The time domain **technique** has the ability to reconstruct an accurate proﬁle of electrical properties as in the FBTS **technique**, compared to frequency domain based **technique**. Thus, the FBTS **technique** has been used for the detection of breast cancer [6, 7, 35], tumors in the lung [36], and tumors in the brain [37]. Following the advancement of FBTS, ﬁlters such as Elliptic ﬁlter [38] and Chebyshev ﬁlter [39] are incorporated to solve the non-linearity problem. In addition, regularization techniques are also integrated into FBTS as described in [40, 41] to handle the quantitative information with an ill-posed or ill-conditioning problem. Regularization techniques are able to provide higher accuracy of electrical properties proﬁle, object shapes, and location. Other promising techniques introduced in [42] propose the inversion method that eﬃciently handles the strong non-linearity of **inverse** **scattering** problem in the inhomogeneous medium using diﬀerence Lippmann-Schwinger integral equation (D-LSIE) and diﬀerence new integral equation (D-NIE).

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A two-dimensional (2D) Forward-Backward Time-Stepping (FBTS) **inverse** **scattering** **technique** has been applied to the breast and demonstrated good results in detecting tumours in the breast [12–15]. Therefore, the focus of this paper is on a brain tumour detection utilizing **inverse** **scattering** **technique** in 2D head model. A homogenous head model of an MRI in 3D (.mnc format) is obtained from [16, 17]. A slice from the head model is selected to get the 2D transverse plane view, and this slice is then used as an object under test (OUT). For numerical analysis purposes, we consider 4 signiﬁcant tissue types: skin, skull, grey matter (GM) and white matter (WM). The head model contains dielectric properties with high contrast between the skin and skull, and low contrast between GM and WM. This paper demonstrates the ability of the **inverse** **scattering** **technique** to detect an embedded tumour of diﬀerent sizes in WM region. Safety was taken into account for microwave imaging even though electromagnetic wave is non-ionizing wave and has a certain impact on biological beings as it can increase the temperature at the area of incident wave. However, limiting the frequency to less than 6 GHz for a certain amount of time of exposure on tissues at a distance of 200 mm and below helps to prevent an adverse thermal eﬀect [18, 19]. Therefore, a near-ﬁeld EMI is used for this research.

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The Forward-Backward Time-Stepping (FBTS) **inverse** **scattering** **technique** is utilized for breast composition reconstruction of an extremely dense breast model at different center frequencies. A numerical extremely dense breast phantom is used and resized to suit the Finite-Difference Time-Domain (FDTD) lattice environment utilizing two- dimensional (2-D) FBTS **technique**. The average value of fibro glandular region for reconstruction with Frequency- hopping approach applied is much closer to average value of the actual image compared to the reconstruction without Frequency-approach applied. Hence, the composition of the extremely dense breast model can be reconstructed with Frequency-hopping approach is applied and the details of the reconstruction is also enhanced.

Abstract—In this paper, a Frequency-Dependent Forward-Backward Time-Stepping (FD-FBTS) **inverse** **scattering** **technique** is used for reconstruction of homogeneous dispersive object. The aim of the **technique** is to reconstruct the relative permittivity at inﬁnite frequency, static relative permittivity and static conductivity of the homogeneous dispersive object simultaneously. The **technique** utilizes iterative ﬁnite-diﬀerence time-domain (FDTD) method for solving **inverse** **scattering** problem in time domain. The minimization of the cost functional is carried out utilizing Dai-Yuan nonlinear conjugate-gradient algorithm. The Fr´ echet derivatives of the augmented cost functional are derived analytically with respect to scatterer properties. Numerical results for reconstruction of two-dimensional homogeneous dispersive illustrate the performance of the proposed **technique**.

**Inverse** **scattering** plays a very important role in radar, remote sensing, non-destructive testing, etc. For a conducting target, the **inverse** **scattering** techniques are basically divided into two categories. The ﬁrst category is based on the physical optics approximation and discrete Fourier transformation, such as [1–5]. The main advantage of this category is that the computation is very eﬃcient. However, the **inverse** **scattering** **technique** of this category is limited to only a convex target with the smooth surface The second category is to solve nonlinear **scattering** integral equations directly by numerical methods, such as [6–10]. The main advantage of this category is that there is no limitation on the target shape. However, the computation is time-consuming and even diﬃcult due to the nonlinearity and ill-posed problems. In [11], the **technique** of the second category is modiﬁed and further transformed into a nonlinear optimization problem. Thus the **inverse** **scattering** becomes a nonlinear optimization problem. This will make the **inverse** **scattering** scheme clear and easy since numerical techniques for solving nonlinear integral equations have been replaced by optimization algorithms

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The practical situation when ultrasonic NDE is used is in fact an **inverse** problem, i.e. based on the signals from transmitting and receiving probes an interpretation is performed (see Figure 1). This interpretation is then often based on earlier experiences or by comparing with experimental work or computer simulations. Analytical solutions to the **inverse** problem have up today only been found for very simple situations and are often based upon a linearization of the **inverse** problem performed by the Born approximation (an extensive review discuss- ing this is available by Bates et al. [4]). This linearization limits the applicability since it, at least in principle, restricts the problem to weak penetrable scatterers or low frequencies. Even so this assumption has actually been successfully applied for more complex ultrasonic situations [5]. A slightly different approach, still based on the Born approximation, has been to retrieve a large amount of point source information and address the inversion by various time domain back propagation techniques (Synthetic Aperture Focussing Techniques described in e.g. [6] [7]). Degtyar et al. [8] introduced an inversion procedure based on a nonlinear least-squares method to de- termine elastic constants from group or phase velocity data in orthotropic and transversely isotropic materials. Corresponding approach has also been used in order to retrieve viscoelastic material properties based on ultra- sonic experimental data (Castaings et al. [9] [10]).

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Kaup [16], Caudrey [18, 20] and Deift et al. [21] studied the **inverse** problem for certain third order spectral equations. We adapt the results obtained by these authors to the present problem and describe a procedure for using the IST to ﬁnd the N -soliton solution to the transformed VE, and hence to the VE itself.

We introduce a novel iterative method for solving nonlinear **inverse** **scattering** problems. Inspired by the theory of nonlocality, we formulate the **inverse** **scattering** problem in terms of reconstructing the nonlocal unknown **scattering** potential V from scattered field measurements made outside a sample. Utilizing the one- to-one correspondence between V and T, the T-matrix, we iteratively search for a diagonally dominated **scattering** potential V corresponding to a data compatible T-matrix T. This formulation only explicitly uses the data measurements when initializing the iterations, and the size of the data set is not a limiting factor. After introducing this method, named data-compatible T-matrix completion (DCTMC), we detail numerous improvements the speed up convergence. Numerical simulations are conducted that provide evidence that DCTMC is a viable method for solving strongly nonlinear ill-posed **inverse** problems

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In Chapter 2 of this dissertation, we look at different ways of regularizing Gauss- Newton steps based on a priori information available for particular models. We also study an iterative approach to the selection of a regularization parameter and propose a new a posteriori stopping rule to terminate Gauss-Newton iterations “just in time” before the noise propagation can potentially destroy an approximate solution. Numer- ical experiments for both linear and nonlinear models are conducted to illustrate this **technique**.

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Recently, there has been an increased interest in Bayesian principles for **inverse** problems, see e.g., [1, 32, 34, 35]. One possibility is to include the Tikhonov regularization parameter in a Bayesian framework and devise efficient algorithms to determine this parameter from the data, see [1, 32, 34, 35]. These approaches, and the related derivations leading to useful algorithms, usually relies on some form of linearization, such as e.g., with the distorted Born iterative method [4], etc. However, the approach taken here is different. A similar connection is exploited as in e.g., [34], i.e., the connection between the Maximum A Posteriori (MAP) estimate with a Gaussian prior, and the Tikhonov regularization. However, instead of placing a prior on the regularization parameter itself, the MAP criterion is exploited here in a Fisher information analysis setting, which is relating to some known background of interest.

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• In practice, it looks like our result is not the most convenient way to approach the ISM on the full line as we introduce extra **scattering** data (T (k)) only to eliminate it in the end, using the relations (2.52)-(2.53). However, at the conceptual level, our point of view is rather unifying. Firstly, it brings further justification for the use of the terminology “unified” transform. The central idea of a simultaneous spectral analysis of the both half of the Lax pair now also encompasses the historical ISM as a special case, in sharp contrast with the traditional spectral analysis of only one half of the Lax pair. Secondly, as we illustrate in the rest of the paper on the concept of reductions, it allows us to cast “new” (nonlocal) reductions as “old” (local) ones (see below for what we mean by this). This produces a framework to tackle the classification of nonlocal reductions, taking advantage of the huge amount of available results for the local case.

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Rather, it is a first step for future developments among which further study of the classical r matrix approach and quantization of the method are important. Let us mention also the contruction of other integrable defects allowing if possible reflection as well. If applicable, the quantization should then be related to existing quantum algebraic frame- works like the Reflection-Transmission algebras [6]. Finally, the complete setup of the direct and **inverse** part of the method for the actual construction of the solutions, espe- cially of soliton type, should shed new light on the results already obtained by the more direct approach of [11, 12].

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As is well known, unfortunately, both the absorption and the refractive index of materials depend very strongly on wavelength; the variation of these quantities for water, for example, is depicted in Figure 3.2. It is easy to see from this figure that, for imaging of biological bodies—in which water is a main component— for example, an **inverse** problem solver must only rely upon a range of frequencies restricted to the narrow wavelength band around the visible band—for which the absorption is very small, and for which the index of refraction is virtually constant. Indeed, use of light outside this narrow band (for which the combined effects of the orders-of-magnitude larger absorption losses and the uncertainties caused by the fast and large variations of the refractive indexes—which, in view of Figure 3.2 are certain to occur, but which, because of the presence of other materials in combination with water, are actually unknown), cannot provide any useful information about the internal structure of a water-based sample.

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In this paper, we study the **inverse** nodal problem of the problem (1.1) and (1.2). Nodes are the zeros of eigenfunctions. **Inverse** nodal problem is to reconstruct potential function by nodal set, in some experiments, nodal set is easier to be observed and measured than the other spectral data. **Inverse** nodal problem for Sturm-Liouville operators with Dirichlet boundary conditions, see the original paper by McLaughlin [23]. and some generalizations were made in [4, 5, 8, 11, 13, 14, 15, 21, 26, 28, 29, 30, 31] and so on. For the other applications of nodal sets, for example, the paper [17] considered the zeros of Bessel functions and their application to the uniqueness of **inverse** acoustic **scattering** problem; the papers [18, 19, 20] showed that nodal sets of the Laplacian eigenfunctions play a critical role in establishing the uniqueness results for the **inverse** **scattering** problems.

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INRIA-IRISA, Campus de Beaulieu, Rennes Cedex 35042, France Abstract—As studied by Jaulent in 1982, the **inverse** problem of lossy electric transmission lines is closely related to the **inverse** **scattering** of Zakharov-Shabat equations with two potential functions. Focusing on the numerical solution of this **inverse** **scattering** problem, we develop a fast one-shot algorithm based on the Gelfand-Levitan- Marchenko equations and on some differential equations derived from the Zakharov-Shabat equations. Compared to existing results, this new algorithm is computationally more efficient. It is then applied to the synthesis of non uniform lossy electric transmission lines.

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The **inverse** problem is solved iteratively with the Newton-Kantorovich method [21, 22]. To constitute the data set, emitters also function as receivers. The proﬁle cannot be estimated directly from that data set. The **inverse** problem is thus recasted as an optimization problem. The solution is built up iteratively by successively solving the forward problem and a local linear **inverse** problem. With the locally rough plane approach, the forward problem can be accurately modelized despite the fact that the emitters footprints on the proﬁle are much wider than the sampled part of the proﬁle. The local linear **inverse** problem relies on the calculus of the Fr´ echet derivative of the non-linear **scattering** operator. Based on the reciprocity theorem, this calculus was ﬁrst proposed in far-ﬁeld conﬁguration [23]. It is here adapted to localized emitters and receivers. The **inverse** problem is detailed in Section 3.

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We started this thesis by introducing the necessary motivation for studying the problem of recon- structing the refractive index distribution and why we use the BPM model to solve it. We also note that in practice one uses diffraction tomography based on the Born or the Rytov approximation. Then, we discussed the necessary elements of the BPM model which was required for modeling the **inverse** problem. It was mainly important in calculating the gradient, which was the main tool in all of our reconstruction methods.

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For decades now, integrable partial differential equations (PDEs), and more generally integrable systems, have fuelled research and important discoveries in Mathematics and Physics, and still do. Comparatively more recently, graphs and dynamical systems on (quantum) graphs have emerged as a successful framework to model a large variety of (complex) systems. It is therefore not surprising to see a fast growing interest in developing a theory of integrable systems on graphs, which would combine the power of integrable systems with the flexibility of graphs to model more realistic situations. The review [1] for instance gives a flavour and references for this fast growing area in the context of nonlinear Schr¨ odinger (NLS) equations (not restricted to integrable cases). Originally, integrable PDEs were treated as initial value problems for functions of one space variable x ∈ R and one time variable t ≥ 0. The invention of the **inverse** **scattering** method (ISM) [2] and its refinements [3, 4] through the systematic use of a Lax pair [5] represents a cornerstone of modern integrable PDEs. The first departure from this setup to solve an initial-boundary value (IBV) problem for an integrable PDE on the half-line [6, 7] or a finite interval [8] can be viewed in retrospect as the beginning of the study of integrable PDEs on metric graphs. Indeed, a half-line is nothing but a half-infinite edge attached to a vertex and a finite interval is a finite edge connecting two vertices. The next big step in this natural evolution was the study of integrable PDEs on the line with a defect/impurity at a fixed site (or possibly several such defects). The vast literature on this problem 1 [9, 10, 11, 12, 13, 14, 15, 16, 17, 18] shows both its interest and its difficulty. To date however, despite some impressive results on the behaviour of certain solutions [19, 20], the general problem of formulating an ISM for a problem with defects is still open.

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To obtain **scattering** data of a single RBC, different methods have been used in the scientific literature: the finite-difference time- domain (FDTD) approach, the multilevel fast multipole algorithm (MLFMA) and others [4]. In this paper, we consider the discrete dipole approximation (DDA) [8, 9] that has shown good performance [3, 7, 8]. DDA is a method to compute **scattering** and absorption of electromagnetic wave by particles of arbitrary geometry and composition. In more detail, DDA replaces the solid particle by an array of M oscillating dipoles. Each dipole has an oscillating polarization in response both to an incident plane wave and to the electric fields due to all of the other dipoles in the array; the self- consistent solution for the dipole polarizations can be obtained as the solution to a set of coupled linear equations [7–9]. The optical cell shape model is the same of [7, 10], where a 4-parametric shape model has been proposed. The cell shape is described by as follows:

There are two microwave imaging methods which can be applied for the detection of tumour and abnormalities inside the human body, which are tomography [12] and radar-based **technique** [13]. In microwave tomography, EM radiation is used and the **inverse** **scattering** algorithm is applied to reconstruct shape, location, and the DP of the interest object. Meanwhile, radar-based approach utilizes simpler and faster computational to identify the presence and location of the tumour using significant backscattered signal. Here, this study will focus on the radar-based MI. Detail concept on microwave radar imaging is presented in the following section.

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