Volume 27, Issue 4

The present paper is devoted to the investigation of a source identification problem that describes the flow in capillaries in the case when an unknown pressure acts on the system. First and second order of accuracy difference schemes are presented for the numerical solution of this problem. Almost coercive stability estimates for these difference schemes are established. Additionally, some numerical results are provided by testing the proposed methods on an example.

Let X be an open subset of ℝ2{\mathbb{R}^{2}}. We study the dynamic operator, 𝒜{\mathcal{A}}, integrating over a family of level curves in X when the object changes between the measurement. We use analytic microlocal analysis to determine which singularities can be recovered by the data-set. Our results show that not all singularities can be recovered as the object moves with a speed lower than the X-ray source. We establish stability estimates and prove that the injectivity and stability are of a generic set if the dynamic operator satisfies the visibility, no conjugate points, and local Bolker conditions. We also show this results can be implemented to fan beam geometry.

Nonlinear ill-posed problems arise in many inverse problems in Hilbert space. We investigate the homotopy method, which can obtain global convergence to solve the problems. The “homotopy with Tikhonov regularization” and “homotopy without derivative” are developed in this paper. The existence of the homotopy curve is proved. Several numerical schemes for tracing the homotopy curve are given, including adaptive tracing skills. Compared to the regularized Newton method, the numerical examples show that our proposed methods are stable and effective.

We study inverse nodal problems for Sturm–Liouville operator perturbed by a Volterra integral operator with a constant delay. We have estimated nodal points and nodal lengths for this operator. Moreover, by using these data, we have shown that the potential function of this operator can be established uniquely.

This paper concerns unique continuation for a reaction-diffusion system with cross diffusion, which is a drug war reaction-diffusion system describing a simple dynamic model of a drug epidemic in an idealized community. We first establish a Carleman estimate for this strongly coupled reaction-diffusion system. Then we apply the Carleman estimate to prove the unique continuation, which means that the Cauchy data on any lateral boundary determine the solution uniquely in the whole domain.

We first give a constructive answer to the attenuated tensor tomography problem on simple surfaces. We then use this result to propose two approaches to produce vector-valued integral transforms, which are fully injective over tensor fields. The first approach is by construction of appropriate weights, which vary along the geodesic flow, generalizing the moment transforms. The second one is by changing the pairing with the tensor field to generate a collection of transverse ray transforms.

In this paper, we consider the iteratively regularized Gauss–Newton method, where regularization is achieved by Ivanov regularization, i.e., by imposing a priori constraints on the solution. We propose an a posteriori choice of the regularization radius, based on an inexact Newton/discrepancy principle approach, prove convergence and convergence rates under a variational source condition as the noise level tends to zero and provide an analysis of the discretization error. Our results are valid in general, possibly nonreflexive Banach spaces, including, e.g., L∞{L^{\infty}} as a preimage space. The theoretical findings are illustrated by numerical experiments.

This paper deals with an inverse source problem for an elliptic equation, using interior measurements. Its motivation lies in the seawater intrusion phenomenon, where we are interested in identifying point sources representing illegal wells. A cost function transforming our inverse problem into an optimization one is proposed, and numerical results are performed for a rectangular domain.

Measuring the error by an ℓ1{\ell^{1}}-norm, we analyze under sparsity assumptions an ℓ0{\ell^{0}}-regularization approach, where the penalty in the Tikhonov functional is complemented by a general stabilizing convex functional. In this context, ill-posed operator equations A⁢x=y{Ax=y} with an injective and bounded linear operator A mapping between ℓ2{\ell^{2}} and a Banach space Y are regularized. For sparse solutions, error estimates as well as linear and sublinear convergence rates are derived based on a variational inequality approach, where the regularization parameter can be chosen either a priori in an appropriate way or a posteriori by the sequential discrepancy principle. To further illustrate the balance between the ℓ0{\ell^{0}}-term and the complementing convex penalty, the important special case of the ℓ2{\ell^{2}}-norm square penalty is investigated showing explicit dependence between both terms. Finally, some numerical experiments verify and illustrate the sparsity promoting properties of corresponding regularized solutions.

Numerical issues for the 3D travel time tomography problem with non-overdetemined data are considered. Truncated Fourier series with respect to a special orthonormal basis of functions depending on the source position is used. In addition, truncated trigonometric Fourier series with respect to two out of three spatial variables are used. First, the Lipschitz stability estimate is obtained. Next, a globally convergent numerical method is constructed using a Carleman estimate for an integral operator.