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 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.

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 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]

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.

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

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

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