TY - JOUR A1 - Böckmann, Christine A1 - Brückner, Axel T1 - 100 years of the Runge-Kutta method : a brief editing for schools BT - Hundred years of the Runge-Kutta method : a brief editing for schools N2 - This paper reports on the historical development of the Runge-Kutta methods beginning with the simple Euler method up to an embedded 13-stage method. Moreover, the design and the use of those methods under error order, stability and computation time conditions is edited for students of numerical analysis at undergraduate level. The second part presents applications in natural sciences, compares different methods and illustrates some of the difficulties of numerical solutions. Y1 - 2001 SN - 0268-3679 ER - TY - JOUR A1 - Bösenberg, Jens A1 - Alpers, Matthias A1 - Böckmann, Christine A1 - Jäger, Horst A1 - Matthias, Volker A1 - Trickl, Thomas A1 - Wandinger, Ulla A1 - Wiegner, Matthias T1 - A Lidar Network for the Establishment of an Aerosol Climatology Y1 - 1998 ER - TY - JOUR A1 - Böckmann, Christine T1 - A modification of the trust-region Gauss-Newton method to solve separable nonlinear least squares problems Y1 - 1995 ER - TY - BOOK A1 - Böckmann, Christine T1 - A modification of the Trust-Region Gauss-Newton method to solve separable nonlinear least squares problems T3 - Preprint / Universität Potsdam, Fachbereich Mathematik Y1 - 1992 VL - 1992, 17 PB - Univ. CY - Potsdam ER - TY - JOUR A1 - Pornsawad, Pornsarp A1 - Sapsakul, Nantawan A1 - Böckmann, Christine T1 - A modified asymptotical regularization of nonlinear ill-posed problems JF - Mathematics N2 - In this paper, we investigate the continuous version of modified iterative Runge–Kutta-type methods for nonlinear inverse ill-posed problems proposed in a previous work. The convergence analysis is proved under the tangential cone condition, a modified discrepancy principle, i.e., the stopping time T is a solution of ∥𝐹(𝑥𝛿(𝑇))−𝑦𝛿∥=𝜏𝛿+ for some 𝛿+>𝛿, and an appropriate source condition. We yield the optimal rate of convergence. KW - nonlinear operator KW - regularization KW - discrepancy principle KW - asymptotic method KW - optimal rate Y1 - 2019 U6 - https://doi.org/10.3390/math7050419 SN - 2227-7390 VL - 7 PB - MDPI CY - Basel, Schweiz ET - 5 ER - TY - GEN A1 - Pornsawad, Pornsarp A1 - Sapsakul, Nantawan A1 - Böckmann, Christine T1 - A modified asymptotical regularization of nonlinear ill-posed problems T2 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe N2 - In this paper, we investigate the continuous version of modified iterative Runge–Kutta-type methods for nonlinear inverse ill-posed problems proposed in a previous work. The convergence analysis is proved under the tangential cone condition, a modified discrepancy principle, i.e., the stopping time T is a solution of ∥𝐹(𝑥𝛿(𝑇))−𝑦𝛿∥=𝜏𝛿+ for some 𝛿+>𝛿, and an appropriate source condition. We yield the optimal rate of convergence. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1335 KW - nonlinear operator KW - regularization KW - discrepancy principle KW - asymptotic method KW - optimal rate Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-473433 SN - 1866-8372 IS - 1335 ER - TY - BOOK A1 - Böckmann, Christine A1 - Niebsch, Jenny T1 - A mollifier method for aerosol size T3 - Preprint / Universität Potsdam, Institut für Mathematik Y1 - 1996 VL - 1996, 07 PB - Univ. CY - Potsdam ER - TY - JOUR A1 - Böckmann, Christine A1 - Wandinger, Ulla A1 - Ansmann, Albert A1 - Bösenberg, Jens A1 - Amiridis, Vassilis A1 - Boselli, Antonella A1 - Delaval, Arnaud A1 - De Tomasi, Ferdinando de A1 - Frioud, Max A1 - Grigorov, Ivan Videnov A1 - Hagard, Arne A1 - Horvat, Matej A1 - Iarlori, Marco A1 - Komguem, Leonce A1 - Kreipl, Stephan A1 - Larchevque, Gilles A1 - Matthias, Volker A1 - Papayannis, Alexandros A1 - Pappalardo, GGelsomina A1 - Rocadenbosch, Francesc A1 - Rodrigues, Jose António A1 - Schneider, Johannes A1 - Shcherbakov, Valery A1 - Wiegner, Matthias T1 - Aerosol lidar intercomparison in the framework of the EARLINET project : 2. Aerosol backscatter algorithms N2 - An intercomparison of aerosol backscatter lidar algorithms was performed in 2001 within the framework of the European Aerosol Research Lidar Network to Establish an Aerosol Climatology (EARLINET). The objective of this research was to test the correctness of the algorithms and the influence of the lidar ratio used by the various lidar teams involved in the EARLINET for calculation of backscatter-coefficient profiles from the lidar signals. The exercise consisted of processing synthetic lidar signals of various degrees of difficulty. One of these profiles contained height- dependent lidar ratios to test the vertical influence of those profiles on the various retrieval algorithms. Furthermore, a realistic incomplete overlap of laser beam and receiver field of view was introduced to remind the teams to take great care in the nearest range to the lidar. The intercomparison was performed in three stages with increasing knowledge on the input parameters. First, only the lidar signals were distributed; this is the most realistic stage. Afterward the lidar ratio profiles and the reference values at calibration height were provided. The unknown height- dependent lidar ratio had the largest influence on the retrieval, whereas the unknown reference value was of minor importance. These results show the necessity of making additional independent measurements, which can provide us with a suitable approximation of the lidar ratio. The final stage proves in general, that the data evaluation schemes of the different groups of lidar systems work well. (C) 2004 Optical Society of America Y1 - 2004 SN - 0003-6935 ER - TY - JOUR A1 - Böckmann, Christine A1 - Biele, Jens A1 - Neuber, Roland T1 - Analysis of multi-wavelength lidar data by inversion with mollifier method Y1 - 1998 ER - TY - JOUR A1 - Böckmann, Christine T1 - Auswertung von multispektralen Lidarmeßdaten Y1 - 1998 ER - TY - THES A1 - Böckmann, Christine T1 - Bestimmung atmosphärischer Aerosolparameter mit Hilfe von regularisierenden Inversionsverfahren Y1 - 2002 ER - TY - JOUR A1 - Kammanee, Athassawat A1 - Böckmann, Christine T1 - Boundary value method for inverse Sturm-Liouville problems N2 - In this paper we present a method to recover symmetric and non-symmetric potential functions of inverse Sturm- Liouville problems from the knowledge of eigenvalues. The linear multistep method coupled with suitable boundary conditions known as boundary value method (BVM) is the main tool to approximate the eigenvalues in each iteration step of the used Newton method. The BVM was extended to work for Neumann-Neumann boundary conditions. Moreover, a suitable approximation for the asymptotic correction of the eigenvalues is given. Numerical results demonstrate that the method is able to give good results for both symmetric and non-symmetric potentials. Y1 - 2009 UR - http://www.sciencedirect.com/science/journal/00963003 U6 - https://doi.org/10.1016/j.amc.2009.04.002 SN - 0096-3003 ER - TY - JOUR A1 - Böckmann, Christine A1 - Kammanee, Athassawat T1 - Broyden method for inverse non-symmetric Sturm-Liouville problems JF - BIT : numerical mathematics ; the leading applied mathematics journal for all computational mathematicians N2 - In this paper, we propose a derivative-free method for recovering symmetric and non-symmetric potential functions of inverse Sturm-Liouville problems from the knowledge of eigenvalues. A class of boundary value methods obtained as an extension of Numerov's method is the major tool for approximating the eigenvalues in each Broyden iteration step. Numerical examples demonstrate that the method is able to reduce the number of iteration steps, in particular for non-symmetric potentials, without accuracy loss. KW - Inverse Sturm-Liouville problem KW - Non-symmetric potential KW - Broyden's method KW - Boundary value method Y1 - 2011 U6 - https://doi.org/10.1007/s10543-011-0317-5 SN - 0006-3835 VL - 51 IS - 3 SP - 513 EP - 528 PB - Springer CY - Dordrecht ER - TY - JOUR A1 - Böckmann, Christine A1 - Sarközi, Janos A1 - Althausen, Dietrich T1 - Collocation Methods for Ill-posed Inverse Problems of Multiwavelength Lidar Measurements and Applications on LACE 98-Data Y1 - 1999 ER - TY - GEN A1 - Pornsawad, Pornsarp A1 - Sungcharoen, Parada A1 - Böckmann, Christine T1 - Convergence rate of the modified Landweber method for solving inverse potential problems T2 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe N2 - In this paper, we present the convergence rate analysis of the modified Landweber method under logarithmic source condition for nonlinear ill-posed problems. The regularization parameter is chosen according to the discrepancy principle. The reconstructions of the shape of an unknown domain for an inverse potential problem by using the modified Landweber method are exhibited. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1034 KW - nonlinear operator KW - regularization KW - modified Landweber method KW - discrepancy principle KW - logarithmic source condition Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-471942 SN - 1866-8372 IS - 1034 ER - TY - JOUR A1 - Pornsawad, Pornsarp A1 - Sungcharoen, Parada A1 - Böckmann, Christine T1 - Convergence rate of the modified Landweber method for solving inverse potential problems JF - Mathematics : open access journal N2 - In this paper, we present the convergence rate analysis of the modified Landweber method under logarithmic source condition for nonlinear ill-posed problems. The regularization parameter is chosen according to the discrepancy principle. The reconstructions of the shape of an unknown domain for an inverse potential problem by using the modified Landweber method are exhibited. KW - nonlinear operator KW - regularization KW - modified Landweber method KW - discrepancy principle KW - logarithmic source condition Y1 - 2020 U6 - https://doi.org/10.3390/math8040608 SN - 2227-7390 VL - 8 IS - 4 PB - MDPI CY - Basel ER - TY - JOUR A1 - Böckmann, Christine T1 - Curve fitting and identification of physical spectra Y1 - 1996 ER - TY - JOUR A1 - Böckmann, Christine T1 - Curve fitting and identification of physical spectra Y1 - 1996 ER - TY - BOOK A1 - Böckmann, Christine T1 - Curve fitting and identification of physical spectra T3 - Preprint / Universität Potsdam, Fachbereich Mathematik Y1 - 1992 VL - 1992, 18 PB - Univ. CY - Potsdam ER - TY - JOUR A1 - Böckmann, Christine T1 - Evaluation of Multi-spectral Lidar Measurements of the Tropo- and Stratosphere via Modern Mathematical Methods for Inverse III- posed Problems to Determine the Aerosol Size Distribution Y1 - 2000 ER -