.In both reconstructions two types of artifacts appear, one type due to the singular support of f andthe other due to the singular support of a . Observe how the effect of the latter type of singularitiesdiminishes with an increase in the smoothness of a .At the level of singular support, the image is similar to that in the non-scattering case ( k “ ).This is expected due to the smoothing effect of scattering. A quantitative understanding of the effectof singularities in the attenuation, in the presence of scattering is subject to future work. FOURIER APPROACH TO THE INVERSE SOURCE PROBLEM 23 A CKNOWLEDGMENT
The authors thank D. Kawagoe for useful discussions of regularity of solution of the forwardproblem. The work of H. Fujiwara was supported by JSPS KAKENHI Grant Numbers 16H02155,18K18719, and 18K03436. The work of K. Sadiq was supported by the Austrian Science Fund(FWF), Project P31053-N32.A
PPENDIX : N
EW MAPPING PROPERTIES OF THE INTEGRATING OPERATOR e ˘ G In this section we prove Proposition 2.1.
Proof.
Since a P C ,s p Ω q , s ą { , it follows from Lemma 2.1 that e ˘ h P C ,s p Ω ˆ S q . For z P Ω ,let α p z q “ x α p z q , α p z q , ... y , and B α p z q “ xB α p z q , B α p z q , ... y , where B is the derivative in thespatial variable. By the proof in Bernstein’s lemma [19, Chap. I, Theorem 6.3], sup z P Ω ÿ k “ k | α k p z q | ď sup z P Ω ›› e ´ h p z, ¨q ›› C ,s ă 8 , and sup z P Ω ÿ k “ k | B α k p z q | ď sup z P Ω ›› B e ´ h p z, ¨q ›› C ,s ă 8 . Similarly, β , B β P l , p Ω q .To prove the case p “ “ q , let u P l p N ; L p Ω qq . By using Young’s inequality ›› e ´ G u ›› “ ÿ j “ ›› p e ´ G u q ´ j ›› L p Ω q “ ż Ω } α ˚ u } l ď ż Ω } α } l } u } l ď sup Ω } α } l ż Ω } u } l “ sup Ω } α } l } u } , (88)where the discrete convolution (with respect to the index) is defined in (21).To prove the case p “ and q “ , let u P l , p N ; L p Ω qq . We estimate the norm ›› e ´ G u ›› , “ ÿ j “ p ` j q ›› p e ´ G u q ´ j ›› L p Ω q “ ÿ j “ ›› p e ´ G u q ´ j ›› L p Ω q ` ÿ j “ j ›› p e ´ G u q ´ j ›› L p Ω q ď sup Ω } α } l ż Ω } u } l ` ż Ω ÿ j “ j | p e ´ G u q ´ j | , (89)where in the first inequality we use estimate (88) from the case p “ “ q .To estimate the last term in (89), we need to account for both positive and negative Fourier modes.Let p u , and y e ´ h denote the sequence valued of the Fourier modes of u , and e ´ h , respectively, i.e., p u “ x¨ ¨ ¨ , u ´ n , ¨ ¨ ¨ , u ´ ,u , u , ¨ ¨ ¨ , u n , ¨ ¨ ¨ y , (90a) y e ´ h “ x¨ ¨ ¨ , , ¨ ¨ ¨ ¨ ¨ ¨ , , α , α , ¨ ¨ ¨ , α n , ¨ ¨ ¨ y . (90b)By using Plancherel we estimate ż Ω ÿ j “ j | p e ´ G u q ´ j | ď ż Ω ÿ j “´8 j |p y e ´ h ˚ p u q j | “ ż Ω ż S | B θ p e ´ h u q | ď ż Ω ż S | e ´ h B θ u | ` ż Ω ż S | u B θ p e ´ h q | . (91) To estimate the last two terms in (91), let x B θ u denote the (double) sequence of the Fourier modes of B θ u , and let z B θ e ´ h denote the sequence valued of the Fourier modes of B θ e ´ h , i.e., ´ i x B θ u “ x¨ ¨ ¨ , ´ nu ´ n , ¨ ¨ ¨ , ´ u ´ , ,u , u , ¨ ¨ ¨ ¨ ¨ ¨ , nu n , ¨ ¨ ¨ y , (92a) ´ i z B θ e ´ h “ x¨ ¨ ¨ ¨ ¨ ¨ , , ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ , , , α , α , ¨ ¨ ¨ ¨ ¨ ¨ , nα n , ¨ ¨ ¨ y . (92b)By using Plancherel the expression in (91) becomes ż Ω ÿ j “ j | p e ´ G u q ´ j | ď ż Ω ›››y e ´ h ˚ x B θ u ››› l p Z q ` ż Ω ››› z B θ e ´ h ˚ p u ››› l p Z q ď sup Ω ›››y e ´ h ››› l p Z q ż Ω ÿ j “´8 |p x B θ u q j | ` sup Ω ››› z B θ e ´ h ››› l p Z q ż Ω ÿ j “´8 | p u j | “ sup Ω } α } l ż Ω ÿ j “´8 j | p u j | ` sup z P Ω ˜ ÿ k “ k | α k p z q | ¸ ż Ω ÿ j “´8 | p u j | “ Ω } α } l ż Ω ÿ j “ j | u ´ j | ` } α } l , p Ω q ż Ω } u } l , where in the second inequality we use Young’s inequality, in the first equality we use sup Ω ›››y e ´ h ››› l p Z q “ sup Ω } α } l , and sup Ω ››› z B θ e ´ h ››› l p Z q “ sup z P Ω ˜ ÿ k “ k | α k p z q | ¸ , and in the last equality we use the norm in(28) and u ´ n “ u n as u is real valued.From the above estimate of (91), the expression in (89) becomes ›› e ´ G u ›› , ď sup Ω } α } l ˜ż Ω } u } l ` ż Ω ÿ j “ j | u ´ j | ¸ ` } α } l , p Ω q ż Ω } u } l ď Ω } α } l } u } , ` } α } l , p Ω q } u } ď } α } l , p Ω q } u } , . (93)To prove the case p “ and q “ , let u P l p N ; H p Ω qq , then u , B u P l p N ; L p Ω qq . For u P l p N ; L p Ω qq , e ˘ G u P l p N ; L p Ω qq from the case p “ “ q , so it suffices to show that Bp e ´ G u q P l p N ; L p Ω qq .The discrete convolution below is with respect to the index, while B acts on the spatial variable. ›› Bp e ´ G u q ›› “ ż Ω }Bp α ˚ u q} l ď ż Ω } α ˚ B u } l ` ż Ω }B α ˚ u } l ď ż Ω } α } l }B u } l ` ż Ω }B α } l } u } l ď sup Ω } α } l }B u } ` sup Ω }B α } l } u } , (94)where in the second inequality we use discrete Young’s inequality.From (94) and estimate (88) from the case p “ “ q , we estimate the norm ›› e ´ G u ›› , “ ›› e ´ G u ›› ` ›› Bp e ´ G u q ›› ď sup Ω } α } l ` } u } ` }B u } ˘ ` sup Ω }B α } l } u } ď sup Ω } α } l } u } , ` sup Ω }B α } l } u } ď ´ } α } l , p Ω q ` }B α } l , p Ω q ¯ } u } , . (95) FOURIER APPROACH TO THE INVERSE SOURCE PROBLEM 25
To prove the case p “ “ q , let u P l , p N ; H p Ω qq , then u , B u P l , p N ; L p Ω qq . Since u P l , p N ; L p Ω qq , e ´ G u P l , p N ; L p Ω qq by case p “ and q “ . Thus suffices to show that Bp e ´ G u q P l , p N ; L p Ω qq . We estimate the norm ›› Bp e ´ G u q ›› , “ ›› Bp e ´ G u q ›› ` ÿ j “ j ›› Bp e ´ G u q ´ j ›› L p Ω q ď sup Ω } α } l }B u } ` sup Ω }B α } l } u } ` ż Ω ÿ j “ j | pBp α ˚ u qq ´ j | , (96)where in the first inequality we have used estimate (94) from the case p “ and q “ .To estimate the last term in (96), we need to account for both positive and negative Fourier modes.Let p u , and y e ´ h denote the sequence valued of the Fourier modes of u , and e ´ h , respectively, as in(90), and let x B θ u and z B θ e ´ h denote the sequence valued of the Fourier modes of B θ u , and B θ e ´ h ,respectively, as in (92). Moreover, let x B u , and z B θ B u denote the sequence valued of the Fouriermodes of B u , respectively, B θ B u , i.e. x B u “ x¨ ¨ ¨ ¨ ¨ ¨ , B u ´ n , ¨ ¨ ¨ ¨ ¨ ¨ , B u ´ , B u , B u , ¨ ¨ ¨ , B u n , ¨ ¨ ¨ y , ´ i z B θ B u “ x¨ ¨ ¨ , ´ n B u ´ n , ¨ ¨ ¨ , ´B u ´ , , B u , ¨ ¨ ¨ , n B u n , ¨ ¨ ¨ y . Similarly, let z B e ´ h , and { B θ B e ´ h denote the sequence valued of the Fourier modes of B e ´ h , respec-tively, B θ B e ´ h , and recall that their negative Fourier modes vanish: z B e ´ h “ x¨ ¨ ¨ , , ¨ ¨ ¨ , B α , B α , B α , ¨ ¨ ¨ , B α n , ¨ ¨ ¨ y , ´ i { B θ B e ´ h “ x¨ ¨ ¨ ¨ , , ¨ ¨ ¨ , , B α , B α , ¨ ¨ ¨ , n B α n , ¨ ¨ ¨ y . Using Plancherel and the notations above, we estimate the last term in (96) as follows. ż Ω ÿ j “ j | pBp α ˚ u qq ´ j | ď ż Ω ÿ j “ j | pB α ˚ u q ´ j | ` ż Ω ÿ j “ j | p α ˚ B u qq ´ j | ď ż Ω ÿ j “´8 j |p z B e ´ h ˚ p u q j | ` ż Ω ÿ j “´8 j |p y e ´ h ˚ x B u q j | “ ż Ω ż S | B θ ppB e ´ h q u q | ` ż Ω ż S | B θ p e ´ h B u q | ď ż Ω ż S | pB θ B e ´ h q u | ` ż Ω ż S | Bp e ´ h q B θ u | ` ż Ω ż S | B θ e ´ h B u | ` ż Ω ż S | e ´ h B θ pB u q | . (97)Next we estimate each term on the right hand side of (97) separately.I. Estimate the first term in (97): ż Ω ż S | pB θ B e ´ h q u | “ ż Ω ››› { B θ B e ´ h ˚ p u ››› l p Z q ď sup Ω ››› { B θ B e ´ h ››› l p Z q ż Ω } p u } l p Z q “ sup Ω ˜ ÿ k “ k | B α k p z q | ¸ ż Ω ÿ j “´8 | p u j | “ }B α } l , p Ω q ż Ω } u } l , where in the first equality we use Plancherel, in the first inequality we use Young’s inequality,in the second equality we have used the fact that sup Ω ››› { B θ B e ´ h ››› l p Z q “ sup z P Ω `ř k “ k | B α k p z q | ˘ ,and in the last equality we use the norm in (28) and the fact that u ´ n “ u n as u is real valued.II. Estimate the second term in (97): ż Ω ż S | B e ´ h B θ u | “ ż Ω ›››z B e ´ h ˚ x B θ u ››› l p Z q ď sup Ω ›››z B e ´ h ››› l p Z q ż Ω ››› x B θ u ››› l p Z q “ sup Ω ˜ ÿ k “ | B α k p z q | ¸ ż Ω ÿ j “´8 |p x B θ u q j | “ Ω }B α } l ż Ω ÿ j “ j | u ´ j | , where in the first equality we have use Plancherel, in the first inequality we use Young’s in-equality, in the second equality we have used the fact that sup Ω ›››z B e ´ h ››› l p Z q “ sup z P Ω `ř k “ | B α k p z q | ˘ ,and in the last equality we use the fact that u ´ n “ u n as u is real valued.III. Estimate the third term in (97): ż Ω ż S | B θ e ´ h B u | “ ż Ω ››› z B θ e ´ h ˚ x B u ››› l p Z q ď sup Ω ››› z B θ e ´ h ››› l p Z q ż Ω ›››x B u ››› l p Z q “ sup Ω ˜ ÿ k “ k | α k p z q | ¸ ż Ω ÿ j “´8 |p x B u q j | “ } α } l , p Ω q ż Ω }B u } l , where in the first equality we have use Plancherel, in the first inequality we use Young’s in-equality, in the second equality we have used the fact that sup Ω ››› z B θ e ´ h ››› l p Z q “ sup z P Ω `ř k “ k | α k p z q | ˘ ,and in the last equality we use the norm in (28) and the fact that u ´ n “ u n as u is real valued.IV. Estimate the last term in (97): ż Ω ż S | e ´ h B θ B u | “ ż Ω ›››y e ´ h ˚ z B θ B u ››› l p Z q ď sup Ω ›››y e ´ h ››› l p Z q ż Ω ›››z B θ B u ››› l p Z q “ sup Ω ˜ ÿ k “ | α k p z q | ¸ ż Ω ÿ j “´8 |p z B θ B u q j | “ Ω } α } l ż Ω ÿ j “ j | B u ´ j | , where in the first equality we use Plancherel, in the first inequality we use Young’s inequality,in the second equality we have used the fact that sup Ω ›››y e ´ h ››› l p Z q “ sup z P Ω `ř k “ | α k p z q | ˘ ,and in the second to last equality we use u ´ n “ u n as u is real valued.The last term of (96) is thus estimated by ÿ j “ j (cid:107) pBp e ´ G u qq ´ j (cid:107) L p Ω q ď }B α } l , p Ω q ż Ω } u } l ` Ω }B α } l ż Ω ÿ j “ j | u ´ j | ` } α } l , p Ω q ż Ω }B u } l ` Ω } α } l ż Ω ÿ j “ j | B u ´ j | . FOURIER APPROACH TO THE INVERSE SOURCE PROBLEM 27
Therefore the estimate (96) yields, ›› Bp e ´ G u q ›› , ď }B α } l , p Ω q } u } ` sup Ω }B α } l ˜ż Ω } u } l ` ż Ω ÿ j “ j | u ´ j | ¸ ` } α } l , p Ω q }B u } ` sup Ω } α } l ˜ż Ω }B u } l ` ÿ j “ j | B u ´ j | ¸ ď } α } l , p Ω q }B u } ` Ω }B α } l } u } , ` } α } l , p Ω q }B u } ` Ω } α } l }B u } , ď } α } l , p Ω q }B u } , ` } α } l , p Ω q }B u } , . Using the above estimate of ||Bp e ´ G u q|| , and estimate (93), the norm ›› e ´ G u ›› , “ ›› e ´ G u ›› , ` ›› Bp e ´ G u q ›› , ď ´ } α } l , p Ω q ` }B α } l , p Ω q ¯ } u } , ` } α } l , p Ω q }B u } , ď ´ } α } l , p Ω q ` }B α } l , p Ω q ¯ } u } , . (98)The mapping property for the case p “ and q “ , follows from interpolation and the continu-ous embeddings l , p N ; H p Ω qq ã ÝÑ l , p N ; H p Ω qq ã ÝÑ l p N ; H p Ω qq . (cid:3) R EFERENCES [1] D. S. Anikonov, A. E. Kovtanyuk and I.V. Prokhorov,
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