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Louis,Joyce McLaughlin,William Rundell Inverse difficulties are eager about deciding upon explanations for saw or wanted results. The mathematical modelling of inverse difficulties often results in ill-posed difficulties, i. This quantity comprises twelve survey papers approximately answer tools for inverse and ill-posed difficulties and approximately their software to precise kinds of inverse difficulties, e.

The papers were written by means of top specialists within the box and supply an up to date account of resolution tools for inverse problems. Iterative methods for ill-posed equations need to be stopped after a finite number of iterations to keep the ill-posedness under control; compare, for example, [ 23 , 85 ].

The main idea of the decomposition methods is to split full nonlinear shape reconstruction problem given by 18 into a linear ill-posed equation which is solved first and a nonlinear well-posed equation to be solved in a second step. In order to discuss the Kirsch-Kress method [ 56 — 58 ] via potential approach let us employ an initial boundary as an approximation of the actual boundary.

We assume that is in the interior of the true scatterer. In this case, the approximate total field can be approximated by the sum of the incident wave and a single-layer potential, where is the fundamental solution to the Helmholtz equation in two dimensions and is a continuous density source function defined over. The far field pattern of the single-layer potential should be measured far field pattern , which leads to the linear equation with some constant depending on the dimension of the space under consideration.

The first step consists of solving the ill-posed linear equation 23 to calculate. Then, when we consider a Dirichlet boundary condition, the shape is found in a second nonlinear step as the zero curve of the total field. To find this zero curve, we introduce an operator , which maps to the values of the approximate total field , on such that Then the problem is reduced to the solution of the following optimization problem: Linear sampling [ 70 — 74 ] is based on solving the far field equation Then, the density satisfies that is, when the source point approaches the boundary.

This behaviour can be used to visualize the shape of the scatterer from the knowledge of the far field pattern for all directions. We also refer to the orthogonality sampling method, which has recently been suggested in [ 86 ]. It is particularly suited to deal with multifrequency data as is naturally obtained when acoustic pulses are used to probe an object or region in space. The method has been independently suggested by Ito et al.

A convergence analysis of the method in the limit of small scatterers has recently been achieved by Griesmaier [ 89 ]. The inverse problem considered in this section is finding a continuous function which is defined on the boundary of the obstacle from the knowledge of the scattered acoustic waves for a given shape in two dimensions.

We discuss the methods whose aims are to reconstruct , in the impedance boundary condition 6 , and , in the conductive boundary condition 7. Note that conductivity function is employed to model the inhomogeneity which might exist on the boundary of an object in a more realistic way. In [ 90 — 92 ], 3D obstacles with impedance boundary condition are studied for acoustic case, where in [ 91 ] electromagnetic case is included. In the same field, the papers [ 64 , 93 — 96 ] are focused on 2D geometries for the less complexity of governing numerical experiments. To this aim, approximate Green's functions are used to reduce the nonlinear problem to two linear moment problems.

On the other hand, the study [ 92 ] is devoted for the reconstruction of impedance functions via the Kirsch-Kress and Colton-Monk decomposition methods. Furthermore, some interesting papers appeared on the impedance reconstructions, recently [ 97 , 98 ]. The paper [ 93 ] introduced a new method for impedance reconstructions in the spirit of the Kirsch-Kress decomposition method. That is, the scattered field is represented via single-layer potential over the known boundary of the impedance cylinder , instead of defining an auxiliary initially guessed curve.

From the knowledge of the density function now the total field and the normal derivative of the total field can be computed on the boundary of the obstacle via jump relations [ 22 ]. Finally, is obtained from 6 in the least squares sense.

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This method is also extended for the reconstructions of the conductivity functions of the obstacles in free space [ 99 ], for the obstacles buried in penetrable cylinders [ ] and for a combination of a shape and conductivity function reconstruction problem [ ], firstly by Yaman [ 6 ]. Moreover, [ 64 , 95 , 96 ] are devoted for the shape and impedance reconstructions of 2D obstacles in acoustics.

To do this, the hybrid method is employed by Serranho [ 64 , 66 ]. In [ 95 ], a level set algorithm is combined with boundary integral equations in acoustic case to reconstruct the shape and impedance of 2D obstacles from multi-illuminations, and in [ 96 ], it has been shown that the knowledge of the scattered fields corresponding to three incident waves can be used for the determination of the shape and the impedance via integral equation methods and conformal mapping techniques.

Let be a 3D space variable, and let be a 1D time variable; let be the displacement vector function of an inhomogeneous anisotropic elastic material characterizing by density and the elastic moduli. The density and elastic moduli are varying functions of position. Combining the properties of the strain-energy function with Hooke's law we find [ ] that satisfy the following property and strong convexity for any nonzero real symmetric matrix.

Equations for motion in inhomogeneous anisotropic elastic materials are, in our notation see, e. Let us examine as an approximate solution of 30 for. Near a wavefront , we assume that components of are fluctuating much more rapidly than or , and the successive derivatives and are fluctuating still more rapidly. Substituting 31 into 30 we find see, e. Thus, the left-hand side of 32 must be much smaller than.

We conclude that the matrix of coefficients of must be singular: This equation determines the possible wavefronts in an elastic medium, since it gives a constraint on the function. In an inhomogeneous isotropic medium, where is the Kronecker symbol; that is, if and if ; moreover, , are known as the Lame functions. In an inhomogeneous isotropic medium, the special form 34 of makes it possible to get 33 as follows: This is, satisfies the eikonal equation or eikonal equation where is the local -wave speed and is the local -wave speed.

Suppose that a point source at position becomes active at a time chosen to be the origin,. In homogeneous isotropic medium, wavefronts emanate from the source as ever-expanding spheres, with radius for -waves and for -waves, arriving at the general position at time and. We introduce the function as the travel time required for the wavefront to reach from. The function satisfies 36 for -waves and 37 for -waves.

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One of the first inverse problem, stated and studied in geophysics, was the inverse kinematic problem. The physical interpretation of this problem is the following. Let us assume that Earth is an isotropic inhomogeneous elastic medium and the measurements of the seismic waves, arising from a point source and propagating in Earth, are given for points on its surface.

These measurements contain data of the travel time of seismic waves between the point of the source and any point of the Earth's surface. The inverse kinematic problem is to find the speed of the seismic waves inside of Earth using the measurement data. Mathematically we can state the inverse kinematic problem as follows. Let be a domain bounded by the surface , and let be the function of the travel time required by a signal with unknown speed to reach from. Find for all from if the function is given for all points and , where and are subsets of.

Herglotz [ ] and Wiechert and Zoeppritz [ ] were the first who studied the inverse kinematic problem in assumptions if is known for any from. Gerver and Markushevich [ ] have showed that the condition can be illiminated, but in this case the inverse kinematic problems have many solutions and the set of these solutions has been described. The first theoretical study of the inverse kinematic problem for a horizontal inhomogeneous medium has been made by Lavrenti'ev and Romanov [ ]. The first result of the study of the multidimentional inverse kinematic problem in a linear approximation, when the function depends on 3D space variable , was obtained by Romanov [ ].

In a recent time, the Earth is modeled as an anisotropic elastic medium which is located in the given 3D domain. The wave speed is given by a symmetric positive definite matrix , that is, a Riemannian metric in mathematical terms see, e. The problem is to determine the metric in a given domain from the lengths of geodesics joining points on the boundary of the domain. The linearization of this problem leads to a problem of the integral geometry [ — ].

The regular study of the problems of finding the isotropic and anisotropic Riemannian metrics and integral geometry problems has been made in the works [ , — ]. The modern numerical algorithms for the computation of the inverse kinematic problems of seismic have been developed in the works [ , ].

Let us note that isotropic inhomogeneous elastic medium is completely characterized by three functions: Using the point source at position of the boundary of the given domain , which becomes active at the time , we measure the function of the travel time required for the fronts of - and -waves to reach from. We use these information for solving two inverse kinematic problems for - and -waves. The solutions of these problems are speeds ,.

To complete the identification of unknown isotropic inhomogeneous medium we need to determine the last unknown function after finding ,. An inverse problem to recover in a given bounded domain , containing an isotropic inhomogeneous elastic medium, has been solved by Yakhno [ — ]. In these works, the displacement fields have been measured for all points and running the boundary of for all times from a time interval containing the time of arriving of the -waves.

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The vertical inhomogeneous model of Earth is one of the popular models of geophysics [ ], and the inverse problems of recovering the density and Lame functions and , depending on one variable and appearing in equations of elastodynamics 30 for the case of inhomogeneous isotropic elastic medium, have been studied by many authors [ 1 , 2 , 7 — 15 , 21 , 22 , 24 , 26 — 28 , 36 — 46 , 53 — 55 , 59 — 61 , 70 — 73 , 78 — 81 , 85 , 87 — 91 , 97 , 98 , , , , , , , , — ]. Because the unknown functions depend on one variable, the inverse problems of their recovering are called one-dimensional inverse problems although all differential equations of elasticity contain 3D space variable and 1D time variable.

Alekseev and Dobrinsky [ , ] were the first who described the importance of one-dimensional inverse problems of elastodynamics in geophysics and studied them as problems of the recovery of smooth functions , , and of one variable. The uniqueness of the solutions of these inverse problems has been studied firstly by Blagoveschenskii [ ] and Romanov [ ] for the isotropic elastic media and then by Volkova and Romanov [ ] for anisotropic elastic media. The regular study of the theory, methods, and applications of one-dimensional inverse problems for dynamical differential equations of isotropic and anisotropic elastic media has been made in works [ , , , , — ] and others.

The recent development of theory, methods, and applications of one-dimensional inverse problems of dynamic elasticity can be found in the works [ , , , ]. We note that a model of Earth as a composite medium consisting of a finite number of different elastic layers is very popular in geophysics.

A Survey on Inverse Problems for Applied Sciences

In this case, the one-dimensional inverse problem consists of finding and the Lame functions and as functions of one variable with piecewise constant values. The computation of solutions of this type of one-dimensional inverse problems has been studied in [ , , ]. The modern theory and methods of the construction of solutions of one-dimensional inverse problems for the equations of elastodynamics in elastic composite media can be found in the works [ , , — ].

Linearized multidimensional inverse dynamic problems or inverse problems in the Born approximation take an important place through all statements of multidimensional inverse problems for equations of elastodynamics. The statements of these problems have natural physical and mathematical sense. From the physical point of view, an isotropic inhomogeneous elastic body, which is characterized by the Lame functions and and density , is included in a vertical inhomogeneous or homogeneous elastic medium.

Let, for example, the half space contain this medium, and let the characteristics , , and of the elastic body be unknown functions. The linearized inverse problem is to find these unknown functions if we measure the first act of scattering the displacement field on the surface arising from the forces located on the same surface. From the mathematical point of view, we consider the differential equations of elastodynamics in a half space with boundary conditions on. We assume that the Lame functions and and density appearing in differential equations and boundary conditions can be presented in the form where , , and are functions depending on and characterizing vertical inhomogeneous medium and , , and are functions of 3D space variable characterizing the elastic body which is included in the vertical inhomogeneous medium.

We assume that the displacement field is presented in the form , where is the displacement field of the vertical inhomogeneous medium arising from the given forces, and is the first act of scattering on the inhomogeneous inclusion with characteristics , , and. The equations of elastodynamics with boundary conditions are linearized around , , , and.

The unknown functions , , and appear in inhomogeneous terms of linearized equations for. We need to recover , , and if we know for see, e. The uniqueness of the solution of a linearized multidimensional inverse problem has been studied by Romanov [ ]. The existence theorem and computation of a solutions of a linearized multidimensional inverse problems of elastodynamics have been obtained in the works [ , ].

The recovery of the function characteristics of an elastic body in linear approximation was a subject of the works [ , , , ]. The linearized inverse problems of determining the function characteristics of transversally isotropic elastic media from the measurements of reflected waves have been developed by Sharafutdinov [ , ].

The linearized inverse problems for nonhomogeneous bodies have been stated and developed by Steinberg [ ]. The inverse problems of determining the elastic moduli and density as functions of the space variables in a bounded domain from observed data of the solution on the boundary or a part of the boundary of this domain are geophysical motivated. One important class of these problems is inverse problems for equations of elastodynamics in terms of the Dirichlet-to-Neumann map.

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  4. The Dirichlet-to-Neumann map models surface measurements by giving the correspondence between a displacement at the boundary of the given bounded domain and surface traction where is the unit outer normal to , is the observed time interval, and , are components of the displacement vector function satisfying The details of the use of the Dirichlet-to-Neumann map in modeling surface measurements in inverse problems can be found in [ ]. The inverse problems in the Dirichlet-to-Neumann map statements are successfully applied to study the unique determination of the solutions of the inverse problems of elasticity as in the static isotropic and anisotropic cases [ — ] as well as in dynamic case [ ].

    For the study of the inverse problems for the scalar partial differential equations with a finite number of observation, Bukhgeim and Klibanov [ ] proposed a remarkable method based on a Carleman estimate. Later, the Carleman estimate method has been generalized to study the uniqueness and stability estimate of the solutions of the inverse problems for equations of elastodynamics by Isakov [ ], Ikehata et al.

    In the papers [ — , ] the authors assume some geometric constraints on the surface under observation for proving the uniqueness and stability of the solutions of the multidimensional inverse problems using the Carleman estimates.

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    Later, the stability estimate theorem for solutions of a multidimensional inverse problem for equations of elastodynamics has been proven for an arbitrary subboundary by Bellassoued et al. In the following, we present a general overview on some inversion-based application areas. We then give more details about particular selected applications in the subsequent sections. Further, different applications whose details are skipped in this section such as remote sensing, nuclear science, and geophysics can be found in [ , , — ].

    Underwater Acoustics and Traveltime Tomography. Inverse problems related to underwater acoustics are a critical research area due to their wide range of important practical applications. In water-type medium, propagating acoustic waves are collected at a selected number of separate hydrophones to obtain measured field data. Generally, reconstructions of desired parameters from the knowledge of scattered acoustic waves lead to nonlinear and ill-posed inverse problems.

    Therefore, it is always a complicated issue to find unique and stable solutions, and one has to apply some additional techniques, that is, regularizations, for the proper treatment of the ill-posedness. The inspection of an object without touching or without changing its characteristic properties is a general definition of nondestructive testing NDT in the literature.

    Therefore, approaches which are used for inverse problems, that is, synthetic aperture focussing technique SAFT, [ , ] , diffraction tomography DT, [ — , — ] , multiple signal classification MUSIC, [ , , , , ] , linear sampling method LSM, [ 70 — 74 ] , factorization [ 75 , 76 , ], point source [ 23 , 68 , 69 ], no response test [ 82 — 84 ], and so forth, are also employed for NDT problems [ ].

    More specifically, SAFT is an algorithm which uses the collection of echo signals over a specific aperture to obtain a reconstruction by performing time shifting and superposition of adjacent signals. DT is based on a linear solution of the wave equation which can be obtained via the Born or Rytov approximations. Here, the linearization approaches define mainly the success and the solution space of the inverse problem. MUSIC was initially employed as a direct imaging algorithm to obtain locations of point scatterers [ ] and extended to find also the geometry of targets [ ].

    The method employs the eigenvalue structure of time-reversal matrix which is obtained from measured data at different receiver antennas. The main idea of linear sampling method is to find an indicator function such that its value provides whether an arbitrarily tested space coordinate lies inside or outside of the object.

    Nowadays medical imaging, which can be considered as one of the most developed area of inverse problems in practice, provides high-resolution reconstructions in the order of millimeters. In the last decays, the main effort is given for the implementation of harmless, fast, cheap, robust, and reliable techniques to use in practice for obtaining high-resolution images in real time. In the similar direction, early studies of bioimaging started with the reconstructions of 2D images of human body parts via the inverse Radon transform [ , , ] of measured X-rays which were attenuated inside the body [ , ].

    Afterwards, 3D images were assembled via computed tomography CT-scan from a series of X-ray data measured on different planes sinograms in 2D [ , , , , ]. Even though satisfactory results were obtained with the X-ray radiology especially for the bone structures [ , ] the method was found not sufficiently efficient due to attenuation characteristics of X-rays and harm risks of using ionized radiation on humans. On the other hand, electrical impedance tomography EIT , after its main idea and formulation were introduced by Calderon in [ ] and D.

    Isaacson [ ], has gained high interest both from theoretical and physical point of view. In principle, in EIT low-frequency electrical currents are applied to the body part under investigation. Then electrical properties of body tissues are computed from the measurements of electric currents and voltage at the boundary [ , , , , , , , — ]. EIT is successfully applied for diagnosis of breast cancer, monitoring brain and gastrointestinal functions, detection of blood clots in the lungs, and so forth [ ].

    Furthermore, electroencephalography EEG and magnetoencephalography MEG are used for passive monitoring of neuron activities in the brain from the weak electric or magnetic fields, respectively, [ , , , , , ].

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    A different approach which is based on using properties of subatomic particles with the connection to electromagnetism opened a new area for obtaining high spatial resolutions in bioimaging, for example, magnetic resonance imaging MRI , positron emission tomography PET , and single-photon emission computed tomography SPECT.

    In MRI, the patient stays in a tunnel under a strong magnetic field typically 0. This large static magnetic field aligns protons of many atoms either parallel or antiparallel existing in the body. In the meantime, weak radio frequency fields are applied systematically to the patient for altering the alignment of the magnetization. As a result of this procedure, rotating magnetic fields induce a voltage at the receiver coils of the magnet which is used to reconstruct the image of the scanning area [ , , — ].

    Gamma rays, which occur when an electron and an emitted positron annihilate in PET, and photons which are released in SPECT can be visualized out of body by using scintigraphic detectors for clinical applications of oncology, cardiology, pharmacology, and so forth [ , , — ].

    Another group of techniques such as microwave tomography, ultrasound, and optical imaging, which are commonly used for solutions of inverse problems in different areas, are also applied to biomedical applications especially for investigating soft body tissues [ , , , , , — ]. For instance, optical tomography is used for the detection of cancerous cells in breast and brain.

    Acoustic waves are employed for the imaging of liver, kidney, fetus in pregnant women, and so forth, and microwaves are used in mammography and diagnosis of leukemia [ , ]. Over the past two decades, it has became feasible to simulate atmospheric and geophysical processes from large-scale atmospheric flow down to convective processes on a kilometer scale. This led to the need to determine initial conditions for simulations and forecasts from a collection of diverse direct and indirect measurements, and the field of data assimilation arose.

    Inverse Problems in Biological and Environmental Applications.

    A Survey on Inverse Problems for Applied Sciences

    Inverse problems are of growing importance in many parts of medicine or biology as well as in environmental applications. Here, we will provide a brief introduction into two areas, first into recent results of neural field theory and second into the basic setup of data assimilation as it is, for example, used in numerical weather prediction or for climate projections , which usually incorporates various inverse problems. Neural activity is often modelled by the activity potential in some domain. The activity potential satisfies some integro-differential equations, which in its simplest form has been suggested by Wilson and Cowan [ , ] and Amari [ ]: The kernel models the strength of the influence of an excitation at point to the neural field at point.

    The first term of the right-hand side of 41 generates the decay of the activity potential in the absence of excitation or inhibition. Neural fields have been widely studied in recent years, with applications to a wide range of medical phenomena starting from electroencephalogram EEG and magnetoencephalogram MEG rhythms to robotic behaviour—for an extensive literature list we refer to [ ]. Neural network, that is, the discrete version of neural fields, has attracted strong interest over many years and is a standard tool today in many applied parts of science.

    Neural networks have also been used to solve inverse problems, compare, for example, [ , ] for some recent papers and further citations. However, here we want to look into inverse problems which arise in the modelling of neural activities themselves. Training of neural networks or neural fields is, in general, an ill-posed inverse problem, as we will see in due course. Inverse neural field theory investigates the construction of connectivity kernels given some dynamics for and some time intervals.

    This so-called full-field neural inverse problem is linear and ill-posed in the sense of Hadamard, as can be readily seen by the following transform. We define If we further define the operator equation 41 obtains the form The task is to find the operator given the family of states and corresponding images for. Changing our perspective slightly, introducing the operator we transform 41 or 44 into for each fixed. For each , 46 is an integral equation of the first kind for the function.

    Usually we need some smoothness of the potential with respect to its arguments and. In this case, the operator is an integral operator with continuous kernel, which is known to be compact in either or cf. Thus, the inverse neural field problem is ill-posed. The particular form 46 provides a basis for kernel construction, that is, for the solution of the neural field problem. We may apply spectral regularization schemes as described in [ 1 , 22 , 85 ], for example, Tikhonov regularization in a similar sense of 21 with regularization parameter as regularized inverse to calculate from the knowledge of and.

    For smooth dynamics, the problem is exponentially ill-posed. We refer to [ — , ] for the analysis and many examples for the inverse neural field problem. The problem of the ill-posedness of the neural inverse problem is addressed in [ ], where a dimensional reduction approach is suggested to decompose the large and strongly ill-posed full problem into more stable individual tasks.

    In inverse problems, we are usually given measured data, and the task is to gain insight into some unknown parameter functions insight of an inaccessible body or area of space or to reconstruct the shape of scatterers or inclusions, but often, the quantities under reconstruction are not static but dynamic and change over time. Then, the inverse task is not only carried out once but the reconstruction is repeated over time with some cycling frequency given by a time interval.

    Let for be an element of a Hilbert space representing our measurement data.

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    The task is to reconstruct some state in a Hilbert space , where the measurement is described by an observation operator. An underlying dynamical system is given by some operator , mapping the state at time onto the state at time.