Volume 23, Issue 1

In this paper, we study inverse problems for multi-dimensional linear degenerate parabolic equations and strongly coupled systems. In particular we discuss the Lipschitz type stability results for the inverse source problems which determine a source term by boundary data on an appropriate sub-boundary and the data on any fixed time. Our arguments are based on the Carleman estimate. Here we prove and use the Carleman estimate with the x -independent weight function for linear degenerate parabolic equations and systems.

We consider Schrödinger operators on noncompact star-shaped graphs. The following topics will be treated under suitable decay conditions on the potential: characterization of the set of eigenvalues and expression of each eigen-projection; low-energy behavior of the resolvent, spectral representations of the operator restricted on infinite rays; determination of the Marchenko equations and the inverse scattering problems.

We prove stability estimates of Hölder-type for Burgers-type equations u t = ( a ( x , t ) u x ) x - d ( x , t ) u u x + f ( x , t ), ( x , t ) ∈ (0,1)×(0, T ), u (0, t ) = g 0 ( t ), u (1, t ) = g 1 ( t ), 0 ≤ t ≤ T , backward in time, with a ( x , t ), d ( x , t ), g 0 ( t ), g 1 ( t ), f ( x , t ) being smooth functions, under relatively weak conditions on the solutions.

The problem of the recovery of a real-valued potential in the two-dimensional Schrödinger equation at positive energy from the Dirichlet-to-Neumann map is considered. It is know that this problem is severely ill-posed and the reconstruction of the potential is only logarithmic stable in general. In this paper a new stability estimate is proved, which is explicitly dependent on the regularity of the potentials and on the energy. Its main feature is an efficient increasing stability phenomenon at sufficiently high energies: in some sense, the stability rapidly changes from logarithmic type to Hölder type. The paper develops also several estimates for a non-local Riemann–Hilbert problem which could be of independent interest.

We investigate a generalization of the well-known iteratively regularized Gauss–Newton method where the Newton equations are regularized variationally using general data fidelity and penalty terms. To obtain convergence rates, we use a general error assumption which has recently been shown to be useful for impulsive and Poisson noise. We restrict the nonlinearity of the forward operator only by a Lipschitz-type condition and compare our results to other convergence rates results proven in the literature. Finally we explicitly state our convergence rates for the aforementioned case of Poisson noise to shed some light on the structure of the posed error assumption.

The inverse problem of determining the unknown spatial load distribution F ( x ) in the cantilever beam equation m ( x ) u tt = -( E I ( x ) u xx ) x x + F ( x ) H ( t ), with arbitrary but separable source term, from the measured data u T ( x ) := u ( x , T ), x ∈ (0, l ), at the final time T > 0 is considered. Some a priori estimates of the weak solution u ∈ H ° 2,1 (Ω T ) of the forward problem are obtained. Introducing the input-output map, it is proved that this map is a compact operator. The adjoint problem approach is then used to derive an explicit gradient formula for the Fréchet derivative of the cost functional J ( F ) = ∥ u (·, T ; F ) - u T (·) ∥ L 2 (0, l ) 2 . The Lipschitz continuity of the gradient is proved. The collocation algorithm combined with the truncated singular value decomposition (TSVD) is used to estimate the degree of ill-posedness of the considered inverse source problem. The conjugate gradient algorithm (CGA), based on the explicit gradient formula, is proposed for numerical solution of the inverse problem. The algorithm is examined through numerical examples related to reconstruction of various spatial loading distributions F ( x ). The numerical results illustrate bounds of applicability of proposed algorithm, also its efficiency and accuracy.