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The nonlinear model has been developed and implemented to describe gas emission from coal slack placed in a sealed container ('canister test'). The model accounts for initial gas content S, coefficients of diffusion D, mass transfer β and desorption kinetics γ, as well as for fractional composition of the sample. Using the developed analytical method of the initial boundary value problem solution, it is found that the pressure P (t) of gas in the container tends to a constant value with time. The inverse problem of determining the kinetic parameters S, D, β and γ of gas by the pressure P (t) readouts in the experiments has been formulated and subjected to solvability tests. The introduced objective function (discrepancy between P (t) and the calculated pressure) is almost independent of β. Based on the modification of the conjugate gradient method, the inverse problem algorithm is proposed. Using the data of in situ measurements of pressure in three containers, the authors perform quantification of gas content, as well as diffusion and desorption kinetics coefficients in terms of a coal bed in Berezovskaya Mine (Kuznetsk Coal Basin). The range of the mass transfer coefficient is estimated by the results of the grain-size analysis of the test samples. [ABSTRACT FROM AUTHOR]

In this paper, we use asymptotic expansion of the velocity field to reconstruct small deformable droplets (i.e. their forms and locations) immersed in an incompressible Newtonian fluid. Here the fluid motion is assumed to be governed by the non-stationary linear Stokes system. Taking advantage of the smallness of the droplets, our asymptotic formula and identification methods extend those already derived for rigid inhomogeneity and for stationary Stokes system. Our derivations, based on dynamical boundary measurements, are rigorous and proved by involving the notion of viscous moment tensor VMT. The viability of our reconstruction approach is documented by numerical results. [ABSTRACT FROM AUTHOR]

We are concerned with the reconstruction of the heat capacity coefficient of a one-dimensional heat equation from a sequence of observations of solutions at a single fixed point. This new direct method reconstructs the principal eigenfunction by extracting its Fourier coefficients from the observations of the solution. The algorithm presented here is an alternative to Krein's inverse spectral theory, minimization and iterative procedures. Error estimates and numerical experiments are presented. [ABSTRACT FROM AUTHOR]

This work proposes an identification method for reconstructing the characteristic source χ D in the Helmholtz equation Δ u + μ u = χ D from boundary measurements. We formulate the inverse source problem into a shape optimization problem by introducing a least-squares cost function. Using the shape optimization techniques, we prove the existence of an optimal solution to the considered shape optimization problem and we calculate the gradient of the cost function with respect to the shape D . By using the Level set method, we present an iterative algorithm to recover numerically the shape D . We develop a new technique to initialize the level set algorithm, which permits capturing different hidden shapes. To examine the validity of the proposed method, we illustrate several numerical experiments with different hidden shapes. By adding a level of noise to the measured data, we evaluate the robustness of our reconstruction algorithm. [ABSTRACT FROM AUTHOR]

This paper is concerned with the inverse scattering problem which aims to determine the spatially distributed dielectric constant coefficient of the 2D Helmholtz equation from multifrequency backscatter data associated with a single direction of the incident plane wave. We propose a globally convergent convexification numerical algorithm to solve this nonlinear and ill-posed inverse problem. The key advantage of our method over conventional optimization approaches is that it does not require a good first guess about the solution. First, we eliminate the coefficient from the Helmholtz equation using a change of variables. Next, using a truncated expansion with respect to a special Fourier basis, we approximately reformulate the inverse problem as a system of quasilinear elliptic PDEs, which can be numerically solved by a weighted quasi-reversibility approach. The cost functional for the weighted quasi-reversibility method is constructed as a Tikhonov-like functional that involves a Carleman Weight Function. Our numerical study shows that, using a version of the gradient descent method, one can find the minimizer of this Tikhonov-like functional without any advanced a priori knowledge about it. [ABSTRACT FROM AUTHOR]

In this paper, we proposed a fractional diffusion model to simulate the movement of chloride in concrete. In such complex porous structure some of the free chlorides, which are affected by the surrounding heterogeneous physical environment, will be bounded physically and chemically. Furthermore, experiments reveal that the interesting heavy-tailed phenomena appear in diffusion process. The time and spatial fractional derivatives are introduced to explain the anomalous diffusion and the dependence of medium properties. The numerical research shows that present model could better fit the experimental data and the diffusion of chloride in concrete is time-dependent and space-dependent. [ABSTRACT FROM AUTHOR]

In this paper, the inverse problem for identifying the initial value of time-fractional diffusion wave equation on spherically symmetric region is considered. The exact solution of this problem is obtained by using the method of separating variables and the property the Mittag–Leffler functions. This problem is ill-posed, i.e. the solution(if exists) does not depend on the measurable data. Three different kinds landweber iterative methods are used to solve this problem. Under the priori and the posteriori regularization parameters choice rules, the error estimates between the exact solution and the regularization solutions are obtained. Several numerical examples are given to prove the effectiveness of these regularization methods. [ABSTRACT FROM AUTHOR]

In this paper, we study two identification problems related to the mud filtrate invasion phenomenon. We want to determine a parameter (the invasion rate) in the coefficients of the parabolic equation that describes the mud filtrate invasion phenomenon. In the first problem, we determine this parameter starting from the observed values of the mud filtrate dispersion. We reduce the problem to an optimal control problem and prove the existence of the optimal control. In the second problem, we determine the invasion rate imposing the minimum condition of the quantity of mud filtrate that diffuses into the oil reservoir. We also reduce the identification problem to an optimal control problem. We prove the existence of the optimal control and we obtain a simple explicit form of this optimal control. A numerical example is presented for the second problem. [ABSTRACT FROM AUTHOR]

In this article, we consider an inverse source problem for Poisson equation in a strip domain. That is to determine source term in the Poisson equation from a noisy boundary data. This is an ill-posed problem in the sense of Hadamard, i.e., small changes in the data can cause arbitrarily large changes in the results. Before we give the main results about our proposed problem, we display some useful lemmas at first. Then we propose a modified quasi-reversibility regularization method to deal with the inverse source problem and obtain a convergence rate by using an a priori regularization parameter choice rule. Numerical examples are provided to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, the problem of unknown source identification for the space-time fractional diffusion equation is studied. In this equation, the time fractional derivative used is a new fractional derivative, namely, Caputo-Fabrizio fractional derivative. We have illustrated that this problem is an ill-posed problem. Under the assumption of a priori bound, we obtain the optimal error bound analysis of the problem under the source condition. Moreover, we use a modified quasi-boundary regularization method and Landweber iterative regularization method to solve this ill-posed problem. Based on a priori and a posteriori regularization parameter selection rules, the corresponding convergence error estimates of the two regularization methods are obtained, respectively. Compared with the modified quasi-boundary regularization method, the convergence error estimate of Landweber iterative regularization method is order-optimal. Finally, the advantages, stability and effectiveness of the two regularization methods are illustrated by examples with different properties. [ABSTRACT FROM AUTHOR]

Intravascular photoacoustic tomography (IVPAT) is a newly developed imaging modality for the diagnosis and intervention of coronary artery diseases. It is an ill-posed nonlinear least squares (NLS) problem to recover the absorbed optical energy density (AOED) and optical absorption coefficient (OAC) distribution in the vascular cross sections from pressure photoacoustically generated by tissues with variable speed of sound (SoS). The prior knowledge of the SoS is usually unavailable before IVPAT scanning. The ideal assumption of a constant SoS leads to degraded image quality. This paper focuses on improvement of image quality for IVPAT in tissues with variable SoS by simultaneously recovering the SoS, AOED and OAC from the measured time-dependent pressure series. The joint recovery is implemented by alternately minimizing the errors between the measured and theoretical pressure by forward simulation. The demonstration results indicate that the normalized mean square absolute distance (NMSAD) of the reconstructions produced by this method is decreased by about 15% in comparison to that of the reconstructions with a fixed SoS. Comparison results show that this method outperforms the delay compensation method in recovering the AOED and the two-step algorithm in estimating the OAC by about 20% and 25% in NMSAD respectively. [ABSTRACT FROM AUTHOR]

We proved the unique determination of the variable order in a two-scale mobile–immobile variable-order time-fractional partial differential equation with a variable diffusivity tensor imposed on a general multi-dimensional domain, with the observations of the unknown solutions on any arbitrarily small spatial domain over a sufficiently small time interval. The proved theorem provides a guidance where the measurements should be performed and ensures that with these observations the uniqueness of the identification is theoretically guaranteed. [ABSTRACT FROM AUTHOR]

The total variation minimization is proposed for use in recovering the mass density of a fluid medium from back-scattered acoustic waves by the dynamical version of the boundary control method. This may be of particular interest to underwater acoustic imaging or ultrasound tomography. In particular, an analogue of the regularized mean curvature flow equation is proposed and developed to obtain numerical solution of the system of linear equations with the ill-conditioned Gram matrix. To demonstrate the computational effectiveness of the proposed numerical technique, we conduct a series of numerical experiments with the inverse problem for a 1 + 1 dimensional acoustic wave equation. [ABSTRACT FROM AUTHOR]

This article is devoted to applying a local meshless method for specifying an unknown control parameter in one- and multi-dimensional inverse problems which are considered with a temperature overspecification condition at a specific point or an energy overspecification condition over the computational domain. Finding the unknowns in inverse problems is a challenge because these problems are modeled as non-classical parabolic problems and also have a significant role in describing physical phenomena of the real world. In this study, a combination of the meshless method of radial basis functions and finite difference method (called radial basis function-finite difference method) is used to solve inverse problems because this method has two important features. First it does not require any mesh generation. Consequently, it can be exerted to handle the high-dimensional inverse problems. Secondly, since this method is local, at each time step, a system with a sparse coefficient matrix is solved. Hence, the computational time and cost will be much low. Various numerical examples are examined, and also the accuracy and computational time required are presented. The numerical results indicate that the mentioned procedure is appropriate for the identification of the unknown parameter of inverse problems. [ABSTRACT FROM AUTHOR]

The time conformable heat equation is a generalization of classical heat equation involved local and limit-based derivative, which is called conformable fractional derivative. In this paper, we study a backward problem for the time conformable fractional heat equation defined in cylindrical coordinates for the axis-symmetric case which is a severely ill-posed problem. By using a modified quasi-boundary value method, the problem is regularized and some new error estimates are obtained. The numerical experiment shows that the method is feasible and effective. [ABSTRACT FROM AUTHOR]

We study the three-dimensional inverse problem of elastography, that is finding the Young's modulus of a biological tissue from known values of its vertical displacements. In this way, one can find inclusions with Young's modulus several times higher than its known background value. Such inclusions are interpreted as tumours. A quasistatic statement of the problem is used in which the fragment of the tissue is considered as a linearly elastic body. It is assumed that the geometry of inclusions is specified parametrically, and the Young's modulus inside and outside the inclusions is constant. The task is reduced to finding the number of inclusions, parameters defining their shape and the Young's modulus inside the inclusions. To solve the problem, a special algorithm is proposed and justified. The results of numerical experiments on solving a three-dimensional model problems are presented. A comparison is made of the solutions to the inverse problem in the three-dimensional domain and two-dimensional inverse problems in selected cross-sections of this domain. It is found that 2D inverse problems do not always allow one to find a true 3D solution. For one of solved inverse problems, a-posteriori error estimates for Young modulus and geometric parameters of inclusions are obtained. [ABSTRACT FROM AUTHOR]

In this article, for a degenerate parabolic equation we study an inverse problem for restoration of source temperature from the information of final temperature profile. The uniqueness of this inverse problem is first established by taking an integral transform and using Liouville's theorem (complex analysis). With aids of an integral identity, a Lipschitz stability for the inverse problem is further constructed. Finally, several numerical experiments are presented to show the accuracy and efficiency of the algorithm. [ABSTRACT FROM AUTHOR]

Measurements of alternating current potential drop at the surface of a conductor can be predicted based on knowledge of the electromagnetic material properties of the conductor and their variation with position. Here we consider the inverse problem of finding the variations of the material properties with depth from multi-frequency potential drop data, which are generated by injecting alternating currents into the conductor. Measurements are carried out using a four-point probe. A possible application of this inverse problem is that of determining the depth dependence of case-hardening in steel. In this article, we define a mathematical framework for the inverse problem and propose a linearization method for solving it. The linearization is based on the assumption that the variations of the material properties with depth are small. The proposed algorithm is tested against both simulated and real experimental data. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel reconstruction scheme for the low-frequency near-field electromagnetic imaging of high-contrast conductivity distributions inside shielded regions using the system of Maxwell's equations in 3D. In our novel scheme, we focus on estimating the shape characteristics of the electrical conductivity profile inside these regions from low-frequency electromagnetic data measured at external locations for a single frequency. We introduce a colour level set regularization scheme which is a shape-based approach focusing on the simultaneous reconstruction of several shape-like distributions of different conductivity values in the same region of interest. Using two numerical experiments addressing a three-value reconstruction problem related to the imaging of shielded boxes or cargo containers, we compare this novel approach with results obtained from standard voxel-based reconstruction schemes on the one hand and the more established two-value shape-based approach on the other hand. We demonstrate that, depending on the particular situation of the imaging setup, this three-value (or in general multiple-value) shape-based reconstruction technique has the potential to provide superior reconstruction results in many situations, in particular regarding reconstruction of the correct shapes. We also discuss particular challenges of this novel methodology. [ABSTRACT FROM AUTHOR]

In this paper, we consider the inverse problem of recovering an isotropic elastic tensor from the Neumann-to-Dirichlet map. To this end, we prove a Lipschitz stability estimate for Lamé parameters with certain regularity assumptions. In addition, we assume that the Lamé parameters belong to a known finite subspace with a priori known bounds and that they fulfil a monotonicity property. The proof relies on a monotonicity result combined with the techniques of localized potentials. To numerically solve the inverse problem, we propose a Kohn-Vogelius-type cost functional over a class of admissible parameters subject to two boundary value problems. The reformulation of the minimization problem via the Neumann-to-Dirichlet operator allows us to obtain the optimality conditions by using the Fréchet differentiability of this operator and its inverse. The reconstruction is then performed by means of an iterative algorithm based on a quasi-Newton method. Finally, we give and discuss several numerical examples. [ABSTRACT FROM AUTHOR]

An approach is developed to read and simulate experimental data using inverse problems. The idea is founded on a locally sequential refinement of the reconstruction. The regularization that occurs under a scheme of separate matching with observations is the basis for a numerical solution. The simultaneous identification of several unknowns using a set of mathematical models and functional representations of the desired quantities combines the simulation with inverse problem solutions. This unified approach is applied to the investigation of the enzyme kinetics. The object in question is simulated in the absence of information about the functional type of sought kinetic parameters, the noise structure and its level, and also details of observations. Nevertheless, the composition and volume of the samples under consideration prove to be sufficient not only for determining all of the kinetic parameters but also for clarifying the observation conditions. The obtained results demonstrate the effectiveness of the sequential regularized solutions to the inverse problems for the case of acting at once all the known features of ill-posed problems. [ABSTRACT FROM AUTHOR]

In this paper, the acoustic full waveform inversion based on the wavelet method is proposed. It allows to determine density and velocity parameters simultaneously in the time domain. The forward problem is solved based on the wavelet method. The numerical schemes for modelling are constructed in detail. The stability condition is derived for the first time. The inversion is an optimization process for minimizing the mismatch between the synthetic data and the observed data. The computational scheme for the gradient of the objective function is constructed based on the matrix analysis technique. The inversion is implemented from low frequency to high frequency successively. Furthermore, the inversion result of current low frequency is served as the initial guess of next high frequency. This hierarchic strategy includes the smaller wavelengths progressively and improves the inversion robustness. Numerical computations for the benchmark Marmousi model demonstrate that the method has good ability to reconstruct media parameters in complex structures. [ABSTRACT FROM AUTHOR]

In this paper we solve numerically some geometric inverse problems for linear and semilinear elliptic equations with the method of fundamental solutions (MFS). More precisely, we focus our work on the numerical reconstruction of the unknown domain where the equation holds. We first consider two dimensional problem and we present several numerical experiments with different type of geo metries to be recovered (for example, circular, polygon domains and domains formed by two circular unknown regions). We also present numerical results for three dimensional case. Finally, we analyze semininear equation and then we establish some interesting open problems related to the subject. [ABSTRACT FROM AUTHOR]

In this paper we study inverse problems of electric conductivity that arise in the design of spherical shielding or cloaking shells and other functional devices used to control static electric fields. The shells are considered consisting of a finite number of layers filled with homogeneous isotropic or anisotropic medium. The inverse problems under study are reduced to control problems with the layers electric conductivities taken as controls. A different choice of minimized functionals allows us to solve a wide range of design problems using one approach. A numerical algorithm to solve these problems is based on particle swarm optimization. Various results of numerical experiments are discussed in order to find the most effective designs. The findings obtained in this study describe a broad set of specific easy-to-manufacture structures that have the highest cloaking or shielding performance in the class of layered shells. [ABSTRACT FROM AUTHOR]

The monotonicity method for the inverse acoustic scattering problem is to understand the inclusion relation between an unknown object and artificial one by comparing the far field operator with the artificial operator. This paper introduces the development of this method to the inverse crack scattering problem. Our aim is to give the following two indicators: One (Theorem 1.1) is to determine whether an artificial small arc is contained in the unknown arc. The other one (Theorem 1.2) is whether an artificial large domain contains the unknown one. Finally, numerical examples based on Theorem 1.1 are given. [ABSTRACT FROM AUTHOR]

Tikhonov regularization is an usual method to solve an ill-posed problem, recommended when the input data are contaminated with noise. However, in some cases, the use of this technique is not sufficient to provide good solutions. In this work, an improvement of the Tikhonov regularization method was proposed and tested in the inverse black body radiation problem. The method proposed consisted in including the norm of the fractional-order derivative of the solution in the original functional proposed by Tikhonov. In the present framework, the regularized solution depends on the regularization parameter, λ, and on the fractional derivative order, α. For α assuming real value between 0 and 2, the solution obtained is more precise than those from the usual Tikhonov regularization method. [ABSTRACT FROM AUTHOR]

This paper concerns a one-phase inverse Stefan problem in one-dimensional space. The problem is ill-posed in the sense that the solution does not depend continuously on the data. We also consider noisy data for this problem. As such, we first regularize the proposed problem by the discrete mollification method. We apply the integration matrix using Bernstein basis polynomials for the discrete mollification method. Through this method, the execution time was gradually decreased. We then extend the space marching algorithm for solving our problem. Moreover, proofs of stability and convergence of the process are given. Finally, the results of this paper have been illustrated and examined by some numerical examples. Numerical examples confirm the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

This paper studies an inverse problem of recovering an unknown diffusion coefficient in a parabolic equation. We adopt a total variation regularization method to deal with the ill-posedness. This method has the advantage to solve problems that the solution is non-smooth or discontinuous. By transforming the problem into an optimal control problem, we derive a necessary condition of the control functional. Through some prior estimates of the direct problem, the uniqueness and stability of the minimizer are obtained. In the numerical part, a Gauss–Jacobi iteration scheme is used to deal with the non-linear term. Some numerical examples are presented to illustrate the performance of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This paper is concerned with a reconstruction method for multiple moving point/dipole wave sources. We assume that the number, locations, and magnitudes/moments of wave sources are unknown and consider the problem to reconstruct these parameters from the measurement of the wave field on the boundary. For this problem, we derive algebraic relations between the parameters of wave sources and the reciprocity gap functionals and propose a real-time reconstruction procedure based on these relations. We perform some numerical experiments and show the effectiveness of our reconstruction procedure. [ABSTRACT FROM AUTHOR]

In this paper, we consider the source strength identification problem for the three-dimensional inverse heat conduction equations. The problem is to determine an unknown heat source strength from the measurement data for a specified location. In this process, the direct problem is solved by applying the Green's function method. Then, this problem can be converted into a Volterra integral equation of the first kind. Further, the Tikhonov and truncated singular value decomposition regularization methods are developed to identify the unknown source strength based on the discrepancy principle for choosing the regularization parameter. Finally, numerical examples are presented to show the feasibility and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

For an approximate solution of irregular nonlinear operator equations with a smooth operator in Hilbert space, a class of numerically implementable iteratively regularized Gauss–Newton type methods is constructed and studied. The methods include a general finite dimensional approximation for equations under consideration and cover the projection, collocation, and quadrature discretization schemes. Using an a priori stopping rule for the iterative processes and the standard source condition on the solution, we establish accuracy estimates for the approximations generated by the methods. An iteratively regularized quadrature process is applied to an inverse 2 D problem of the wave tomography in a model with the incomplete data range. [ABSTRACT FROM AUTHOR]

We consider the problem of computing the initial condition for a general parabolic equation from the Cauchy lateral data. The stability of this problem is well-known to be logarithmic. In this paper, we introduce an approximate model, as a coupled linear system of elliptic partial differential equations. Solution to this model is the vector of Fourier coefficients of the solutions to the parabolic equation above. This approximate model is solved by the quasi-reversibility method. We will prove the convergence for the quasi-reversibility method as the measurement noise tends to 0. The convergent rate is Lipschitz. We present the implementation of our algorithm in details and verify our method by showing some numerical examples. [ABSTRACT FROM AUTHOR]

This paper is devoted to the problem of determining the initial data for the backward non-homogeneous time fractional heat conduction problem by the Fourier truncation method. The exact solution for the forward and backward fractional heat problems is expressed in terms of eigen function expansion and Mittag–Leffler function. Due to the instability of determining initial data, a regularized truncated solution is considered. Further, the stability estimate for the exact solution and the convergence estimates for the regularized solution using an á-priori choice rule and an á-posteriori choice rule are derived. [ABSTRACT FROM AUTHOR]

We present a formulation to simultaneously invert for a heterogeneous shear modulus field and traction boundary conditions in an incompressible linear elastic plane stress model. Our approach utilizes scalable deterministic methods, including adjoint-based sensitivities and quasi-Newton optimization, to reduce the computational requirements for large-scale inversion with partial differential equation (PDE) constraints. We address the use of regularization for such formulations and explore the use of different types of regularization for the shear modulus and boundary traction. We apply this PDE-constrained optimization algorithm to a synthetic dataset to verify the accuracy in the reconstructed parameters, and to experimental data from a tissue-mimicking ultrasound phantom. In all of these examples, we compare inversion results from full-field and sparse data measurements. [ABSTRACT FROM AUTHOR]

This paper considers the reconstruction problem in acousto-electrical tomography, i.e. the problem of estimating a spatially varying conductivity in a bounded domain from measurements of the internal power densities resulting from different prescribed boundary conditions. Particular emphasis is placed on the limited-angle scenario, in which the boundary conditions are supported only on a part of the boundary. The reconstruction problem is formulated as an optimization problem in a Hilbert space setting and solved using Landweber iteration. The resulting algorithm is implemented numerically in two spatial dimensions and tested on simulated data. The results quantify the intuition that features close to the measurement boundary are stably reconstructed and features further away are less well reconstructed. Finally, the ill-posedness of the limited-angle problem is quantified numerically using the singular value decomposition of the corresponding linearized problem. [ABSTRACT FROM AUTHOR]

In this contribution we consider the problem of flux identification in a scalar conservation law modelling the phenomenon of sedimentation. The experimental observation data used for the calibration consist of a solid concentration profile at a fixed time. The identification problem is formulated as an optimization one, where the distance between the profiles of the model simulation and observation data is minimized by a least squares cost function. The direct problem is approximated by a monotone finite volume scheme. The numerical solution of the calibration problem is obtained by a continuous genetic algorithm. Numerical results are presented in order to validate the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, an inverse source problem for the Helium Production- Diffusion Equation on a columnar symmetric domain is investigated. Based on an a priori assumption, the optimal error bound analysis and a conditional stability result are given. This problem is ill-posed and Landweber iteration regularization method is used to deal with this problem. Convergence estimates are presented under the priori and the posteriori regularization choice rules. For the a priori and the a posteriori regularization parameters choice rules, the convergence error estimates are all order optimal. Numerical examples are given to show that the regularizationmethod is effective and stable for dealing with this ill-posed problem. [ABSTRACT FROM AUTHOR]

In this study,we present existence and uniqueness theorems of a quasi solution to backward time-fractional diffusion equation. To do that, we consider amethodology, involving minimization of a least squares cost functional, to identify the unknown initial data. Firstly, we prove the continuous dependence on the initial data for the corresponding forward problem and then we obtain a stability estimate. Based on this, we give the existence theorem of a quasi solution in an appropriate class of admissible initial data. Secondly, it is shown that the cost functional is Fréchet-differentiable and its derivative can be formulated via the solution of an adjoint problem. These results help us to prove the convexity of cost functional and subsequently the uniqueness theorem of the quasi solution. In addition, in order to approximate the quasi solution, WEB-spline finite element method is used. Since the obtained system of linear equations is ill-posed, we apply the Levenberg-Marquardt regularization. Finally, a numerical example is given to show the validation of the introduced method. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical identification of diffusion parameters in a non-linear convection–diffusion equation, which arises as the saturation equation in the fractional flow formulation of the two-phase porous media flow equations. In order to overcome the defect of the local convergence of traditional methods, an adaptive homotopy perturbation method is applied to solve this parameter identification inverse problem. The adaptive homotopy perturbation method provides a simple way to adapt computational refinement to the choice of the homotopy parameter. Numerical simulations illustrate that the proposed algorithm is globally convergent and computationallyefficient. [ABSTRACT FROM PUBLISHER]

Faults and geological barriers can drastically affect the flow patterns in porous media. Such fractures can be modelled as interfaces that interact with the surrounding matrix. We propose a new technique for the estimation of the location and hydrogeological properties of a small number of large fractures in a porous medium from given distributed pressure or flow data. At each iteration, the algorithm builds a short list of candidates by comparingfracture indicators. These indicators quantify at the first order the decrease of a data misfit function; they are cheap to compute. Then, the best candidate is picked up by minimization of the objective function for each candidate. Optimally driven by the fit to the data, the approach has the great advantage of not requiring remeshing, nor shape derivation. The stability of the algorithm is shown on a series of numerical examples representative of typical situations. [ABSTRACT FROM PUBLISHER]

A novel meshless numerical procedure based on the method of fundamental solutions (MFS) and the heat polynomials is proposed for recovering a time-dependent heat source and the boundary data simultaneously in an inverse heat conduction problem (IHCP). We will transform the problem into a homogeneous IHCP and initial value problems for the first-order ordinary differential equation. An improved method of MFS is used to solve the IHCP and a finite difference method is applied for solving the initial value problems. The advantage of applying the proposed meshless numerical scheme is producing the shape functions which provide the important delta function property to ensure that the essential conditions are fulfilled. Numerical experiments for some examples are provided to show the effectiveness of the proposed algorithm. [ABSTRACT FROM PUBLISHER]

Detecting inhomogeneities in the electrical conductivity is a special case of the inverse problem in electrical impedance tomography, that leads to fast direct reconstruction methods. One such method can, under reasonable assumptions, exactly characterize the inhomogeneities based on monotonicity properties of either the Neumann-to-Dirichlet map (non-linear) or its Fréchet derivative (linear). We give a comparison of the non-linear and linear approach in the presence of measurement noise, and show numerically that the two methods give essentially the same reconstruction in the unit disk domain. For a fair comparison, exact matrix characterizations are used when probing the monotonicity relations to avoid errors from numerical solution to PDEs and numerical integration. Using a special factorization of the Neumann-to-Dirichlet map also makes the non-linear method as fast as the linear method in the unit disk geometry. [ABSTRACT FROM PUBLISHER]

We devise a fast and reliable computational framework for the elasticity imaging inverse problem of detecting cancerous tumors in the human body using an output least-squares (OLS) approach. From a mathematical standpoint, this inverse problem requires identifying a parameter in a mixed variational problem. We develop, in a continuous setting, a first-order adjoint method and two second-order adjoint methods. The continuous formulae are then used to devise a scheme for an efficient computation of the gradient and the Hessian of the OLS objective. We give detailed numerical examples. [ABSTRACT FROM PUBLISHER]

This article deals with an inverse problem of determining the space-dependent diffusion coefficient and the source coefficient simultaneously in the multi-term time fractional diffusion equation (TFDE in short) using measurements at one inner point. From a view point of optimality, solving the inverse problem is transformed to minimize an error functional with the help of the solution operator from the unknown to the additional observation. The solution operator is nonlinear but it is of Lipschitz continuity by which existence of a minimum to the error functional is obtained using Sobolev embedding theorems. The homotopy regularization algorithm is introduced to solve the simultaneous inversion problem based on the minimization problem, and numerical examples are presented. The inversion solutions give good approximations to the exact solutions demonstrating that the homotopy regularization algorithm is efficient for the simultaneous inversion problem arising in the multi-term TFDE. [ABSTRACT FROM PUBLISHER]

In this paper we present the work related to the parameters identification for abrasive waterjet milling (AWJM) model that appears as an ill-posed inverse problem. The necessity of studying this problem comes from the industrial milling applications where the possibility to predict and model the final surface with high accuracy is one of the primary tasks in the absence of any knowledge of the model parameters that should be used. The adjoint approach based on corresponding Lagrangian gives the opportunity to find out the unknowns of the AWJM model and their optimal values that could be used to reproduce the required trench profile. Due to the complexity of the nonlinear problem and the large number of the model parameters, we use an automatic differentiation software tool. This approach also gives us the ability to distribute the research on more complex cases and consider different types of model errors and 3D time dependent model with variations of the jet feed speed. This approach gives us a good opportunity to identify the optimal model parameters and predict the surface profile both with self-generated data and measurements obtained from the real production. Considering different types of model errors allows us to receive the results acceptable in manufacturing and to expect the proper identification of unknowns. [ABSTRACT FROM PUBLISHER]

We study the inverse scattering problem with fixed observation angle data for a non-linear Schrödinger equation in 2D. We show that the main singularities of an unknown potential can be recovered from the Born approximation. Numerical examples are given to illustrate the method. [ABSTRACT FROM AUTHOR]

How to determine adsorption isotherms is an issue of significant importance in chromatography. A modern technique of obtaining adsorption isotherms is to solve an inverse problem so that the simulated batch separation coincides with actual experimental results. In this work, as well as the natural least-square approach, we consider a Kohn–Vogelius type formulation for the reconstruction of adsorption isotherms in chromatography, which converts the original boundary fitting problem into a domain fitting problem. Moreover, using the first momentum regularizing strategy, a new regularization algorithm for both the Equilibrium-Dispersive model and the Transport-Dispersive model is developed. The mass transfer resistance coefficients in the Transport-Dispersive model are also estimated by the proposed inverse method. The computation of the gradients of objective functions for both of the two models is derived by the adjoint method. Finally, numerical simulations for both a synthetic problem and a real-world problem are given to show the robustness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

We present the new method Coupled Multiscenario Full Waveform Inversion (CMFWI) for the solution of the inversion problem arising in seismic applications. As in the case of classical Full Waveform Inversion (FWI), the proposed method is based on seismic reflection signals and it tries to recover the subsoil velocity profile by minimizing a misfit functional between recorded and computed data. CMFWI combines data generated by shooting one source at a time, and shares the effect of this signal with the signals associated with the other sources. Moreover, CMFWI is a new approach with respect to FWI employed with the same sources shot together. We numerically show that CMFWI performs better than FWI. In particular, this comparison focuses on different types of boundary conditions, wave sources, initial guesses of the velocity profile and signal-to-noise ratio. [ABSTRACT FROM AUTHOR]

In this contribution, the reconstruction of a solely time-dependent heat source is studied in an inverse problem arising in one-dimensional thermoelasticity. The missing heat source is recovered from a measurement in time of the average temperature inside the body. Using this observation, the inverse problem is reformulated in a direct setting. The existence and uniqueness of a weak solution are addressed, and a new numerical algorithm based on Rothe’s method is designed. Numerical experiments support the obtained results. [ABSTRACT FROM AUTHOR]

We consider the problem of estimating the 2D vector displacement field in a heterogeneous elastic solid deforming under plane stress conditions. The problem is motivated by applications in quasistatic elastography. From precise and accurate measurements of one component of the 2D vector displacement field and very limited information of the second component, the method reconstructs the second component quite accurately. No a priori knowledge of the heterogeneous distribution of material properties is required. This method relies on using a special form of the momentum equations to filter ultrasound displacement measurements to produce more precise estimates. We verify the method with applications to simulated displacement data. We validate the method with applications to displacement data measured from a tissue mimicking phantom, and in-vivo data; significant improvements are noticed in the filtered displacements recovered from all the tests. In verification studies, error in lateral displacement estimates decreased from about 50% to about 2%, and strain error decreased from more than 250% to below 2%. [ABSTRACT FROM AUTHOR]