Search

Research presented in this paper was initiated by the publication [A. D. Agaltsov and R. G. Novikov, Examples of solving the inverse scattering problem and the equations of the Veselov–Novikov hierarchy from the scattering data of point potentials, Russian Math. Surveys74 (2019), 3, 373–386] and is based on its results. Two sets of the complex monopole scattering coefficients are distinguished among the possible values of these coefficients for nonabsorbing inhomogeneities. These sets differ in phases of the scattering coefficients. In order to analyze the features and possibilities of reconstructing the inhomogeneities of both sets, on the one hand, the inverse problem is solved for each given value of the monopole scattering coefficient using the Novikov functional algorithm. On the other hand, the scatterer is selected in the form of a homogeneous cylinder with the monopole scattering coefficient that coincides with the given one. The results obtained for the monopole inhomogeneity and for the corresponding cylindrical scatterer are compared in the coordinate and spatial-spectral spaces. The physical reasons for the similarities and differences in these results are discussed when the amplitude of the scattering coefficient changes, as well as when passing from one set to another. [ABSTRACT FROM AUTHOR]

The paper contributes to the clarification of the mechanism of one-dimensional pulsating detonation wave propagation for the transition regime with two-scale pulsations. For this purpose, a novel numerical algorithm has been developed for the numerical investigation of the gaseous pulsating detonation wave using the two-stage model of kinetics of chemical reactions in the shock-attached frame. The influence of grid resolution, approximation order and the type of rear boundary conditions on the solution has been studied for four main regimes of detonation wave propagation for this model. Comparison of dynamics of pulsations with results of other authors has been carried out. [ABSTRACT FROM AUTHOR]

In this paper, we consider an inverse backward problem for a nonlinear singularly perturbed parabolic equation of the Burgers' type. We demonstrate how a method of asymptotic analysis of the direct problem allows developing a rather simple algorithm for solving the inverse problem in comparison with minimization of the cost functional. Numerical experiments demonstrate the effectiveness of this approach. [ABSTRACT FROM AUTHOR]

This paper is concerned with the inverse problem for determining the space-dependent source and the initial value simultaneously in a parabolic equation from two over-specified measurements. By means of transforming information of the initial value into the source term and obtaining a combined source term, the parabolic equation problem is converted into a parabolic problem with homogeneous conditions. Then the considered inverse problem is formulated into a regularized minimization problem, which is implemented by the finite element method based on solving a sequence of well-posed direct problems. The uniqueness of inverse solutions are proved by the solvability of the corresponding variational problem, and the conditional stability as well as the convergence rate of regularized solutions are also provided. Then the error estimate of approximate regularization solutions is presented in the finite-dimensional space. The proposed method is a very fast non-iterative algorithm, and it can successfully solve the multi-dimensional inverse problem for recovering the space-dependent source and the initial value simultaneously. Numerical results of five examples including one- and two-dimensional cases show that the proposed method is efficient and robust with respect to data noise. [ABSTRACT FROM AUTHOR]

In this paper a problem of specifying HIV-infection parameters and immune response using additional measurements of the concentrations of the T-lymphocytes, the free virus and the immune effectors at fixed times for a mathematical model of HIV dynamics is investigated numerically. The problem of the parameter specifying of the mathematical model (an inverse problem) is reduced to a problem of minimizing an objective function describing the deviation of the simulation results from the experimental data. A genetic algorithm for solving the least squares function minimization problem is implemented and investigated. The results of a numerical solution of the inverse problem are analyzed. [ABSTRACT FROM AUTHOR]

Total variation (TV) based models have been used widely in multiplicative denoising problem. However, these models are always accompanied by an unsatisfactory effect named staircase due to the property of BV space. In this paper, we present two high-order variational models based on total generalized variation (TGV) for two kinds of multiplicative noises. The proposed models reduce the staircase while preserving the edges. In the meantime we develop an efficient algorithm which is called Prediction-Correction proximal alternative direction method of multipliers (PADMM) to solve our models. Moreover, we show the convergence of our algorithm under certain conditions. Numerical experiments demonstrate that our high-order models outperform the classical TV-based models in PSNR and SSIM values. [ABSTRACT FROM AUTHOR]

A new combined numerical algorithm for solving inverse problems of epidemiology is described in this paper. The combined algorithm consists of optimization and iterative methods, and determines the parameters specific to a particular population by using the statistical information for a few previous years. The coefficients of the epidemiology model describe particular qualities of the population and the development of the disease. The inverse problem of parameter identification in a mathematical model is reduced to the problem of minimizing an objective function characterizing the square deviation of the statistical data from the experimental data. The combination of statistical and optimization algorithms demonstrates the identification of parameters with an appropriate relative accuracy of 30%. The results can be used by public health organizations to predict the infectious disease epidemic in a given region by comparing the data of simulation with historical data. [ABSTRACT FROM AUTHOR]

In this paper, a regularized sampling algorithm for band-limited signals is presented in the twodimensional case. The convergence of the regularized sampling algorithm is studied and compared with Shannon's sampling theorem and the Tikhonov regularization method by some examples. [ABSTRACT FROM AUTHOR]

This paper investigates the inverse scattering problem for a discrete Dirac system on the entire line with coefficients that stabilize to zero in one direction.We develop an algorithm for solving the inverse problem of reconstruction of coefficients. We derive a necessary and a sufficient condition on the scattering data so that the inverse problem is uniquely solvable. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the use of simple models for solving the inverse problem in electrocardiography (IPE), which aims at recovering the heart condition from a set of remote voltages measurements. Specifically, we consider here the problem of estimating the shape, size and location of cardiac ischemic regions. The forward problem to generate the data (voltage measurements) is formulated by using the Luo-Rudy model, which provides a detailed description of the electrical behavior of cardiac cells. As for the inversion process, we use the two-current phenomenological model. The inversion procedure also incorporates a semi-automatic stage to characterize the conduction properties of the cardiac tissue. The ischemic regions are modeled by using standard level set techniques. Numerical results show that the algorithm is capable of estimating the position, size and shape of cardiac ischemic regions from noisy voltage measurements, for both 2D and 3D geometries. Our inverse procedure is benchmarked against zero-order Tikhonov regularization. This work is a proof of principle demonstrating the possibility of using simple models in the IPE towards realistic situations. [ABSTRACT FROM AUTHOR]

We investigate the inverse spectral problem of the interior transmission eigenvalue problem for an anisotropic medium supported in D := ... with the boundary conditions u = v, αv ·▽u = v ·▽v for x ε ∂D, where α and n are physical parameters. In the spherical symmetry case, we consider the case α ≠ 1, whereas most previous work deals with α = 1 only. In this paper we prove that all transmission eigenvalues (including multiplicity) uniquely determine n and α under the condition a := ... ≤ 1, and provide construction algorithms. In particular, when a = 1 one needs an additional condition for unique recovery and reconstruction. [ABSTRACT FROM AUTHOR]

In this paper we present an algorithm to bound parameters estimated from very small data sets and use this method to suggest optimal times for data collection. The method is deterministic in that we assume that one has error bars on the data. The basic idea is based on bounded error parameter identification methods and we use this framework to assign a measure of quality to the estimated parameter. Exploiting the dynamics of the underlying mathematical model allows us to describe the quality of the estimated parameter in a way that, due to the small number of data points, is not appropriate for traditional statistical techniques. The algorithm in this paper differs from traditional bounded error parameter identification techniques in that one computes the membership set by solving an inverse problem rather than a forward interval problem. [ABSTRACT FROM AUTHOR]

Linear ill-posed problems written as the operator equation Az = ū on sets of functions z convex on multi-dimensional sets Ω are considered in the paper. A regularizing algorithm zη = R(Ah, uδ, η), where |Ah - A| ≤ h, |uδ - ū| ≤ δ, η = (h,δ), obtained in the previous papers for line segments is generalized such that the approximate solution zη tends to the exact one z uniformly on some subsets of the domain Ω. The algorithms to estimate an error of finite dimensional approximation and to find a lower zl and an upper zu functions that bound all approximation solutions are provided. [ABSTRACT FROM AUTHOR]

In this paper, we aim to calibration pricing models from market prices. We investigate the problem of calibrating the parameters using the trust region method from given price data. We start with the Hull-White model and formulate the problem by obtaining the first kind integral equation, and then consider the parameter recovery problem of the Black-Scholes model. We apply trust region algorithm for numerical retrieval problems. Numerical simulations are given to illustrate the feasibility of our proposed method. [ABSTRACT FROM AUTHOR]

In the matrix completion problem, most methods to solve the nuclear norm model are relaxing it to the nuclear norm regularized least squares problem. In this paper, we propose a new unconstraint model for matrix completion problem based on nuclear norm and indicator function and design a proximal point algorithm (PPA-IF) to solve it. Then the convergence of our algorithm is established strictly. Finally, we report numerical results for solving noiseless and noisy matrix completion problems and image reconstruction. [ABSTRACT FROM AUTHOR]

In this paper we study a linear inverse problem with a biological interpretation, which is modeled by a Fredholm integral equation of the first kind. When the kernel in the Fredholm equation is represented by step functions, we obtain identifiability, stability and reconstruction results. Furthermore, we provide a numerical reconstruction algorithm for the kernel, whose main feature is that a non-regular mesh has to be used to ensure the invertibility of the matrix representing the numerical discretization of the system. Finally, a second identifiability result for a polynomial approximation of degree less than nine of the kernel is also established. [ABSTRACT FROM AUTHOR]

This paper investigates an adaptive algorithm for identification of source terms in the linear parabolic equation and Robin boundary condition from the measured final data and the measurement of the temperature in a subregion. This is based on the minimization of Tikhonov functional. We pursue the approach iteratively regularized gradient method. Results with the adaptive algorithm are presented. [ABSTRACT FROM AUTHOR]

This is a review paper of the role of Carleman estimates in the theory of multidimensional coefficient inverse problems since the first inception of this idea in 1981. [ABSTRACT FROM AUTHOR]

In this paper, we combine concepts from two different mathematical research topics: adaptive wavelet techniques for well-posed problems and regularization theory for nonlinear inverse problems with sparsity constraints. We are concerned with identifying certain parameters in a parabolic reaction-diffusion equation from measured data. Analytical properties of the related parameter-to-state operator are summarized, which justify the application of an iterated soft shrinkage algorithm for minimizing a Tikhonov functional with sparsity constraints. The forward problem is treated by means of a new adaptive wavelet algorithm which is based on tensor wavelets. In its general form, the underlying PDE describes gene concentrations in embryos at an early state of development. We implemented an algorithm for the related nonlinear parameter identification problem and numerical results are presented for a simplified test equation. [ABSTRACT FROM AUTHOR]

This paper proposes an approach to reconstructing initial tsunami waveforms in tsunami source areas based on the inversion of remote measurements of water-level data. This inverse problem is treated as an ill-posed problem of hydrodynamic inversion with tsunami tide gauge records. Tsunami wave propagation is considered within the scope of the linear shallow-water theory. The direct problem is approximated by a finite-difference technique. The ill-posed inverse problem of reconstructing initial tsunami waveforms is regularized by means of a least square inversion using the truncated SVD approach. A so-called r-solution is numerically obtained. The algorithm is verified with real bathymetry data of the Peru subduction zone and synthetic records. It is found that the accuracy of tsunami source reconstruction strongly depends on the signal-to-noise ratio, the azimuthal and temporal coverage of tide gauge stations relative to the target area and the bathymetric features along the wave path. [ABSTRACT FROM AUTHOR]

In this paper, we show the uniqueness and local stability of an inverse source problem for the quasi-static Maxwell equation in a layered domain, where the source consists of multiple point dipoles. Also, an algebraic algorithm is proposed to identify the number, locations, and moments of the dipoles from boundary measurements of tangential components of the electric and magnetic fields. The proposed algorithm is numerically verified. [ABSTRACT FROM AUTHOR]

In this paper, the possibility of using the reverse connection control for iteratively regularized Gauss–Newton-type algorithms is analyzed. It is shown that for the generalized Gauss–Newton iterations the diameter of the set containing the control elements can tend to ∞ as n → ∞. To further improve the stabilizing properties of the Gauss–Newton scheme, a new penalty term is proposed that can depend nonlinearly on the current iteration. As a test numerical example, a nonlinear inverse scattering problem of computing a 2D obstacle shape is considered. [ABSTRACT FROM AUTHOR]

The preconditioned iteratively regularized Gauss–Newton algorithm for the minimization of general nonlinear functionals was introduced by Smirnova, Renaut and Khan (Inverse Problems 23: 1547–1563, 2007). In this paper, we establish theoretical convergence results for an extended stabilized family of Generalized Preconditioned Iterative methods which includes ℳ-times iterated Tikhonov regularization with line search. Numerical schemes illustrating the theoretical results are also presented. [ABSTRACT FROM AUTHOR]

In this paper we construct and investigate localization algorithms for isolated singularities of a function which is a solution to a linear convolution equation of the first kind whose right-hand side is given with an error. We consider two types of singularities: δ-functions and discontinuities of the first kind. A problem for singularities localization is an ill-posed problem having perturbation. We use averaging methods defined by an averaging functional to obtain the singularities. We formulate conditions that the averaging functional must satisfy. For convolution equations, using the Fourier transform, we obtained upper estimates of precision of the singularities localization, separation threshold and other important characteristics of the supposed methods. [ABSTRACT FROM AUTHOR]

The Magnetic resonance electrical impedance tomography (MREIT) is a new medical imaging method combining electrical impedance tomography (EIT) and current injection MRI technique. In this paper, we show the uniqueness in MREIT problem with an orthotropic conductivity under the hypothesis that the ratios of conductivities are known. Based on an effective numerical differentiation method and an approach to detect discontinuity, we also propose an iterative reconstruction algorithm for the orthotropic conductivity reconstruction. The resulting numerical algorithm is accurate and stable against the noise, and numerical examples are made to illustrate the performance of our algorithm. [ABSTRACT FROM AUTHOR]

The problem to determine the location and shape of a perfect conductor within a conducting homogeneous host medium from measured current and voltages on the accessible exterior boundary of the host medium can be modelled by an inverse Dirichlet boundary value problem for the Laplace equation. For this, recently Kress and Rundell suggested a novel inverse algorithm based on nonlinear integral equations arising from the reciprocity gap principle. The present paper extends this approach to the problem to recover the location and shape of a rigid body immersed in a fluid from the measured velocity and traction at the exterior boundary of the fluid, that is, to an inverse boundary value problem for the Stokes equation. The mathematical foundation of this extension is provided and numerical examples illustrate the feasibility of the method. [ABSTRACT FROM AUTHOR]

The article analyzes the development of the variational Tikhonov regularization method for ill-posed problems. A. N. Tikhonov published papers on the development of efficient solution methods for a broad class of ill-posed problems in 1963. He was also credited for coming up with the generalization of the notion of approximate solution. Definitions of the regularizing algorithm and regularized solution are provided.

In this paper, a boundary integral method is used to solve an inverse heat conduction problem. An algorithm for the inverse problem of the one dimensional case is given by using the fundamental solution. Numerical results show that our algorithm is effective. [ABSTRACT FROM AUTHOR]

In this paper we discuss an inversion algorithm which combined with the L-curve criterion for the selection of the regularization parameter, effectively yields shape reconstructions of penetrable obstacles. The required scattered elastic field is generated by either a P or S-incident wave. In particular the improved variant of the linear sampling method (LSM), the so called (F*F)1/4-method is studied for the two dimensional elastic transmission case. For our reconstructions we assume that the far field data are noisy and we employ the L-curve for the selection of the regularization parameter. The location of the vertex of the L-curve yields an appropriate value of the regularization parameter. Furthermore, the L-curve approach does not require a priori knowledge of the noise level, and hence combined with the LSM can be used for real world reconstructions, in which noise in the data is unknown. [ABSTRACT FROM AUTHOR]

In this paper we show that an additional a priori information about a sourcewise representation of the solution of nonlinear ill-posed problem can be used for constructing regularizing algorithms with a posteriori error estimates. We describe these algorithms. Algorithms for the case when operators are linear can be obtained from these general algorithms. [ABSTRACT FROM AUTHOR]

An algorithm for the simultaneous determination of two coefficients in an inverse problem for equations of the parabolic type is studied. The model under investigation has been proposed by H. de Vriend and M. Capobianco to describe the long-time coastal profile evolution. The iterative inversion procedure is based on the minimization of a suitable cost functional. Results of numerical tests are shown to illustrate the performance of the algorithm. The real measured data of the coastal profile evolution have been numerically processed to validate the aforementioned diffusion type model. The paper is targetly oriented to attract attention of professionals, who study Inverse Problems, to such new area of applications as Coastal Engineering. [ABSTRACT FROM AUTHOR]

In this paper, an algorithm for solving a 3D inverse kinematic problem of seismics for continuous-inhomogeneous media with zones of considerable horizontal velocity gradients is proposed. This method is a further development of the "nonclassical equation" method proposed earlier for 2D media. [ABSTRACT FROM AUTHOR]

In the paper we numerically solve an inverse problem of determination of medium parameters of a member of thin anisotropic layers for an elasticity system. We also investigate a case of a transversely isotropic medium with the symmetry axis lying in the plane Oxy. For example, this model can describe a vertically fractured medium. We present one of the possible ways of numerical solution of the problem. Examples of the numerical reconstruction show efficiency of the algorithm offered. [ABSTRACT FROM AUTHOR]

Modeling of large lakes or oceans for environmental studies or deformable blood vessels for mini-invasive surgery involve dynamical free boundaries governed by their interaction with their immediate environment. The object of this paper is to further Shape Analysis techniques in the formulation and analysis of dynamical free boundary problems defined over a time-varying domain (noncylindrical problems). The generic example of a potential incompressible fluid flow with a pressure condition (Bernouilli type) on the dynamical free boundary is presented along with a discrete time algorithm to follow the evolution of the time-space differential equation for the normal component of the velocity field on the time-varying free boundary. These ideas and constructions can be extended to more complex and realistic models. [ABSTRACT FROM AUTHOR]

In this paper we propose an iterative algorithm for the solution of Volterra equations of the first kind whose kernel is a square matrix. The algorithm, essential the Lavrientev method coupled with discretization, is "direct in the sense that preliminary numerical computation of the derivative of the observed variable is not required. We assume boundedness of the input μ and mild regularity conditions of the kernel. We prove convergence of the algorithm in L[supP] (0,T), 1 ≤ p < +∞ and uniform convergence on the intervals where the input is continuous. Under additional information on μ we give both integral and pointwise convergence estimates. The observed variable is read with errors, at discrete time instants. [ABSTRACT FROM AUTHOR]

This paper is concerned with the ill-posed problem of identifying a hydraulic transmissivity in an isotropic and confined aquifer in two space dimensions. To define a regularization by adaptive discretization of the parameter, we use refinement and coarsening indicators, which give the first order effect on the optimal data misfit of adding or removing degrees of freedom to a current set of parameters. The direct problem is discretized by a mixed hybrid finite element method. A combination of the direct problem discretization with the adaptive discretization for the transmissivity allows us to prove convergence of the algorithm as the mesh size goes to zero, stability of the finite-dimensional approximation, and convergence for noisy data with an appropriate stopping rule. [ABSTRACT FROM AUTHOR]