An Inverse Problem for One-dimensional Diffusion Equation in Optical Tomography

In this paper, we study the one-dimensional inverse problem for the diffusion equation based optical tomography. The objective of the present work is a mathematical and numerical analysis concerning one-dimensional inverse problem. In the first stage, the forward diffusion equation with boundary conditions is solved using an intermediate elliptic equation. We give the existence and the uniqueness results of the solution. An approximation of the photon density in frequency-domain is proposed using a Splines Galerkin method. In the second stage, we give theoretical results such as the stability and lipschitz-continuity of the forward solution and the Fréchet differentiability of the Dirichlet-to-Neumann nonlinear map with respect to the optical parameters. The Fréchet derivative is used to linearize the considered inverse problem. The Newton method based on the regularization technique will allow us to compute the approximate solutions of the inverse problem. Several test examples are used to verify high accuracy, effectiveness and good resolution properties for smooth and discontinuous optical property solutions.


Introduction
This paper deals with the inverse problem in one-dimensional optical tomography.This problem consists of the estimation of the optical properties, namely the absorption coefficient µ a , the diffusion coefficient D and the refractive index ν, from the flow measurements at the boundary of the computational domain.The signal of measurements is assumed to be measured at the boundary using the photon density solution of the forward diffusion equation with the modified Robin boundary conditions.The estimation of the solution of the inverse problem requires an investigation of the forward problem.Such investigations must be of physical and mathematical aspects, numerical methods, software and computer simulations.In the corresponding forward problem, we consider the diffusion transport equation.This equation arises in many scientific applications such as in optical tomography.The principle of the optical tomography is to use multiple movable light sources and detectors attached to the tissue surface, such as human tissue, to collect information on light attenuation and to reconstruct the internal absorption and scattering distributions.The time-dependent diffusion transport equations have been studied by several authors in the recent years; see [12,13,14,17].The radiative transfer transport equation (also derived from the Boltzmann equation) has been used successfully as a standard model for describing light transport in scattering media, see [11,13,14,17].The variational method proves that the corresponding forward problem has a unique solution for the given optical properties and the sources [2,3].In the literature, many numerical techniques have been proposed to solve such problem.Our approaches for solving the proposed inverse problem is based on the Tikhonov regularization [18].In the current study, we present some theoretical results concerning the continuity of the forward solutions and Dirichlet-to-Neumann (DtN) map with respect to the optical parameters, Lipschitz-continuous property of the DtN map, the adjoint diffusion problem and the Fréchet derivative.Such derivative allows us to obtain a linearized problem from the nonlinear inverse problem.So, the discretized problem by using the spline basis functions may be written in the vectorial form and the obtained problem is solved by using the Newton iterations.The goal of the inverse problem is to determine unknown coefficients that appear in the diffusion equation from measurements at the boundary of the solution of the associated forward problem.The paper is organized as follows.In Section 2, we recall some results of the forward diffusion problem.In Section 3, we set the inverse problem of 1D-diffusion equation.We give some theoretical results concerning the Lipschitz-continuity of the Dirichlet-to-Neumann linear map and of its Fréchet Derivative.This guarantees the stability of the solution of the forward diffusion problem in terms of the Robin boundary conditions.Section 3 is ended by some results concerning the adjoint diffusion problem.In Section 4, using the spline basis functions and the Fréchet derivatives, we formulate the discretized inverse problem and its equivalent optimization problem.The Newton method combined with the Tikhonov regularization technique is used to solve such an optimization problem, which require the computation of the jacobian and the hessian matrices.In Section 5, we give some numerical tests to illustrate our proposed methods.

Statement of the forward diffusion problem 2.1. One-dimensional diffusion problem
To facilitate the readability of this paper, we recall some notations and results already given in [3].Let φ(x, t) be the photon density at the position x ∈]a, b[⊂ R and at the time t ∈ [0, T ], where T is the time period.Let µ a , µ s and ν be the absorption coefficient, the scattering coefficient and the refractive index coefficient, respectively.The reduced scattering coefficient is given by μs (x) = (1 − ϑ)µ s (x) where ϑ ∈ [0, 1] is the anisotropic parameter.The diffusion potential function D is related to µ a and μs by the following relation: Let us denote by c(x) = c 0 /ν(x) the speed of light in the domain [a, b], where c 0 is the speed of light in the vacuum [3,15] and ν(x) ≥ 1 for all x ∈ [a, b].We assume that µ a , µ s and ν are real, bounded and nonnegative functions on [a, b].For (x, t) in ]a, b[×[0, T ] we denote by φ(x, t) the photon density located at position x at time t.The time-dependent diffusion equation is described by the following parabolic partial differential equation (see [3,4]) for every (x, t) ∈]a, b[×[0, T ] with the initial condition φ(x, 0) = φ 0 (x) for every x ∈ (a, b), where φ 0 is a given function.The derivative operations given in (2.2) are used in the distribution sense.Let ζ a ≥ 0 and ζ b ≥ 0 be the refractive indexes mismatched at the boundaries a and b, respectively.We introduce the following constants β a and β b given for all ξ ∈ {a, b} by β ξ = 0 if ζ ξ = 0 and In this work, we consider the Robin boundary conditions given as follows, (2. 3) The purposed forward diffusion problem is (2.2)-(2.3)which consists of finding the photon density φ(x, t) for a given data f , g a , g b , D, µ a and ν, and the inverse problem for one-dimensional diffusion equation is to compute the optical parameters s = (ν, D, µ a ) for a given source terms f , g a , g b and the flow measurements at M. ADDAM the boundary of our computational domain.
Throughout this paper, we assume that the real-valued functions D, µ a and ν are bounded functions and belongs to L ∞ ([a, b], R), and D, µ a and ν are positive functions and there is α > 0 such that: (2.4)

Frequency-domain framework
To understand the forward problem and its corresponding inverse problem treated in this work, we need to introduce the following classical spaces [1,8].b[, C).The space L 2 is endowed by its natural topology defined by the following inner product and its associated norm ) be the usual Sobolev space, with the classical norm .H k given by where ∂ ℓ u ∂x ℓ stands for the derivative of u, in the distribution sense, of order ℓ.The classical space L The space H k is equipped with the classical norm .H k given as in (2.6).For 1 ≤ p < ∞, the spaces L p (R; H k ) and L p (R; H k ) endowed with the norms We assume that the signal function t −→ φ(x, t) and the derivative function We also assume that the given sources terms f , g a and g b have extensions, which are still denoted by f , g a and We recall that the Fourier transform g of a function g in L 1 (R, R) is given by Inverse Problem

163
By using the Lebesgue theorem of the derivation under the integral sign, we obtain We apply the Fourier transform to the equation (2.2) together with the nonhomogeneous Robin boundary conditions (2.3) and we use the classical relation ∂g ∂t = iω g.Then we obtain the following Sturm-Liouville equation satisfied by the photon density u(x, ω) in the frequency domain (2.8) with the following non-homogeneous boundary conditions where u(., ω) = φ(., ω), F (., ω) = f (., ω), G a (ω) = g a (ω) and G b (ω) = g b (ω), are the Fourier transforms with respect to the time variable of φ(x, .),f (x, .),g a and g b , respectively.In the literature, u is called the photon density in frequency domain and F , G a and G b are also called the sources terms in frequency domain.

Weak variational formulation
Let u * (., ω) be a sufficiently smooth function satisfying the problem (2.8)-(2.9),for instance u * (., ω) is in H 1 .Multiplying the diffusion equation (2.8) by an arbitrary function v ∈ H 1 , and integrating over [a, b] with respect to the nonhomogeneous Robin boundary conditions (2.9), provides the following weak variational formulation: where the sesquilinear form respectively.Therefore, the problem (2.8)-(2.9) is equivalent to the weak variational formulation (2.10).
From the Sobolev continuous embedding theorem, there exists a constant C > 0 (depending only on a and b) such that (2.11) Using the Schwarz inequality and the Sobolev continuous embedding theorems, it follows that there exists a constant C 0 > 0 (depending only on a and b) such that where Therefore, the operator L ω : H 1 → C is a continuous semi-linear form on H 1 .Since the functions D, µ a and ν satisfies (2.4) and using again the Sobolev continuous imbedding theorem, it follows that there exists a constant C 1 > 0 (not depending on the frequency ω) such that where the positive constants α and M (ω) are given in (2.4) and (2.12), respectively.Furthermore, if ) is the unique solution of the problem (2.8)-(2.9) in the usual sense.

Discretized forward diffusion problem
We discretize the interval [a, b] using equally spaced knots x i = a + ih, for i = 0, 1, 2 . . ., n with x 0 = a, x n = b and h = (b − a)/n.For i = −3, . . ., n + 1, let us consider a partition of (n + 5) nodes of the interval

16)
Inverse Problem 165 So, for a fixed frequency ω in R, let us denote by V h the finite dimensional subspace of H 2 , given by (2.17) The classical Galerkin approximation consists of finding an approximation u h (., ω) of the analytic solution u * (., ω) as a solution in V h of the following discrete variational problem: The solution u h (., ω) of the problem (2.18) is written in the following form where the z h,i (ω) are unknown complex coefficients depending on the frequency ω and (B i ) 1≤i≤n are the functions given by with B 3 is the classical cubic Spline with the support embedding in [0,4].For more details on the Spline we refer to [2,5].The space V h of solution given in (2.17) is spanned by the Spline family (B i ) 1≤i≤N .By using the test function v h = B j in the weak variational formulation (2.18), we obtain with for all (i, j) ∈ {1, . . ., N } 2 .Let A(ω) = [A ij (ω)] be the N × N matrix whose entries are the complex coefficients , let b h (ω) = (b h,1 (ω), . . ., b h,N (ω)) T be the complex vector whose coefficients are b h,j (ω) = L ω (B j ) and let z h (ω) = (z h,1 (ω), . . ., z h,N (ω)) T be the vector formed

Inverse problem for diffusion problem in optical tomography
In the following sections, we consider the Schwartz space S(R; L 2 ) of functions φ : t ∈ R −→ φ(., t) belonging to L 2 and satisfying the conditions: The space S(R; L 2 ) is endowed with the topology defined by the set of the seminorms family (N ℓ,m ) ℓ,m .The topological dual space S ′ (R; L 2 ) of S(R; L 2 ) is the Schwartz space of tempered distributions, namely the space of linear continuous forms from S(R; L 2 ) into R.
For every s ∈ R, the non-homogeneous Sobolev spaces H s (R; R) defined by is endowed with the topology defined by the norm given by The non-homogeneous Sobolev spaces H s (R; L 2 ) given by is endowed with the topology defined by the norm given by For every (r, s) ∈ R 2 , the non-homogeneous Sobolev spaces H s (R; H r ) is defined by replacing the space L 2 and the norm (3.3) by H r and respectively.
The product space L ∞ (a, b)×L ∞ (a, b)×L ∞ (a, b) endowed with the L ∞ -topology defined by the norm given by is a Banach space.
In the following sections, we will study the inverse problem which consists of the estimation of the optical properties from the flow measurements of analytic and computed solutions of the direct problem (2.8)-(2.9),at the boundary of the domain.For given sources functions f , g a and g b , we denote then by φ(x, s, t) := φ[s](x, t) the solution of the forward diffusion problem (2.2)-( 2.3) and by u(x, s, ω) := u[s](x, ω) the solution of the forward diffusion problem in frequency domain (2.8)-(2.9),with respect to the optical parameters s = (ν, D, µ a ).We will demonstrate theoretical results concerning the inverse problem and describe a numerical approximation of the parameters s = (ν, D, µ a ) using the values of the signal at the boundary of our computational domain.

Dirichlet-to-Neumann map
where ∂ϕ ∂x is the derivative of ϕ ∈ H 1 in the distribution sense.The subspace B is equipped by the inner product and its associated norm induced by the Hilbert space The subspace B is equipped by the inner product and its associated norm induced by the Hilbert space It is clear that the subspaces (B, .B ) and ( B, , .B ) are Hilbert spaces.Define the nonlinear operator Λ by where where t ∈ [0, T ].So, by using the boundary conditions (2.3) and (3.9) we obtain Now, define the non-linear operator Λ by where where ω ∈ R be a frequency value and u(., s, .)denote the solution the problem (2.8)-(2.9).So, by using (3.12) and the boundary conditions (2.9) we prove that Proposition 3.1.For a given frequency ω ∈ R, the linear map Λ s is characterized by the following duality product for all (v, z a , z b ) ∈ B and for all source functions , where u(., s, ω) ∈ H 1 is the solution of (2.8) − (2.9) corresponding to the source functions F (., ω), G a (ω) and G b (ω).
Proof.Let us denote by G, the following duality product By using the boundary conditions (3.7), we obtain the following equality From the variational formulation (2.10) we have Using the boundary conditions corresponding to the solution u(., s, ω), we get Substituting the identities (3.17In the following theorem, we will show the continuity of the linear operators Λ s and Λ s with respect to optical parameters s = (ν, D, µ a ).Let us denote it by where the space of the linear and continuous operators from and where the space of the linear and continuous operators from We denote by .L2 and .L2 the usual norms of L 2 and L 2 , respectively.So, we introduce the following notations. 3with their components satisfying the condition (2.4) and let u 1 := φ 1 and u 2 := φ 2 be the solutions of the diffusion problem (2.8) − (2.9) with respect to the parameters s 1 and s 2 , respectively.Then, there exists a constant C > 0, independent of s 1 and s 2 , such that Proof.Let ω be a given real frequency.
(iii) As in Item (i), we consider two solutions u 2 (., ω) and u 1 (., ω) in H 1 of the diffusion problem (2.8) − (2.9) with respect to the parameters s 1 and s 2 , respectively.Let Y = (v, z a , z b ) where v is a function in H 1 satisfying the trace boundary conditions (3.7).According to Proposition 3.1, we get Taking into account of (2.11), there exists a constant C 1 > 0 such that It follows that Since ν, D and µ a belong to L ∞ ([a, b], R), then there exists ̺ > 0 such that Inverse Problem 175 and using the norm (3.5) we obtain Using again the inequality t 1 t 2 + t 3 t 4 ≤ (t 1 + t 3 )(t 2 + t 4 ) for positive real numbers t i (i = 1, . . ., 4), we obtain From the Cauchy-Schwarz inequality, we obtain Using (2.12), (2.15) and (3.27) of Item (i), there exists a constant C 3 > 0 such that Then, the following inequality holds.Using again the inequality (t 3 ) for reals numbers t i (i = 1, 2, 3), it follows that Integrating the inequality (3.30) over R with respect to ω holds the following inequality Now, by using the Parseval identity, we may write Thus, there exists a constant which concludes the proof of Item (iii).✷

Fréchet derivatives and adjoint diffusion problem
Now, we will study the Fréchet derivatives of the DtN.Let s = (ν, D, σ a ), h = (h 1 , h 2 , h 3 ) and ŝ = (ν 0 , D 0 , σ a0 ) are available such that s = ŝ + h.Let's consider φ and φ 0 the analytic solutions of the problem (2.2)-(2.3)with respect to the parameters s and ŝ, respectively, and with the source functions X(t) = (f (., t), g a (t), g b (t)).The difference ψ = φ−φ 0 belongs to H 2 ⊂ H 1 .Let's consider u and u 0 the exact solutions of the problem (2.8)-(2.9)with respect to the parameters s and ŝ, respectively, and with the source functions X(ω) = (F (., ω), G a (ω), G b (ω)).So, the difference w = u−u 0 belongs to H 2 ⊂ H 1 .Let us consider Y = (ϕ, r a , r b ) and Y = (v, z a , z b ) belongs to B and B respectively.We consider the non-linear maps respectively.We have −D(a)w(a, ω) ∂v ∂x (a).

M. ADDAM
We notice that the function w = u − u 0 ∈ H 2 satisfies the non-homogeneous equation with the non-homogeneous Robin boundary conditions (3.36) with the notations and assumptions of Proposition 3.1 and Theorem 3.2.There exists a constant C > 0 such that ŝ is the Fréchet derivative at ŝ = (ν 0 , D 0 , σ a0 ) of Λ s with respect to (ν, D, σ a ).
(iii) The derivatives Λ ′ ŝ and Λ ′ ŝ have the following Lipschitz-continuous property, , for all ξ in {a, b}, then the linearization Λ ′ s h is given by where w is the solution of Eq.(3.35)-(3.36).
Proof.Let ω ∈ R be a fixed frequency.
(i) We have for ξ = a or b, we use similar arguments as in the proof of Theorem 3.2.We get Using the Cauchy-Schwarz inequality and according to (2.11), there exists a constant According to the inequality (3.26), there exists a constant C 2 > 0 such that where C 3 = 2 max{1, C1C2̺ α } is independent of ω, s and ŝ and M. ADDAM (ii) By using the inequality (3.28), we obtain where 2 .Integrating over R with respect to ω and using the Parseval identity, we obtain It follows that By similar arguments, we prove that ⋆ of Λ ′ ŝh is given by the following duality product where u ⋆ is the unique solution of the adjoint equation: with the non-homogeneous boundary conditions where Y ⋆ (ω) = (u ⋆ (., ω), z ⋆ a (ω), z ⋆ b (ω)) ∈ B. Furthermore, the adjoint of Λ ′ s h is given by Proof.The adjoint of this linearized equation Eq.(3.35) can be derived using the integration by part Using the assumption (3.41), we have h 2 (a) = h 2 (b) = 0, then the equation (3.34) implies An immediate consequence of the theorem 3.3 is as follows which achieves the proof.✷

Discretized inverse problem, Tikhonov regularization and Newton method
The inverse problem treated here, consists of an estimation the parameters s * = (ν * , D * , µ * a ) around an observed optical parameter ŝ = (ν 0 , D 0 , µ a0 ).The nominal valued vector function ŝ is composed by the observed optical parameters ν 0 , D 0 and µ a0 .We assume that N s source terms X j (., ω) = (f j (., ω), g aj (ω), g bj (ω)), for 1 ≤ j ≤ N s are available.Consider the cost functional J given by for all s = (ν, D, µ a ) ∈ (L ∞ ([a, b], R)) 3 .For every 1 ≤ j ≤ N s , we have where Let u j (., s, .)and u j (., ŝ, .)denote the exact solutions of the problem (2.8)-(2.9)with respect to the sources term X j (., ω) and the parameters s and ŝ, respectively.For j = 1, . . ., N s , we have where ε aj and ε bj are the noise perturbations due to the beam measured at the boundary a and b for a given source term X j (., ω).In general, measurement of u j (x, s, ω) may be not possible, only some observable part Γ 2j−1 (s) and Γ 2j (s) of the actual state u j (x, s, ω) may be measured.Also, the cost functional J is given by 2) The objective of the inverse problem is to find the vector parameter s * in the domain [a, b], that minimizes the cost functional (4.2) over all possible s in [a, b] subject to u j (x, s, ω) satisfying the diffusion problem (2.8)-(2.9)with the source terms X j (., ω) for all j = 1, . . ., N s .So, we consider s * as the solution of the following minimization problem Consider the following space S h ′ given by where P 3 is the space of polynomials of degree ≤ 3. It is well known that the dimension of the subspace S h is m = n ′ + 3. We use similar notations by considering the classical cubic Spline as a basis B 1 , . . ., B m of the subspace S h ′ .Here, (B i ) 1≤i≤m denote the Splines associated with the given partition.We have where x i−4 = a + (i − 4)h ′ , for i = 1, . . ., m.We use the piecewise approximations for s = (ν, D, µ a ) in the cubic spline basis (B ℓ ) 1≤ℓ≤m as follows The functions ν h ′ , D h ′ and µ ah ′ belong to the space S h ′ .Let s m = (ν 1 , . . ., ν m , d 1 , . . ., d m , µ a1 , . . ., µ am ) T be a vector in R 3m with the components are the coefficients given in the discrete compositions (4.7).Thus, we get a relationship between the continuous and the discrete cases Now, we consider the discrete cost functional s m → J(s m ) given as follows where R(s for j = 1, . . ., N s and u hj (., s m , .) is a solution of the direct problem with respect to the sources term X j (., ω), and u j (a, ŝm , ω) and u j (b, ŝm , ω) are the data measurements at the boundary a and b for given source terms X j (., ω).The norm R(s m ) 2 is the residual at s m .The discrete minimization problem associated with the problem (4.3) is The problems (4.3) and (4.11) are ill-posed problems.In order to diminish the effects of the noise in the data, we replace the original problem (4.11) by a better conditioned one.One of the most popular regularization methods is due to Tikhonov [18].The method replaces the problem (4.11) by the following unconstrained optimization problem where the typical cost functional J λ is and the discrete parameter ŝm is a nominal vector.The coefficient λ > 0 is the regularization parameter to be chosen and ŝm is the coefficient of an observed nominal parameter ŝ in the Spline basis functions.Since the function J λ is Gâteaux-differentiable and convex it follows that s m * is a solution of the unconstrained optimization problem (4.12) if and only if The Hessian ∇ 2 J λ (s m ) is positive semi-definite for all s m ∈ R 3m .The Newton iterations consist of generating a vector sequence (s m k ) k from a guess s m 0 in a neighborhood of s m * and the next iterations are computed from the scheme where η k is backtracking strategy coefficient.Let d k be the decent direction, d k is a solution of the system We get the following iterations scheme

Computation of the Jacobian and the Hessian
Taking into account the expressions in (4.7), then ∀ (i, j) ∈ {1, . . ., N } 2 , the coefficients A ij (ω) of the matrix A(ω) may be written in the following form where D (ℓ) ij and U (ℓ) ij are given as respectively.Let D ℓ and U ℓ be the matrices of size m × m with coefficients D (ℓ) ij and U (ℓ) ij respectively.Thus, where Q is a matrix with coefficient Let s m = (s 1 , . . ., . . ., s 3m ) T be a generic vector in R 3m .The first derivatives of (2.21) with respect to the coefficients s ℓ are given as follows and, the second derivatives are given for all ℓ, ℓ ′ = 1, . . ., 3m as and By using the relations (4.19), we compute the derivatives ∂A(ω) ∂s ℓ for all 1 ≤ ℓ ≤ 3m, from the following expressions (4.23) For j = 1, . . ., N s , let X j (., t) = (f j (., t), g aj (t), g bj (t)) be the j th sources term, then the corresponding solution u hj (., s m , ω) is given by By using the Fréchet derivative, we compute the Jacobian matrix K(s m ) of the vector valued function R(s m ) as the 2N s × 3m matrix with the entries are given by K j,ℓ (s m ) = ∂R j (s m ) ∂s ℓ , ℓ = 1, ..., 3m and j = 1, . . ., 2N s .

Let H(s
It is obvious that there exists a constant C > 0 such that for all s m ∈ R 3m and for all h ∈ R 3m such that Then, the Hessian matrix is given for all s m by Hence, the linear systems (4.15) may be written in the following form And come the following iterations for k = 0, 1, 2 . . ., k max , where η k is a backtracking small positif number.

Numerical results and applications
In this section, we give numerical examples to illustrate the effectiveness of our proposed method.All computations were carried out using Matlab 6.5 on an Intel Pentium workstation with about 16 significant decimal digits.Let [a, b] be the computational domain and ω = 2π υ be a modulation frequency parameter where υ is the wavelength number of laser light in the wavelength range between 650 and 1000 nm.Let δ −ts : t → δ(t − t s ) be a Dirac measure function defined for all time variable t in [0, T ] where t s is a given time center in [0, T ].The Fourier transform of δ −ts considered as a time forcing distribution source is given by (for more detail see [16]): (5.1)

Verification tests for the forward solver
To test our forward solver, we first computed the approximate solutions in two cases: Example 5.1.Let c = 1, µ a = 1, µ s = 2 and ϑ ∈ [0, 1], then D = 1 3−2ϑ .For a given time center t s and the source terms F (x, ω), G a (ω) and G b (ω), the analytic solution u e (x, ω) of the diffusion problem in frequency domain is x 2 e −x , ∀x ∈]0, 6[.
We consider the relative L p -norm error function defined as where • L p is the L p -norm, u h and u e are respectively, the computed and analytic solutions.First we check the accuracy of the direct solver with respect to the number of gridpoints used in the computational domain.To this end we summarize in Table 1 the L ∞ -error, L 1 -error and L 2 -error norms at time t s = 0.5 using different values of N .It is evident that increasing the number of gridpoints in the computational domain results in a decrease in all error norms along with an increase in the computational cost.It should be stressed that the main part of the CPU times listed in Table 1 are used by the direct solver for solving the associated linear systems Eq.(2.21).
Table 1: Error norms for Example 1 using different numbers of gridpoints and the wavelength number υ = 750, the anisotropic coefficient ϑ = 0.9 and t s = 0.5.In the example 2, we check the accuracy of the direct solver with respect to the number of gridpoints used in the computational domain.To this end, we summarize in Table 2 the L ∞ -error, L 1 -error and L 2 -error norms using different values of N .It is evident that increasing the number of gridpoints in the computational domain results in a decrease in all error norms along with an increase in the computational cost.It should be stressed that the main part of the CPU times listed in Table 2 are used by the direct solver for solving the associated linear systems Eq.(2.21).

Numerical experiment for the inverse problem solver
To test our inverse problem solver, we compute the approximate solutions in two cases: Example 5.3.We consider the diffusion problem in frequency domain (2.8)-(2.9)with the source terms given, for j = 1, . . ., N s , x s ) is a Dirac's measure function at the boundary ξ ∈ {a, b}, x (j) s ∈]a, b[ is a center position of the j th source and A o is an amplitude of the sources terms.The flow measurements at the boundary of computational domain are computed for j = 1, . . ., N s as follows In this example, we fix a source functions and detectors at the boundary position.In the following computations, we use a Newton scheme (4.32) with the initial estimation s 0 = (1, . . ., 1) ∈ R 3(n ′ +3)×1 , a tolerance (T ol = 10 −8 ), a number of iterations k max = 20, a backtracking number η = 0.5 and Tikhonov regularization parameter λ = 10 −6 .In the left-hand side of Figure 1, we present respectively, the simulation plots of the diffusion coefficient x → D(x), the absorption coefficient x → µ a (x) and the refractive index coefficient x → ν(x) for n = n ′ = 20.And, in the right-hand side of Figure 1, we give respectively, the simulations plots of the diffusion coefficient x → D(x), the absorption coefficient x → µ a (x) and the refractive index coefficient x → ν(x) for n = n ′ = 160.The line corresponds to truth optical parameters (D, µ a , ν) and the star corresponds to the constructed optical parameters (D, µ a , ν).The comparison of the left-hand side of Figure 1 and the right-hand side of Figure 1 demonstrates the accurate convergence of the presented method, along with increasing the number of gridpoints in the computational domain.Therefore, we have provided and investigated the theoretical convergence and implementation of our algorithm with accurate reconstruction of the optical parameters.In the following computations, we use a Newton scheme (4.32) with the initial estimation s 0 = (1, . . ., 1) ∈ R 3(n ′ +3)×1 , a tolerance (T ol = 10 −10 ), a number of Inverse Problem 193 iterations k max = 20, a backtracking number η = 0.5 and Tikhonov regularization parameter λ = 10 −2 .In the left-hand side of Figure 2, we present respectively, the simulation plots of the diffusion coefficient x → D(x), the absorption coefficient x → µ a (x) and the refractive index coefficient x → ν(x), for n = n ′ = 20.And, in the right-hand side of Figure 2, we give respectively, the simulation plots of the diffusion coefficient x → D(x), the absorption coefficient x → µ a (x) and the refractive index coefficient x → ν(x), for n = n ′ = 160.The line corresponds to truth optical parameters (D, µ a , ν) and the star corresponds to the constructed optical parameters (D, µ a , ν).The comparison of the left-hand side of Figure 2 and the right-hand side of Figure 2 demonstrates the accurate convergence of the presented method, along with an increasing the number of gridpoints in the computational domain.Therefore, we have provided and investigated the theoretical convergence and implementation of our algorithm with accurate reconstruction of the optical parameters.

Conclusions
In this paper, we presented an non-linear inverse problem for one-dimensional diffusion transport equation with Robin boundary conditions.For the forward diffusion problem, we gave results such as existence, uniqueness and smoothness solutions of the diffusion problem in the frequency domain.And, for the inverse problem, we gave the same theoretical results such as the continuity, the stability and Fréchet derivatives of the Dirichlet-to-Neumann non-linear map.Also, we discretized the inverse problem using the cubic spline basis functions.The Tikhonov regularization and Newton solver are used to compute the solution of the considered inverse problem.Consequently, an approximate of the optical parameters are obtained in the cubic spline basis functions.Some numerical examples are presented to prove the efficacious of the proposed method for the forward solver and backward solver.From a modeling point of view, the problem considered in this paper, has some limitations such as the applications in mammography, breast cancer treatment, and optical tomography.The 2D-inverse problem or Higher dimensional problem for more realistic applications in optical tomography, needs more investigation and future work.
2 (]a, b[, R) of square integrable real-valued functions on the open interval ]a, b[ is denoted by L 2 := L 2 (]a, b[, R) and H k denotes the usual Sobolev space H and Λ s is the Dirichlet-to-Neumann (DtN) linear map corresponding to s = (ν, D, µ a ) given by , and the Dirichlet-to-Neumann linear map Λ s corresponding to s = (ν, D, µ a ) is given by

. 13 )
The linear map Λ s is an analogue operator of the linear operator Λ s given in (3.9).It describes Fourier's law in the frequency domain.The operators Λ s and Λ s associated to s = (ν, D, µ a ) are the detectors of the output flow measurements at the boundary {a, b} of the domain [a, b].

( 3 .
23) Multiplying the equation (3.21) by w(., ω) and integrating over [a, b], by using the conditions (3.23), we obtain the following equation .29) Inverse Problem 173 To achieve the proof of Item i), we integrate the inequations (3.29) over R with respect to ω by taking into account the definition of the norms in (3.2), (3.3) and (3.5).
modeled the sources with (x s , t s ) = 15a+b 16 , T