The inverse problem of molecular force fields calculation is considered within the theory of regularization. In our strategy, we choose the stabilizing matrix F0 as a result of quantum mechanical calculations. The solution of the inverse problem is finding a matrix 𝐹 which is the nearest by the chosen Euclidean norm to the given ab initio F0. The optimized solution is referred to as regularized quantum

Consider the direct and inverse scattering problem of time harmonic acoustic waves by a two-dimensional elastic obstacle which contains an unknown impenetrable object inside. We apply the boundary integral equation method to solve the direct scattering problem. Since the obtained boundary integral system is a mixed form of scalar and vector equations, we consider the existence of the solution in the

We consider the zeta function ζΩ for the Dirichlet-to-Neumann operator of a simply connected planar domain Ω bounded by a smooth closed curve of perimeter 2⁢π. We name the difference ζΩ-ζD the normalized Steklov zeta function of the domain Ω, where 𝔻 denotes the closed unit disk. We prove that (ζΩ-ζD)′′⁢(0)≥0 with equality if and only if Ω is a disk. We also provide an elementary proof that, for a

In the present paper, we devote our effort to a nonlinear inverse problem for recovering a time-dependent potential term in a multi-term time fractional diffusion equation from an additional measurement in the form of an integral over the space domain. First we study the existence, uniqueness, regularity and stability of the solution for the direct problem by using the fixed point theorem. And we obtain

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

On-line optical monitoring of multilayer coating production requires solving inverse identification problems of determining the thicknesses of coating layers. Regardless of the algorithm used to solve inverse problems, the errors in the thicknesses of the deposited layers are correlated by the monitoring procedure. Studying the correlation of thickness errors is important for the production of the

We consider inverse problems of recovering a source term in initial boundary value problems for linear multidimensional partial differential equations (PDEs) of a general form. A universal stable method suitable for solving such inverse problems is proposed. The method allows one to obtain in the same way approximations to exact sources in different kinds of PDEs using various types of linear supplementary

The problem of continuation of a solution of electrodynamic equations from the time-like half-plane S={x∈R3∣x3=0} inside the half-space R+3={x∈R3∣x3>0} is considered. A regularization method for a solution of this problem with approximate data is proposed, and the convergence of this method for the class of functions that are analytic with respect to space variables is stated.

We propose a new approach to constructing globally strictly convex objective functional in a 1-D inverse medium scattering problem using multi-frequency backscattering data. The global convexity of the proposed objective functional is proved. We also prove the global convergence of the gradient projection algorithm and derive an error estimate. Numerical examples are presented to illustrate the performance

In this paper, we consider a seventh-order generalized Korteweg–de Vries (GKdV) equation and study the boundary stability results concerning the inverse problem of recovering a space-dependent source term. We establish a new boundary Carleman estimate for the seventh-order linear operator with the Dirichlet–Neumann type boundary conditions. Using this crucial estimate along with regularity result of

In a Hilbert space, we consider a class of conditionally well-posed inverse problems for which the Hölder type estimate of conditional stability on a bounded closed and convex subset holds. We investigate a finite-dimensional version of Tikhonov’s scheme in which the discretized Tikhonov’s functional is minimized over the finite-dimensional section of the set of conditional stability. For this optimization

This paper is concerned with the convergence of a series associated with a certain version of the convexification method. That version has been recently developed by the research group of the first author for solving coefficient inverse problems. The convexification method aims to construct a globally convex Tikhonov-like functional with a Carleman weight function in it. In the previous works, the

This paper deals with an inverse coefficient problem of simultaneously identifying the thermal conductivity k⁢(x) and radiative coefficient q⁢(x) in the 1D heat equation ut=(k⁢(x)⁢ux)x-q⁢(x)⁢u from the most available Dirichlet and Neumann boundary measured outputs. The Neumann-to-Dirichlet and Neumann-to-Neumann operators Φ⁢[k,q]⁢(t):=u⁢(ℓ,t;k,q), Ψ⁢[k,q]⁢(t):=-k⁢(0)⁢ux⁢(0,t;k,q) are introduced, and

The problem of identification of coefficients and initial conditions for a boundary value problem for parabolic equations that reduces to a minimization problem of a misfit function is investigated. Firstly, the tensor train decomposition approach is presented as a global convergence algorithm. The idea of the proposed method is to extract the tensor structure of the optimized functional and use it

The differential evolution algorithm is applied to solve the optimization problem to reconstruct the production function (inverse problem) for the spatial Solow mathematical model using additional measurements of the gross domestic product for the fixed points. Since the inverse problem is ill-posed the regularized differential evolution is applied. For getting the optimized solution of the inverse

We consider a new class of inverse problems on the recovery of the coefficients of differential equations from a finite set of eigenvalues of a boundary value problem with unseparated boundary conditions. A finite number of eigenvalues is possible only for problems in which the roots of the characteristic equation are multiple. The article describes solutions to such a problem for equations of the

PSF (point spread function) estimation plays an important role in blind image deconvolution. It has been shown in our previous work that minimization of the Stein’s unbiased risk estimate (SURE) – unbiased estimate of mean squared error (MSE) – could yield an accurate PSF estimate. In this paper, we show that the PSF estimation error is upper bounded by the deconvolution accuracy and the mismatch between

We present a family of non-local variational regularization methods for solving tomographic problems, where the solutions are functions with range in a closed subset of the Euclidean space, for example if the solution only attains values in an embedded sub-manifold. Recently, in [R. Ciak, M. Melching and O. Scherzer, Regularization with metric double integrals of functions with values in a set of vectors

In this paper, we propose the r-d class predictors which are general predictors of the best linear unbiased predictor (BLUP), the principal components regression (PCR) and the Liu predictors in the linear mixed models. Superiorities of the linear combination of the new predictors to each of these predictors are done in the sense of the mean square error matrix criterion. Finally, numerical examples

We explore the effect of sampling rates when measuring data given by Mf for special operators M arising in Thermoacoustic Tomography. We start with sampling requirements on Mf given f satisfying certain conditions. After this we discuss the resolution limit on f posed by the sampling rate of Mf without assuming any conditions on these sampling rates. Next we discuss aliasing artifacts when Mf is known

The monitoring, analysis and prediction of epidemic spread in the region require the construction of mathematical model, big data processing and visualization because the amount of population and the size of the region could be huge. One of the important steps is refinement of mathematical model, i.e. determination of initial data and coefficients of system of differential equations of epidemiologic

An inverse problem for determining the order of the Caputo time-fractional derivative in a subdiffusion equation with an arbitrary positive self-adjoint operator A with discrete spectrum is considered. By the Fourier method it is proved that the value of ∥A⁢u⁢(t)∥, where u⁢(t) is the solution of the forward problem, at a fixed time instance recovers uniquely the order of derivative. A list of examples

We study the solvability and stability for the problem of identifying parameter in a class of fractional differential variational inequalities. Our approach is based on a regularity analysis for fractional diffusion equations and fixed point techniques.

A number of particles in a multiplying medium under rather general conditions is asymptotically exponential with respect to time t with the parameter λ, i.e., with the index of power λ⁢t. If the medium is random, then the parameter λ is the random variable. To estimate the temporal asymptotics of the mean particles number (via the medium realizations), it is possible to average the exponential function

In the paper, for systems described by ordinary differential equations a review of algorithms of dynamical input reconstruction by results of inaccurate observations of its solutions is given. The problem under discussion is referred to the class of dynamical inverse problems. The proposed algorithms are stable with respect to informational noises and computational errors. They are based on the combination

We study a time-reversed hyperbolic heat conduction problem based upon the Maxwell–Cattaneo model of non-Fourier heat law. This heat and mass diffusion problem is a hyperbolic type equation for thermodynamics systems with thermal memory or with finite time-delayed heat flux, where the Fourier or Fick law is proven to be unsuccessful with experimental data. In this work, we show that our recent variational

A novel inverse source problem concerning the determination of a term in the right-hand side of parabolic equations from boundary observation is investigated. The observation is given by an imprecise Dirichlet data on some part of the boundary. The unknown heat source is sought as a function depending on both space and time variables with an a priori information. The problem is reformulated as an optimal

We give a short review of results on inverse spectral problems for second-order differential operators on an interval with non-separated boundary conditions. We pay the main attention to the most important nonlinear inverse problems of recovering coefficients of differential operators from given spectral characteristics. In the first part of the review, we provide the main results and methods related

The half-inverse problem is studied for the Sturm–Liouville operator with an eigenparameter dependent boundary condition on a finite interval. We develop a reconstruction procedure and prove the existence theorem for solution of the inverse problem. Our method is based on interpolation of entire functions.

Because of the ill-posedness of multi-frame super resolution (MSR), the regularization method plays an important role in the MSR field. Various regularization terms have been proposed to constrain the image to be estimated. However, artifacts also exist in the estimated image due to the artificial tendency in the manually designed prior model. To solve this problem, we propose a novel regularization-based

Journal Name: Journal of Inverse and Ill-posed Problems Volume: 28 Issue: 4 Pages: i-iv

In this paper, we analyze the convergence rates for finite-dimensional variational regularization in Banach spaces by taking into account the noisy data and operator approximations. In particular, we determine the convergence rates by incorporating the smoothness concepts of Hölder stability estimates and the variational inequalities. Additionally, we discuss two ill-posed inverse problems to complement

An anharmonic oscillator T⁢(q)=-d2d⁢x2+x2+q⁢(x) on the half-axis 0≤x<∞ with the Neumann boundary condition is considered. By means of transformation operators, the direct and inverse spectral problems are studied. We obtain the main integral equations of the inverse problem and prove that the main equation is uniquely solvable. An effective algorithm for reconstruction of perturbed potential is indicated

In 2012, Jin Qinian considered an inexact Newton–Landweber iterative method for solving nonlinear ill-posed operator equations in the Banach space setting by making use of duality mapping. The method consists of two steps; the first one is an inner iteration which gives increments by using Landweber iteration, and the second one is an outer iteration which provides increments by using Newton iteration

Using the input data – the spectrum of the Dirac operator with discontinuity, the potential on the half-interval and one boundary condition as well as discontinuity conditions – this paper presents a uniqueness theorem and the solvability conditions of the potential on the whole interval, and provides a reconstruction algorithm of the solution to this kind of inverse problems.

The projection data obtained using the computed tomography (CT) technique are often incomplete and inconsistent owing to the radiation exposure and practical environment of the CT process, which may lead to a few-view reconstruction problem. Reconstructing an object from few projection views is often an ill-posed inverse problem. To solve such problems, regularization is an effective technique, in

In this paper, the identification problem of recovering the spatial source F∈L2⁢(0,l) in the wave equation ut⁢t=ux⁢x+F⁢(x)⁢cos⁡(ω⁢t), with harmonically varying external source F⁢(x)⁢cos⁡(ω⁢t) and with the homogeneous boundary u⁢(0,t)=u⁢(l,t)=0, t∈(0,T), and initial u⁢(x,0)=ut⁢(x,0)=0, x∈(0,l), conditions, is studied. As a measurement output g⁢(t), the Neumann-type boundary measurement g⁢(t):=ux⁢(0

We consider the inverse scattering problem to reconstruct the defect in an infinite medium with periodicity in the upper half space from near field data. This paper has two contributions. Firstly, we mention that there is a mistake in the factorization method of the earlier paper [A. Lechleiter, The factorization method is independent of transmission eigenvalues, Inverse Probl. Imaging 3 2009, 1, 123–138]

We consider systems of parabolic equations coupled in zero and first order terms. We establish Lipschitz estimates in Lq-norms, 2≤q≤∞, for the source in terms of the solution in a subdomain. The main tool is a family of appropriate Carleman estimates with general weights, in Lebesgue spaces, for nonhomogeneous parabolic systems.

In this paper, a Sturm–Liouville boundary value problem which includes conformable fractional derivatives of order α, 0<α≤1 is considered. We give some uniqueness theorems for the solutions of inverse problems according to the Weyl function, two given spectra and classical spectral data. We also study the half-inverse problem and prove a Hochstadt–Lieberman-type theorem.

In this paper, partial inverse problems for the quadratic pencil of Sturm–Liouville operators on a graph with a loop are studied. These problems consist in recovering the pencil coefficients on one edge of the graph (a boundary edge or the loop) from spectral characteristics, while the coefficients on the other edges are known a priori. We obtain uniqueness theorems and constructive solutions for partial

Degenerate parabolic partial differential equations (PDEs) with vanishing or unbounded leading coefficient make the PDE non-uniformly parabolic, and new theories need to be developed in the context of practical applications of such rather unstudied mathematical models arising in porous media, population dynamics, financial mathematics, etc. With this new challenge in mind, this paper considers investigating

A seven-layers parabolic model with Stephan–Boltzmann interface conditions and Robin boundary conditions is mathematically formulated to describe the heat transfer process in environment-three layers clothing-air gap-body system. Based on this model, the solution to the corresponding inverse problem of simultaneous determination of triple fabric layers thickness is given in this paper, which satisfies

The heat conduction process in composite medium can be modeled by a parabolic equation with discontinuous radiative coefficient. To detect the composite medium characterized by such a non-smooth coefficient from measurable information about the heat distribution, we consider a nonlinear inverse problem for parabolic equation, with the average measurement of temperature field in some time interval as

This paper deals with an inverse problem of the determination of the fractional order in time-fractional diffusion equations from one interior point observation. We give a representation of the solution via the Mittag-Leffler function and eigenfunction expansion, from which the Lipschitz stability of the fractional order with respect to the measured data at the interior point is established.

In this paper we estimate an unknown space-time dependent force being exerted on the vibrating Euler–Bernoulli beam under different boundary supports, which is obtained with the help of measured boundary forces as additional conditions. A sequence of spatial boundary functions is derived, and all the boundary functions and the zero element constitute a linear space. A work boundary functional is coined

A simple idea of finding a domain that encloses an unknown discontinuity embedded in a body is introduced by considering an inverse boundary value problem for the heat equation. The idea gives a design of a special heat flux on the surface of the body such that from the corresponding temperature field on the surface one can extract the smallest radius of the sphere centered at an arbitrary given point

In this work, we consider the inverse spectral problem for the impulsive Sturm–Liouville problem on (0,π) with the Robin boundary conditions and the jump conditions at the point π2. We prove that the potential M⁢(x) on the whole interval and the parameters in the boundary conditions and jump conditions can be determined from a set of eigenvalues for two cases: (i) the potential M⁢(x) is given on (0

Two main aims of this paper are to develop a numerical method to solve an inverse source problem for parabolic equations and apply it to solve a nonlinear coefficient inverse problem. The inverse source problem in this paper is the problem to reconstruct a source term from external observations. Our method to solve this inverse source problem consists of two stages. We first establish an equation of

Consider the fluid-solid interaction problem for a two-layered non-penetrable cavity. We provide a novel fundamental proof for a uniqueness theorem on the determination of the interface between acoustic and elastic waves from many internal measurements, disregarding the boundary conditions imposed on the exterior non-penetrable boundary. The proof depends on a uniform H1-norm boundedness for the elastic

Journal Name: Journal of Inverse and Ill-posed Problems Volume: 28 Issue: 3 Pages: i-iv

We propose a new numerical method for the solution of the problem of the reconstruction of the initial condition of a quasilinear parabolic equation from the measurements of both Dirichlet and Neumann data on the boundary of a bounded domain. Although this problem is highly nonlinear, we do not require an initial guess of the true solution. The key in our method is the derivation of a boundary value

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

We propose an approach and the numerical algorithm for mapping the electroencephalographic (EEG) data from the scalp to the cortex. The algorithm is based on the solution of ill-posed Cauchy problem for the Laplace’s equation using tetrahedral finite elements. The FEM-based scheme allows to calculate the volumetric distribution of a potential over the head volume. We demonstrate the usage of the the

In this work we investigate numerically the reconstruction approach proposed in [F. O. Goncharov and R. G. Novikov, An analog of Chang inversion formula for weighted Radon transforms in multidimensions, Eurasian J. Math. Comput. Appl. 4 2016, 2, 23–32] for weighted ray transforms (weighted Radon transforms along oriented straight lines) in 3D. In particular, the approach is based on a geometric reduction

The paper aims a logarithmic stability estimate for the inverse source problem of the one-dimensional Helmholtz equation with attenuation factor in a two layer medium. We establish a stability by using multiple frequencies at the two end points of the domain which contains the compact support of the source functions.

This paper studies a backward problem for a time fractional diffusion equation, with the distributed order Caputo derivative, of determining the initial condition from a noisy final datum. The uniqueness, ill-posedness and a conditional stability for this backward problem are obtained. The inverse problem is formulated into a minimization functional with Tikhonov regularization. Based on the series

We present a uniqueness result in dimensions 3 for the inverse fixed angle scattering problem associated to the Schrödinger operator -Δ+q, where q is a small real-valued potential with compact support in the Sobolev space Wβ,2, with β>0. This result improves the known result [P. Stefanov, Generic uniqueness for two inverse problems in potential scattering, Comm. Partial Differential Equations 17 1992