@article{SamarasNicolaeBoeckmannetal.2015,
  author    = {Samaras, Stefanos and Nicolae, Doina and B{\"o}ckmann, Christine and Vasilescu, Jeni and Binietoglou, Ioannis and Labzovskii, Lev and Toanca, Florica and Papayannis, Alexandros},
  title     = {Using Raman-lidar-based regularized microphysical retrievals and Aerosol Mass Spectrometer measurements for the characterization of biomass burning aerosols},
  series = {Journal of computational physics},
  volume    = {299},
  journal   = {Journal of computational physics},
  publisher = {Elsevier},
  address   = {San Diego},
  issn      = {0021-9991},
  doi       = {10.1016/j.jcp.2015.06.045},
  pages     = {156 -- 174},
  year      = {2015},
  abstract  = {In this work we extract the microphysical properties of aerosols for a collection of measurement cases with low volume depolarization ratio originating from fire sources captured by the Raman lidar located at the National Institute of Optoelectronics (INOE) in Bucharest. Our algorithm was tested not only for pure smoke but also for mixed smoke and urban aerosols of variable age and growth. Applying a sensitivity analysis on initial parameter settings of our retrieval code was proved vital for producing semi-automatized retrievals with a hybrid regularization method developed at the Institute of Mathematics of Potsdam University. A direct quantitative comparison of the retrieved microphysical properties with measurements from a Compact Time of Flight Aerosol Mass Spectrometer (CToF-AMS) is used to validate our algorithm. Microphysical retrievals performed with sun photometer data are also used to explore our results. Focusing on the fine mode we observed remarkable similarities between the retrieved size distribution and the one measured by the AMS. More complicated atmospheric structures and the factor of absorption appear to depend more on particle radius being subject to variation. A good correlation was found between the aerosol effective radius and particle age, using the ratio of lidar ratios (LR: aerosol extinction to backscatter ratios) as an indicator for the latter. Finally, the dependence on relative humidity of aerosol effective radii measured on the ground and within the layers aloft show similar patterns. (C) 2015 Elsevier Inc. All rights reserved.},
  language  = {en}
}
@article{BoeckmannBernutatFischer1998,
  author    = {B{\"o}ckmann, Christine and Bernutat, Claudia and Fischer, Silva},
  title     = {The nonlinear LIDAR-equation : an inverse ill-posed problem},
  year      = {1998},
  language  = {en}
}
@article{BoeckmannSarkoezi1999,
  author    = {B{\"o}ckmann, Christine and Sark{\"o}zi, Janos},
  title     = {The Ill-posed Problem of Multiwavelength Lidar Data by a Hybrid Method of Variable Projection},
  year      = {1999},
  language  = {en}
}
@unpublished{BoeckmannSarkoezi1999,
  author    = {B{\"o}ckmann, Christine and Sark{\"o}zi, Janos},
  title     = {The ill-posed inversion of multiwavelength lidar data by a hybrid method of variable projection},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-14847},
  year      = {1999},
  abstract  = {The ill-posed problem of aerosol distribution determination from a small number of backscatter and extinction lidar measurements was solved successfully via a hybrid method by a variable dimension of projection with B-Splines. Numerical simulation results with noisy data at different measurement situations show that it is possible to derive a reconstruction of the aerosol distribution only with 4 measurements.},
  language  = {en}
}
@article{PereraBoeckmann2020,
  author    = {Perera, Upeksha and B{\"o}ckmann, Christine},
  title     = {Solutions of Sturm-Liouville problems},
  series = {Mathematics},
  volume    = {8},
  journal   = {Mathematics},
  number    = {11},
  publisher = {MDPI},
  address   = {Basel},
  issn      = {2227-7390},
  doi       = {10.3390/math8112074},
  pages     = {14},
  year      = {2020},
  abstract  = {This paper further improves the Lie group method with Magnus expansion proposed in a previous paper by the authors, to solve some types of direct singular Sturm-Liouville problems. Next, a concrete implementation to the inverse Sturm-Liouville problem algorithm proposed by Barcilon (1974) is provided. Furthermore, computational feasibility and applicability of this algorithm to solve inverse Sturm-Liouville problems of higher order (for n=2,4) are verified successfully. It is observed that the method is successful even in the presence of significant noise, provided that the assumptions of the algorithm are satisfied. In conclusion, this work provides a method that can be adapted successfully for solving a direct (regular/singular) or inverse Sturm-Liouville problem (SLP) of an arbitrary order with arbitrary boundary conditions.},
  language  = {en}
}
@misc{PereraBoeckmann2019,
  author    = {Perera, Upeksha and B{\"o}ckmann, Christine},
  title     = {Solutions of direct and inverse even-order Sturm-Liouville problems using Magnus expansion},
  series = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  journal   = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  number    = {1336},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-47341},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-473414},
  pages     = {24},
  year      = {2019},
  abstract  = {In this paper Lie group method in combination with Magnus expansion is utilized to develop a universal method applicable to solving a Sturm-Liouville problem (SLP) of any order with arbitrary boundary conditions. It is shown that the method has ability to solve direct regular (and some singular) SLPs of even orders (tested for up to eight), with a mix of (including non-separable and finite singular endpoints) boundary conditions, accurately and efficiently. The present technique is successfully applied to overcome the difficulties in finding suitable sets of eigenvalues so that the inverse SLP problem can be effectively solved. The inverse SLP algorithm proposed by Barcilon (1974) is utilized in combination with the Magnus method so that a direct SLP of any (even) order and an inverse SLP of order two can be solved effectively.},
  language  = {en}
}
@article{PereraBoeckmann2019,
  author    = {Perera, Upeksha and B{\"o}ckmann, Christine},
  title     = {Solutions of Direct and Inverse Even-Order Sturm-Liouville Problems Using Magnus Expansion},
  series = {Mathematics},
  volume    = {7},
  journal   = {Mathematics},
  number    = {6},
  publisher = {MDPI},
  address   = {Basel, Schweiz},
  issn      = {2227-7390},
  doi       = {10.3390/math7060544},
  pages     = {24},
  year      = {2019},
  abstract  = {In this paper Lie group method in combination with Magnus expansion is utilized to develop a universal method applicable to solving a Sturm-Liouville problem (SLP) of any order with arbitrary boundary conditions. It is shown that the method has ability to solve direct regular (and some singular) SLPs of even orders (tested for up to eight), with a mix of (including non-separable and finite singular endpoints) boundary conditions, accurately and efficiently. The present technique is successfully applied to overcome the difficulties in finding suitable sets of eigenvalues so that the inverse SLP problem can be effectively solved. The inverse SLP algorithm proposed by Barcilon (1974) is utilized in combination with the Magnus method so that a direct SLP of any (even) order and an inverse SLP of order two can be solved effectively.},
  language  = {en}
}
@article{BoeckmannOsterloh2014,
  author    = {B{\"o}ckmann, Christine and Osterloh, Lukas},
  title     = {Runge-Kutta type regularization method for inversion of spheroidal particle distribution from limited optical data},
  series = {Inverse problems in science and engineering},
  volume    = {22},
  journal   = {Inverse problems in science and engineering},
  number    = {1},
  publisher = {Routledge, Taylor \& Francis Group},
  address   = {Abingdon},
  issn      = {1741-5977},
  doi       = {10.1080/17415977.2013.830615},
  pages     = {150 -- 165},
  year      = {2014},
  abstract  = {The Runge-Kutta type regularization method was recently proposed as a potent tool for the iterative solution of nonlinear ill-posed problems. In this paper we analyze the applicability of this regularization method for solving inverse problems arising in atmospheric remote sensing, particularly for the retrieval of spheroidal particle distribution. Our numerical simulations reveal that the Runge-Kutta type regularization method is able to retrieve two-dimensional particle distributions using optical backscatter and extinction coefficient profiles, as well as depolarization information.},
  language  = {en}
}
@unpublished{BoeckmannBieleNeuberetal.1997,
  author    = {B{\"o}ckmann, Christine and Biele, Jens and Neuber, Roland and Niebsch, Jenny},
  title     = {Retrieval of multimodal aerosol size distribution by inversion of multiwavelength data},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-14360},
  year      = {1997},
  abstract  = {The ill-posed problem of aerosol size distribution determination from a small number of backscatter and extinction measurements was solved successfully with a mollifier method which is advantageous since the ill-posed part is performed on exactly given quantities, the points r where n(r) is evaluated may be freely selected. A new twodimensional model for the troposphere is proposed.},
  language  = {en}
}
@article{BoeckmannBieleNeuberetal.1997,
  author    = {B{\"o}ckmann, Christine and Biele, Jens and Neuber, Roland and Niebsch, Jenny},
  title     = {Retrieval of Multimodal aerosol size distribution by Inversion of Multiwavelength data},
  isbn      = {0-8194-2525-7},
  year      = {1997},
  language  = {en}
}
@article{PornsawadD'AmicoBoeckmannetal.2012,
  author    = {Pornsawad, Pornsarp and D'Amico, Giuseppe and B{\"o}ckmann, Christine and Amodeo, Aldo and Pappalardo, Gelsomina},
  title     = {Retrieval of aerosol extinction coefficient profiles from Raman lidar data by inversion method},
  series = {Applied optics},
  volume    = {51},
  journal   = {Applied optics},
  number    = {12},
  publisher = {Optical Society of America},
  address   = {Washington},
  issn      = {1559-128X},
  doi       = {10.1364/AO.51.002035},
  pages     = {2035 -- 2044},
  year      = {2012},
  abstract  = {We regard the problem of differentiation occurring in the retrieval of aerosol extinction coefficient profiles from inelastic Raman lidar signals by searching for a stable solution of the resulting Volterra integral equation. An algorithm based on a projection method and iterative regularization together with the L-curve method has been performed on synthetic and measured lidar signals. A strategy to choose a suitable range for the integration within the framework of the retrieval of optical properties is proposed here for the first time to our knowledge. The Monte Carlo procedure has been adapted to treat the uncertainty in the retrieval of extinction coefficients.},
  language  = {en}
}
@article{HoffmannOsterlohStoneetal.2012,
  author    = {Hoffmann, Anne and Osterloh, Lukas and Stone, Robert and Lampert, Astrid and Ritter, Christoph and Stock, Maria and Tunved, Peter and Hennig, Tabea and B{\"o}ckmann, Christine and Li, Shao-Meng and Eleftheriadis, Kostas and Maturilli, Marion and Orgis, Thomas and Herber, Andreas and Neuber, Roland and Dethloff, Klaus},
  title     = {Remote sensing and in-situ measurements of tropospheric aerosol, a PAMARCMiP case study},
  series = {Atmospheric environment : air pollution ; emissions, transport and dispersion, transformation, deposition effects, micrometeorology, urban atmosphere, global atmosphere},
  volume    = {52},
  journal   = {Atmospheric environment : air pollution ; emissions, transport and dispersion, transformation, deposition effects, micrometeorology, urban atmosphere, global atmosphere},
  number    = {3},
  publisher = {Elsevier},
  address   = {Oxford},
  issn      = {1352-2310},
  doi       = {10.1016/j.atmosenv.2011.11.027},
  pages     = {56 -- 66},
  year      = {2012},
  abstract  = {In this work, a closure experiment for tropospheric aerosol is presented. Aerosol size distributions and single scattering albedo from remote sensing data are compared to those measured in-situ. An aerosol pollution event on 4 April 2009 was observed by ground based and airborne lidar and photometer in and around Ny-Alesund, Spitsbergen, as well as by DMPS, nephelometer and particle soot absorption photometer at the nearby Zeppelin Mountain Research Station. The presented measurements were conducted in an area of 40 x 20 km around Ny-Alesund as part of the 2009 Polar Airborne Measurements and Arctic Regional Climate Model Simulation Project (PAMARCMiP). Aerosol mainly in the accumulation mode was found in the lower troposphere, however, enhanced backscattering was observed up to the tropopause altitude. A comparison of meteorological data available at different locations reveals a stable multi-layer-structure of the lower troposphere. It is followed by the retrieval of optical and microphysical aerosol parameters. Extinction values have been derived using two different methods, and it was found that extinction (especially in the UV) derived from Raman lidar data significantly surpasses the extinction derived from photometer AOD profiles. Airborne lidar data shows volume depolarization values to be less than 2.5\% between 500 m and 2.5 km altitude, hence, particles in this range can be assumed to be of spherical shape. In-situ particle number concentrations measured at the Zeppelin Mountain Research Station at 474 m altitude peak at about 0.18 mu m diameter, which was also found for the microphysical inversion calculations performed at 850 m and 1500 m altitude. Number concentrations depend on the assumed extinction values, and slightly decrease with altitude as well as the effective particle diameter. A low imaginary part in the derived refractive index suggests weakly absorbing aerosols, which is confirmed by low black carbon concentrations, measured at the Zeppelin Mountain as well as on board the Polar 5 aircraft.},
  language  = {en}
}
@misc{OsterlohaBoeckmannNicolaeetal.2014,
  author    = {Osterloha, Lukas and B{\"o}ckmann, Christine and Nicolae, Doina and Nemuc, Anca},
  title     = {Regularized inversion of microphysical atmospheric particle parameters: Theory and application (vol 237, pg 79, 2013)},
  series = {Journal of computational physics},
  volume    = {275},
  journal   = {Journal of computational physics},
  publisher = {Elsevier},
  address   = {San Diego},
  issn      = {0021-9991},
  doi       = {10.1016/j.jcp.2014.07.041},
  pages     = {696 -- 696},
  year      = {2014},
  language  = {en}
}
@article{OsterlohBoeckmannNicolaeetal.2013,
  author    = {Osterloh, Lukas and B{\"o}ckmann, Christine and Nicolae, Doina and Nemuc, Anca},
  title     = {Regularized inversion of microphysical atmospheric particle parameters - theory and application},
  series = {Journal of computational physics},
  volume    = {237},
  journal   = {Journal of computational physics},
  number    = {11},
  publisher = {Elsevier},
  address   = {San Diego},
  issn      = {0021-9991},
  doi       = {10.1016/j.jcp.2012.11.040},
  pages     = {79 -- 94},
  year      = {2013},
  abstract  = {Retrieving the distribution of aerosols in the atmosphere via remote sensing techniques is a highly complex task that requires dealing with a wide range of different problems stemming both from Physics and Mathematics. We focus on retrieving this distribution from multi-wavelength lidar data for aerosol ensembles consisting of spherical particles via an iterative regularization technique. The optical efficiencies for spherical scatterers are examined to account for the behavior of the underlying integral equation. The ill-posedness of the problem and the conditioning of the discretized problem are analyzed. Some critical points in the model, like the assumed wavelength-independence of the refractive index and the fixed grid of investigated refractive indices, are studied with regard to their expected impact on the regularized solution. A new Monte-Carlo type method is proposed for retrieval of the refractive index. To validate the results, the developed algorithm is applied to two measurement cases of burning biomass gained from multi-wavelength Raman lidar.},
  language  = {en}
}
@article{KirscheBoeckmann2005,
  author    = {Kirsche, Andreas and B{\"o}ckmann, Christine},
  title     = {Rational approximations for ill-conditioned equation systems},
  year      = {2005},
  abstract  = {In this study we present iterative methods using rational approximations, e.g... Pade approximants, which work very well for strongly ill-conditioned systems. In principle all methods of the family are convergent. One type of those methods has the advantage that their convergence behavior is very fast without additional a-priori information on the optimal relaxation parameter. (c) 2005 Elsevier Inc. All rights reserved},
  language  = {en}
}
@article{Boeckmann1997,
  author    = {B{\"o}ckmann, Christine},
  title     = {Projection method for lidar measurements},
  isbn      = {981-02-2918-6},
  year      = {1997},
  language  = {en}
}
@article{KirscheBoeckmann2006,
  author    = {Kirsche, Andreas and B{\"o}ckmann, Christine},
  title     = {Pade iteration method for regularization},
  series = {Applied mathematics and computation},
  volume    = {180},
  journal   = {Applied mathematics and computation},
  number    = {2},
  publisher = {Elsevier},
  address   = {New York},
  issn      = {0096-3003},
  doi       = {10.1016/j.amc.2006.01.011},
  pages     = {648 -- 663},
  year      = {2006},
  abstract  = {In this study we present iterative regularization methods using rational approximations, in particular, Pade approximants, which work well for ill-posed problems. We prove that the (k,j)-Pade method is a convergent and order optimal iterative regularization method in using the discrepancy principle of Morozov. Furthermore, we present a hybrid Pade method, compare it with other well-known methods and found that it is faster than the Landweber method. It is worth mentioning that this study is a completion of the paper [A. Kirsche, C. Bockmann, Rational approximations for ill-conditioned equation systems, Appl. Math. Comput. 171 (2005) 385-397] where this method was treated to solve ill-conditioned equation systems. (c) 2006 Elsevier Inc. All rights reserved.},
  language  = {en}
}
@article{BoeckmannNiebsch1997,
  author    = {B{\"o}ckmann, Christine and Niebsch, Jenny},
  title     = {Mollifier Methods for Aerosol Size Distribution},
  year      = {1997},
  language  = {en}
}
@article{PornsawadBoeckmann2016,
  author    = {Pornsawad, Pornsarp and B{\"o}ckmann, Christine},
  title     = {Modified Iterative Runge-Kutta-Type Methods for Nonlinear Ill-Posed Problems},
  series = {Numerical functional analysis and optimization : an international journal of rapid publication},
  volume    = {37},
  journal   = {Numerical functional analysis and optimization : an international journal of rapid publication},
  publisher = {Wiley-VCH},
  address   = {Philadelphia},
  issn      = {0163-0563},
  doi       = {10.1080/01630563.2016.1219744},
  pages     = {1562 -- 1589},
  year      = {2016},
  abstract  = {This work is devoted to the convergence analysis of a modified Runge-Kutta-type iterative regularization method for solving nonlinear ill-posed problems under a priori and a posteriori stopping rules. The convergence rate results of the proposed method can be obtained under a Holder-type sourcewise condition if the Frechet derivative is properly scaled and locally Lipschitz continuous. Numerical results are achieved by using the Levenberg-Marquardt, Lobatto, and Radau methods.},
  language  = {en}
}
@unpublished{PornsawadBoeckmann2014,
  author    = {Pornsawad, Pornsarp and B{\"o}ckmann, Christine},
  title     = {Modified iterative Runge-Kutta-type methods for nonlinear ill-posed problems},
  volume    = {3},
  number    = {7},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  issn      = {2193-6943},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-70834},
  pages     = {30},
  year      = {2014},
  abstract  = {This work is devoted to the convergence analysis of a modified Runge-Kutta-type iterative regularization method for solving nonlinear ill-posed problems under a priori and a posteriori stopping rules. The convergence rate results of the proposed method can be obtained under H{\"o}lder-type source-wise condition if the Fr{\´e}chet derivative is properly scaled and locally Lipschitz continuous. Numerical results are achieved by using the Levenberg-Marquardt and Radau methods.},
  language  = {en}
}