TY - JOUR A1 - Dube, Jonas A1 - Böckmann, Christine A1 - Ritter, Christoph T1 - Lidar-Derived Aerosol Properties from Ny-Ålesund, Svalbard during the MOSAiC Spring 2020 JF - Remote sensing / Molecular Diversity Preservation International (MDPI) N2 - In this work, we present Raman lidar data (from a Nd:YAG operating at 355 nm, 532 nm and 1064 nm) from the international research village Ny-Alesund for the time period of January to April 2020 during the Arctic haze season of the MOSAiC winter. We present values of the aerosol backscatter, the lidar ratio and the backscatter Angstrom exponent, though the latter depends on wavelength. The aerosol polarization was generally below 2%, indicating mostly spherical particles. We observed that events with high backscatter and high lidar ratio did not coincide. In fact, the highest lidar ratios (LR > 75 sr at 532 nm) were already found by January and may have been caused by hygroscopic growth, rather than by advection of more continental aerosol. Further, we performed an inversion of the lidar data to retrieve a refractive index and a size distribution of the aerosol. Our results suggest that in the free troposphere (above approximate to 2500 m) the aerosol size distribution is quite constant in time, with dominance of small particles with a modal radius well below 100 nm. On the contrary, below approximate to 2000 m in altitude, we frequently found gradients in aerosol backscatter and even size distribution, sometimes in accordance with gradients of wind speed, humidity or elevated temperature inversions, as if the aerosol was strongly modified by vertical displacement in what we call the "mechanical boundary layer". Finally, we present an indication that additional meteorological soundings during MOSAiC campaign did not necessarily improve the fidelity of air backtrajectories. KW - aerosol KW - Arctic haze KW - lidar KW - microphysical properties KW - backtrajectories; KW - Ny-Alesund KW - Svalbard KW - MOSAiC KW - aerosol-boundary layer interactions Y1 - 2022 U6 - https://doi.org/10.3390/rs14112578 SN - 2072-4292 VL - 14 IS - 11 PB - MDPI CY - Basel 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 - Perera, Upeksha A1 - Böckmann, Christine T1 - Solutions of Direct and Inverse Even-Order Sturm-Liouville Problems Using Magnus Expansion JF - Mathematics N2 - 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. KW - higher-order Sturm–Liouville problems KW - inverse Sturm–Liouville problems KW - Magnus expansion Y1 - 2019 U6 - https://doi.org/10.3390/math7060544 SN - 2227-7390 VL - 7 IS - 6 PB - MDPI CY - Basel, Schweiz 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 - JOUR A1 - Perera, Upeksha A1 - Böckmann, Christine T1 - Solutions of Sturm-Liouville problems JF - Mathematics N2 - 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. KW - Sturm-Liouville problems of higher order KW - singular Sturm-Liouville KW - problems KW - inverse Sturm-Liouville problems Y1 - 2020 U6 - https://doi.org/10.3390/math8112074 SN - 2227-7390 VL - 8 IS - 11 PB - MDPI CY - Basel ER - TY - GEN A1 - Böckmann, Christine A1 - Ritter, Christoph A1 - Cappelletti, David T1 - Mathematical tool for a closure study of aerosol microphysical property retrieval using lidar and photometer data T2 - IGARSS 2018 - 2018 IEEE International Geoscience and Remote Sensing Symposium N2 - We present a project combining lidar, photometer and particle counter data with a regularization software tool for a closure study of aerosol microphysical property retrieval. In a first step only lidar data are used to retrieve the particle size distribution (PSD). Secondly, photometer data are added, which results in a good consistency of the retrieved PSDs. Finally, those retrieved PSDs may be compared with the measured PSD from a particle counter. The data here were taken in Ny Alesund, Svalbard, as an example. KW - Aerosol KW - Raman lidar KW - photometer KW - inversion KW - regularization KW - particle microphysics Y1 - 2018 SN - 978-1-5386-7150-4 U6 - https://doi.org/10.1109/IGARSS.2018.8518674 SN - 2153-6996 SP - 5575 EP - 5578 PB - IEEE CY - New York ER - TY - JOUR A1 - Kirsche, Andreas A1 - Böckmann, Christine T1 - Pade iteration method for regularization JF - Applied mathematics and computation N2 - 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. KW - Pade approximants KW - iterative regularization KW - ill-posed problem Y1 - 2006 U6 - https://doi.org/10.1016/j.amc.2006.01.011 SN - 0096-3003 VL - 180 IS - 2 SP - 648 EP - 663 PB - Elsevier CY - New York ER - TY - JOUR A1 - Pornsawad, Pornsarp A1 - Böckmann, Christine T1 - Modified Iterative Runge-Kutta-Type Methods for Nonlinear Ill-Posed Problems JF - Numerical functional analysis and optimization : an international journal of rapid publication N2 - 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. KW - Nonlinear ill-posed problems KW - Runge-Kutta methods KW - regularization methods KW - Holder-type source condition KW - stopping rules Y1 - 2016 U6 - https://doi.org/10.1080/01630563.2016.1219744 SN - 0163-0563 SN - 1532-2467 VL - 37 SP - 1562 EP - 1589 PB - Wiley-VCH CY - Philadelphia ER - TY - JOUR A1 - Müller, Detlef A1 - Böckmann, Christine A1 - Kolgotin, Alexei A1 - Schneidenbach, Lars A1 - Chemyakin, Eduard A1 - Rosemann, Julia A1 - Znak, Pavel A1 - Romanov, Anton T1 - Microphysical particle properties derived from inversion algorithms developed in the framework of EARLINET JF - Atmospheric measurement techniques : an interactive open access journal of the European Geosciences Union N2 - We present a summary on the current status of two inversion algorithms that are used in EARLINET (European Aerosol Research Lidar Network) for the inversion of data collected with EARLINET multiwavelength Raman lidars. These instruments measure backscatter coefficients at 355, 532, and 1064 nm, and extinction coefficients at 355 and 532 nm. Development of these two algorithms started in 2000 when EARLINET was founded. The algorithms are based on a manually controlled inversion of optical data which allows for detailed sensitivity studies. The algorithms allow us to derive particle effective radius as well as volume and surface area concentration with comparably high confidence. The retrieval of the real and imaginary parts of the complex refractive index still is a challenge in view of the accuracy required for these parameters in climate change studies in which light absorption needs to be known with high accuracy. It is an extreme challenge to retrieve the real part with an accuracy better than 0.05 and the imaginary part with accuracy better than 0.005-0.1 or +/- 50 %. Single-scattering albedo can be computed from the retrieved microphysical parameters and allows us to categorize aerosols into high-and low-absorbing aerosols. On the basis of a few exemplary simulations with synthetic optical data we discuss the current status of these manually operated algorithms, the potentially achievable accuracy of data products, and the goals for future work. One algorithm was used with the purpose of testing how well microphysical parameters can be derived if the real part of the complex refractive index is known to at least 0.05 or 0.1. The other algorithm was used to find out how well microphysical parameters can be derived if this constraint for the real part is not applied. The optical data used in our study cover a range of Angstrom exponents and extinction-to-backscatter (lidar) ratios that are found from lidar measurements of various aerosol types. We also tested aerosol scenarios that are considered highly unlikely, e.g. the lidar ratios fall outside the commonly accepted range of values measured with Raman lidar, even though the underlying microphysical particle properties are not uncommon. The goal of this part of the study is to test the robustness of the algorithms towards their ability to identify aerosol types that have not been measured so far, but cannot be ruled out based on our current knowledge of aerosol physics. We computed the optical data from monomodal logarithmic particle size distributions, i.e. we explicitly excluded the more complicated case of bimodal particle size distributions which is a topic of ongoing research work. Another constraint is that we only considered particles of spherical shape in our simulations. We considered particle radii as large as 7-10 mu m in our simulations where the Potsdam algorithm is limited to the lower value. We considered optical-data errors of 15% in the simulation studies. We target 50% uncertainty as a reasonable threshold for our data products, though we attempt to obtain data products with less uncertainty in future work. Y1 - 2016 U6 - https://doi.org/10.5194/amt-9-5007-2016 SN - 1867-1381 SN - 1867-8548 VL - 9 SP - 5007 EP - 5035 PB - Copernicus CY - Göttingen ER - TY - JOUR A1 - Altundag, Hüseyin A1 - Böckmann, Christine A1 - Taseli, Hasan T1 - Inverse Sturm-Liouville problems with pseudospectral methods JF - International journal of computer mathematics N2 - In this paper a technique to obtain a first approximation for singular inverse Sturm-Liouville problems with a symmetrical potential is introduced. The singularity, as a result of unbounded domain (-infinity, infinity), is treated by considering numerically the asymptotic limit of the associated problem on a finite interval (-L, L). In spite of this treatment, the problem has still an ill-conditioned structure unlike the classical regular ones and needs regularization techniques. Direct computation of eigenvalues in iterative solution procedure is made by means of pseudospectral methods. A fairly detailed description of the numerical algorithm and its applications to specific examples are presented to illustrate the accuracy and convergence behaviour of the proposed approach. KW - 31A25 KW - 65F18 KW - regularization method KW - condition number KW - pseudospectral method KW - regular and singular inverse Sturm-Liouville problems Y1 - 2015 U6 - https://doi.org/10.1080/00207160.2014.939646 SN - 0020-7160 SN - 1029-0265 VL - 92 IS - 7 SP - 1373 EP - 1384 PB - Routledge, Taylor & Francis Group CY - Abingdon ER - TY - JOUR A1 - Samaras, Stefanos A1 - Nicolae, Doina A1 - Böckmann, Christine A1 - Vasilescu, Jeni A1 - Binietoglou, Ioannis A1 - Labzovskii, Lev A1 - Toanca, Florica A1 - Papayannis, Alexandros T1 - Using Raman-lidar-based regularized microphysical retrievals and Aerosol Mass Spectrometer measurements for the characterization of biomass burning aerosols JF - Journal of computational physics N2 - 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. KW - Aerosols KW - Microphysical properties KW - Lidar KW - AMS KW - AERONET Y1 - 2015 U6 - https://doi.org/10.1016/j.jcp.2015.06.045 SN - 0021-9991 SN - 1090-2716 VL - 299 SP - 156 EP - 174 PB - Elsevier CY - San Diego ER - TY - JOUR A1 - Böckmann, Christine A1 - Osterloh, Lukas T1 - Runge-Kutta type regularization method for inversion of spheroidal particle distribution from limited optical data JF - Inverse problems in science and engineering N2 - 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. KW - inverse ill-posed problem KW - iterative regularization KW - integral equation KW - laser remote sensing KW - inverse scattering KW - aerosol size distribution KW - 65R32 KW - 47A52 KW - 65R20 KW - 78A46 Y1 - 2014 U6 - https://doi.org/10.1080/17415977.2013.830615 SN - 1741-5977 SN - 1741-5985 VL - 22 IS - 1 SP - 150 EP - 165 PB - Routledge, Taylor & Francis Group CY - Abingdon ER - TY - GEN A1 - Osterloha, Lukas A1 - Böckmann, Christine A1 - Nicolae, Doina A1 - Nemuc, Anca T1 - Regularized inversion of microphysical atmospheric particle parameters: Theory and application (vol 237, pg 79, 2013) T2 - Journal of computational physics Y1 - 2014 U6 - https://doi.org/10.1016/j.jcp.2014.07.041 SN - 0021-9991 SN - 1090-2716 VL - 275 SP - 696 EP - 696 PB - Elsevier CY - San Diego ER - TY - JOUR A1 - Böckmann, Christine A1 - Kammanee, Athassawat A1 - Braunss, Andreas T1 - Logarithmic convergence rate of Levenberg-Marquardt method with application to an inverse potential problem JF - Journal of inverse and ill-posed problems N2 - We prove logarithmic convergence rate of the Levenberg-Marquardt method in a Hilbert space if a logarithmic source condition is satisfied. This method is applied to an inverse potential problem. Numerical implementations demonstrate the convergence rate. KW - Levenberg-Marquardt method KW - inverse potential problems KW - logarithmic convergence rate KW - discrepancy principle KW - logarithmic source condition Y1 - 2011 U6 - https://doi.org/10.1515/JIIP.2011.034 SN - 0928-0219 VL - 19 IS - 3 SP - 345 EP - 367 PB - De Gruyter CY - Berlin 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 - Pornsawad, Pornsarp A1 - D'Amico, Giuseppe A1 - Böckmann, Christine A1 - Amodeo, Aldo A1 - Pappalardo, Gelsomina T1 - Retrieval of aerosol extinction coefficient profiles from Raman lidar data by inversion method JF - Applied optics N2 - 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. Y1 - 2012 U6 - https://doi.org/10.1364/AO.51.002035 SN - 1559-128X SN - 2155-3165 VL - 51 IS - 12 SP - 2035 EP - 2044 PB - Optical Society of America CY - Washington ER - TY - JOUR A1 - Hoffmann, Anne A1 - Osterloh, Lukas A1 - Stone, Robert A1 - Lampert, Astrid A1 - Ritter, Christoph A1 - Stock, Maria A1 - Tunved, Peter A1 - Hennig, Tabea A1 - Böckmann, Christine A1 - Li, Shao-Meng A1 - Eleftheriadis, Kostas A1 - Maturilli, Marion A1 - Orgis, Thomas A1 - Herber, Andreas A1 - Neuber, Roland A1 - Dethloff, Klaus T1 - Remote sensing and in-situ measurements of tropospheric aerosol, a PAMARCMiP case study JF - Atmospheric environment : air pollution ; emissions, transport and dispersion, transformation, deposition effects, micrometeorology, urban atmosphere, global atmosphere N2 - 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. KW - Arctic KW - Aerosols KW - Lidar KW - Arctic haze Y1 - 2012 U6 - https://doi.org/10.1016/j.atmosenv.2011.11.027 SN - 1352-2310 VL - 52 IS - 3 SP - 56 EP - 66 PB - Elsevier CY - Oxford ER - TY - JOUR A1 - Osterloh, Lukas A1 - Böckmann, Christine A1 - Nicolae, Doina A1 - Nemuc, Anca T1 - Regularized inversion of microphysical atmospheric particle parameters - theory and application JF - Journal of computational physics N2 - 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. KW - Inverse ill-posed problem KW - Regularization KW - Lidar remote sensing KW - Microphysical particle properties Y1 - 2013 U6 - https://doi.org/10.1016/j.jcp.2012.11.040 SN - 0021-9991 VL - 237 IS - 11 SP - 79 EP - 94 PB - Elsevier CY - San Diego ER - TY - JOUR A1 - Rattana, Amornrat A1 - Böckmann, Christine T1 - Matrix methods for computing eigenvalues of Sturm-Liouville problems of order four JF - Journal of computational and applied mathematics N2 - This paper examines and develops matrix methods to approximate the eigenvalues of a fourth order Sturm-Liouville problem subjected to a kind of fixed boundary conditions. Furthermore, it extends the matrix methods for a kind of general boundary conditions. The idea of the methods comes from finite difference and Numerov's methods as well as boundary value methods for second order regular Sturm-Liouville problems. Moreover, the determination of the correction term formulas of the matrix methods is investigated in order to obtain better approximations of the problem with fixed boundary conditions since the exact eigenvalues for q = 0 are known in this case. Finally, some numerical examples are illustrated. KW - Finite difference method KW - Numerov's method KW - Boundary value methods KW - Fourth order Sturm-Liouville problem KW - Eigenvalues Y1 - 2013 U6 - https://doi.org/10.1016/j.cam.2013.02.024 SN - 0377-0427 SN - 1879-1778 VL - 249 IS - 8 SP - 144 EP - 156 PB - Elsevier CY - Amsterdam 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 -