Publications: Recent

[1]

F. Kappel and D. Salamon. Approximation of the algebraic Riccati operator equation. In Proc. 27th IEEE Conference on Decision and Control, December 7–9, 1988, Austin, Texas, pages 2001–2002, 1988.

[2]

K. Ito and F. Kappel. The Trotter-Kato theorem and approximation of PDEs. Math. Computation, 67:21–44, 1998.

 

We present formulations of the Trotter-Kato theorem for approximation of linear C₀-semigroups which provide very useful framework when convergence of numerical approximations to solutions of PDEs are studied. Applicability of our results is demonstrated using a first order hyperbolic equation, a wave equation and Stokes' equation as illustrative examples.

[3]

P. Bachhiesl, H. Hutten, F. Kappel, B. Puswald, and H. Scharfetter. Patient individual analysis and control of exchange processes during hemodialysis. Med. Biol. Eng. Comput., 37, Suppl. 2:1162–1163, 1999.

[4]

P. Bachhiesl, H. Hutten, F. Kappel, and H. Scharfetter. Ein Ansatz zur Optimierung der Prozeßsteuerung bei der Hämodialyse. Automatisierungstechnik, 47:38–48, 1999.

[5]

H. Hutten, F. Kappel, R. Merwa, and H. Scharfetter. Modular programming environment for simulation, identification and validation of complex nonlinear compartment models. Med. Biol. Eng. Comput., 37, Suppl 2:1204–1205, 1999.

[6]

Abd El Rahman Aly Hussein and F. Kappel. The Trotter-Kato theorem and collocation methods for parabolic equations. In Sri Wahyuni, Retantyo Wardoyo, CH. Rini Indrati, and Supama, editors, Proc. of the SEAMS–GMU Internatl. Conf. Math. and its Appl., Yogyakarta, July 26–29, 1999, pages 22–36, Yogyakarta, Indonesia, 2000. Department of Mathematics, Gadjah Mada University. ISBN 979–95118–2--8.

 

Convergence of a C¹ collocation scheme for a parabolic PDE in one space dimension is established for any initial data in the state space L²(0,1). The proof of converngence is based on the Trotter-Kato approximation scheme for C₀-semigroups.

[7]

Abd El Rahman Aly Hussein, K. Ito, and F. Kappel. Some aspects of the Trotter-Kato approximation theorem for semigroups. In I. I. Eremin, I. Lasicka, and V. I. Maksimov, editors, Distributed Systems: Optimization and Economic-Environmental Applications (DSO'2000), Proceedings of an Internatl. Conf., May 30 – June 2, 2000, Ekaterinburg, Russia, pages 36–37, 2000. ISBN 5–7691–1047–3.

 

The classical proof of the Trotter-Kato theorems also provides rate estimates for smooth initial data. We discuss the possibilities to reduce the socalled smoothness gap for special types of semigroups as for instance analytic semigroups. Furthermore, we show that the assumption of stability of the approximating semigroups can be relaxed if at the same time the consistency requirement is replaced by a stronger assumption. Finally, we give a proof for a `folk' theorem for nonhomogeneous problems which repeatedly was proved under stronger assumptions than necessary.

[8]

F. Kappel and A. V. Kuntsevich. An implementation of Shor's r-algorithm. Computational Optimization and Applications, 15:193–205, 2000.

 

Here we introduce a new implementation of well-known Shor's r-algorithm with space dilations along the difference of two successive (sub)gradients for minimization of a nonlinear (non-smooth) function [Sh85]. The modifications made to Shor's algorithm are heuristicand concern the termination criterion and the line search strategy. A large number of test runs indicate that this implementation is to some degree robust, efficient and accurate. We hope that this implementation of Shor's r-algorithm will prove to be useful for solving a wide class of non-smooth optimization problems.

[9]

F. Kappel and V. I. Maksimov. Dinamiceskaja rekonstrukcija sostojanii i garantirujuscee upravlenie sistemoi reakcii-diffuzii. Doklady Akad. Nauk, 370:599–601, 2000.

[10]

F. Kappel and V. I. Maksimov. Robust dynamical input reconstruction for delay systems. Int. J. Appl. Math. Comp. Sci., 10:283–307, 2000.

[11]

F. Kappel, V. I. Maksimov, and E. N. Skuratov. On dynamical reconstruction of control in a system with time delay: Finite-dimensional models. In I. I. Eremin, I. Lasicka, and V. I. Maksimov, editors, Distributed Systems: Optimization and Economic-Environmental Applications (DSO'2000), Proceedings of an Internatl. Conf., May 30 – June 2, 2000, Ekaterinburg, Russia, pages 262–264, 2000. ISBN 5–7691–1047–3.

 

The problem of dynamical reconstruction of an unknown control acting upon a linear system with time delay through inaccurate observation of phase states is discussed. Regularized solving algorithm functioning in real time mode is proposed. This algorithm is stable with respect to informational noises and computational errors.

[12]

P. Bachhiesl, M. Hintermüller, H. Hutten, F. Kappel, and H. Scharfetter. Efficient computation of optimal controls for the exchange processes during the dialysis therapy. Computational Optimization and Applications, 18:161–174, 2001.