Band 22, Heft 4

In this work, we study an ill-posed problem involving two time variables. Using a modified quasi-boundary value method, we perturb the final conditions to form an approximate non-local problem. We show that the approximate problems are well posed and we prove some convergence results.

We consider a pencil of non-self-adjoint matrix Sturm–Liouville operators on the half line and study the inverse problem of constructing this pencil by its Weyl matrix. A uniqueness theorem is proved, and a constructive algorithm for the solution is obtained.

The focus of this paper is on the numerical inversion of two integral transforms, namely the spherical Radon and the cosine transform. Both transforms are frequently used in integral geometry and related fields, and their numerical inversion is needed in several applications. To derive fast regularization schemes, the method of the approximate inverse is utilized. We introduce a new family of mollifiers and calculate the corresponding reconstruction kernels analytically for dimension d = 3 and even dimensions d ≥ 4. Numerical results for the three-dimensional case are presented showing that the new class of mollifiers clearly improves the quality of the reconstruction in comparison to the Gaussian mollifier. Moreover, the regularization theory for the method is extended to a framework for arbitrary dimension d ≥ 3 (the special case d = 3 was already considered in [Inverse Problems 27 (2011), 035015]).

We investigate the problem of sampling a unit great circle on the unit sphere 𝕊 3 as a support of orientation distribution functions on which acts the discrete spherical x-ray transform. The circle's partition subsets are gnomonically mapped onto lines that constitute a convex polygon inside the bounding cubes of hypercube. Thus the problem of the great circle tracing is reduced to the problem of the four-dimensional cube sectioning by the plane containing the circle and the intersection figure (the polygon) vertices finding. In this paper, a fast, non-combinatorial approach for the polygon tracing within the general multi-dimensional frame is proposed.

Standard methods for regularizing ill-posed nonlinear equations rely on derivatives of the nonlinear forward mapping. Thereby stronger structural properties of the concrete problem are neglected and the derived algorithms only show mediocre efficiency. We concentrate on nonlinear mappings with quadratic structure and develop a derivative-free regularization method that allows us to apply classical techniques known from linear inverse problems to quadratic equations. In fact, regularization of a quadratic problem can be reduced to regularization of one linear problem and a downstream inversion of a well-posed quadratic mapping. The motivation for considering problems with quadratic structure in more detail comes from applications in laser optics where kernel-based autoconvolution-type equations have to be solved.

This paper investigates an adaptive algorithm for identification of source terms w:={F(x,t);p(t)}${w:=\lbrace F(x,t);p(t)\rbrace }$ in the linear parabolic equation ut=uxx+F(x,t)$u_t= u_{xx} + F(x,t)$ and Robin boundary condition -ux(l,t)=ν[u(l,t)-p(t)]${-u_x(l,t)= \nu [u(l,t) - p(t)]}$ 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.

Recently, Vasin [J. Inverse Ill-Posed Probl. 21 (2013), 109–123] considered a new iterative method for approximately solving nonlinear ill-posed operator equation in Hilbert spaces. In this paper we introduce a modified form of the method considered by Vasin. This paper weakens the conditions needed in the existing results. We use a center-type Lipschitz condition in our convergence analysis instead of a Lipschitz-type condition used in [J. Inverse Ill-Posed Probl. 21 (2013), 109–123]. This way a tighter convergence analysis is obtained and under less computational cost, since the more precise and easier to compute center-Lipschitz instead of the Lipschitz constant is used in the convergence analysis. Order optimal error bounds are given in case the regularization parameter is chosen a priori and by the adaptive method of Pereverzev and Schock [SIAM J. Numer. Anal. 43 (2005), 2060–2076]. A numerical example of a nonlinear integral equation proves the efficiency of the proposed method.