Band 10, Heft 4

- This paper is devoted to the mathematical analysis and the numerical solution of data-driven construction of fuzzy controllers. We show that for a special class of controllers (so-called Sugeno controllers), the design problem is equivalent to a nonlinear least squares problem, which turns out to be ill-posed. Therefore we investigate the use of regularization in order to obtain stable approximations of the solution. We analyze a smoothing method, which is common in spline approximation, as well as Tikhonov regularization with respect to stability and convergence. In addition, we develop an iterative method for the regularized problems, which uses the special structure of the problem and test it in some typical numerical examples. We also compare the behavior of the iterations for the original and the regularized least squares problems. It turns out that the regularized problem is not only more robust but also favors solutions that are interpretable easily, which is an important criterion for fuzzy systems.

- An identification of a spherically symmetric potential by its phase shifts is an important physical problem. Recent theoretical results assure that such a potential is uniquely defined by a sufficiently large subset of its phase shifts at any one fixed energy level. However, two different potentials can produce almost identical phase shifts. That is, the inverse problem of the identification of a potential from its phase shifts at one energy level k2 is ill-posed, and the reconstruction is unstable. In this paper we introduce a quantitative measure D(k) of this instability. The diameters of minimizing sets D(k) are used to study the change in the stability with the change of k, and the influence of noise on the identification. They are also used in the stopping criterion for the nonlinear minimization method IRRS (Iterative Random Reduced Search). IRRS combines probabilistic global and deterministic local search methods and it is used for the numerical recovery of the potential by the set of its phase shifts. The results of the identification for noiseless as well as noise corrupted data are presented.

- In the work we will introduce some results of numerical experiments, that allowed the author to choose a certain tactic for the numerical solution of onedimensional coefficient inverse problems which are reduced to minimization of the discrepancy functional. Principle distinction from the well-known approach is in the following: we will use the Laplace transform instead of the Fourier transform with respect to the time variable. In the work we will present a new method of derivation of a gradient for the discrepancy functional. The difference of this way is in the following: it is suitable to obtain analytic formulas for the gradient both in the case investigated for one equation and for a system of equations. As was shown in the numerical experiment which results are introduced in the work, represented considerations and recommendations may be useful for numerical solution of the coefficient inverse problem.

- If f(x, k) is the Jost solution and f(k) = f(0, k), then the I-function is I(k) := f(0, k)/f(k). It is proved that I(k) is in one-to-one correspondence with the scattering triple S := {S(k), k j , s j , 1 ≤ j ≤ J} and with the spectral function ρ(λ) of the Sturm-Liouville operator l = −d 2 /dx 2 + q(x) on (0,∞) with the Dirichlet condition at x = 0 and Analytical methods are given for finding S from I(k) and I(k) from S, and ρ(λ) from I(k) and I(k) from ρ(λ). Since the methods for finding q(x) from S or from ρ(λ) are known, this yields the methods for finding q(x) from I(k).

- We prove that in multidimensional potential scattering some familiar methods of reconstructing unknown potential are valid also for the potentials from the Kato space. We also show a new method for recovering such potentials. The proof is based on the new estimates for Schrödinger operator for the continuous spectrum. These estimates allow us for the cases n = 2 and n = 3 to consider the potentials with stronger singularities than in previous publications.

- Inverse spectral problems for ordinary differential equations of an arbitrary order having a non-integrable singularity inside the interval are studied under additional sewing conditions for solutions at the singular point.