Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

IMPACT FACTOR 2017: 0.941
5-year IMPACT FACTOR: 0.953

CiteScore 2017: 0.91

SCImago Journal Rank (SJR) 2017: 0.461
Source Normalized Impact per Paper (SNIP) 2017: 1.022

Mathematical Citation Quotient (MCQ) 2017: 0.49

See all formats and pricing
More options …
Volume 26, Issue 1


A modified coupled complex boundary method for an inverse chromatography problem

Xiaoliang Cheng / Guangliang Lin / Ye Zhang / Rongfang Gong / Mårten Gulliksson
Published Online: 2017-05-05 | DOI: https://doi.org/10.1515/jiip-2016-0057


Adsorption isotherms are the most important parameters in rigorous models of chromatographic processes. In this paper, in order to recover adsorption isotherms, we consider a coupled complex boundary method (CCBM), which was previously proposed for solving an inverse source problem [2]. With CCBM, the original boundary fitting problem is transferred to a domain fitting problem. Thus, this method has advantages regarding robustness and computation in reconstruction. In contrast to the traditional CCBM, for the sake of the reduction of computational complexity and computational cost, the recovered adsorption isotherm only corresponds to the real part of the solution of a forward complex initial boundary value problem. Furthermore, we take into account the position of the profiles and apply the momentum criterion to improve the optimization progress. Using Tikhonov regularization, the well-posedness, convergence properties and regularization parameter selection methods are studied. Based on an adjoint technique, we derive the exact Jacobian of the objective function and give an algorithm to reconstruct the adsorption isotherm. Finally, numerical simulations are given to show the feasibility and efficiency of the proposed regularization method.

Keywords: Chromatography; adsorption isotherm; inverse problem; coupled complex boundary method; Tikhonov regularization

MSC 2010: 65N21; 65F22; 65M32; 65K15; 76R50


  • [1]

    S. W. Anzengruber and R. Ramlau, Morozov’s discrepancy principle for Tikhonov-type functionals with nonlinear operators, Inverse Problems 26 (2009), no. 2, Article ID 025001. Web of ScienceGoogle Scholar

  • [2]

    X. Cheng, R. Gong, W. Han and X. Zheng, A novel coupled complex boundary method for solving inverse source problems, Inverse Problems 30 (2014), no. 5, Article ID 055002. Web of ScienceGoogle Scholar

  • [3]

    M. Enmark, P. Forssén, J. Samuelsson and T. Fornstedt, Determination of adsorption isotherms in supercritical fluid chromatography, J. Chromatogr. A 1322 (2013), 124–133. Web of ScienceGoogle Scholar

  • [4]

    M. Enmark, J. Samuelsson, E. Forss, P. Forssén and T. Fornstedt, Investigation of plateau methods for adsorption isotherm determination in supercritical fluid chromatography, J. Chromatogr. A 1354 (2014), 129–138. Web of ScienceCrossrefPubMedGoogle Scholar

  • [5]

    L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 2015.Google Scholar

  • [6]

    A. Felinger, A. Cavazzini and G. Guiochon, Numerical determination of the competitive isotherm of enantiomers, J. Chromatogr. A 986 (2003), no. 2, 207–225. PubMedWeb of ScienceCrossrefGoogle Scholar

  • [7]

    A. Felinger, D. Zhou and G. Guiochon, Determination of the single component and competitive adsorption isotherms of the 1-indanol enantiomers by the inverse method, J. Chromatogr. A 1005 (2003), no. 1–2, 35–49. PubMedCrossrefGoogle Scholar

  • [8]

    A. V. Fiacco, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Academic Press, New York, 1983. Google Scholar

  • [9]

    T. Fornstedt, Characterization of adsorption processes in analytical liquid–solid chromatography, J. Chromatogr. A 1217 (2010), no. 6, 792–812. PubMedCrossrefWeb of ScienceGoogle Scholar

  • [10]

    P. Forssén, R. Arnell and T. Fornstedt, An improved algorithm for solving inverse problems in liquid chromatography, Comput. Chem. Eng. 30 (2006), no. 9, 1381–1391. CrossrefGoogle Scholar

  • [11]

    P. Forssén and T. Fornstedt, A model free method for estimation of complicated adsorption isotherms in liquid chromatography, J. Chromatogr. A 1409 (2015), 108–115. CrossrefWeb of SciencePubMedGoogle Scholar

  • [12]

    P. Forssén, J. Samuelsson and T. Fornstedt, Relative importance of column and adsorption parameters on the productivity in preparative liquid chromatography. I: Investigation of a chiral separation system, J. Chromatogr. A 1299 (2013), 58–63. CrossrefWeb of SciencePubMedGoogle Scholar

  • [13]

    R. Gong, X. Cheng and W. Han, A coupled complex boundary method for an inverse conductivity problem with one measurement, Appl. Anal. 96 (2017), no. 5, 869–885. Web of ScienceCrossrefGoogle Scholar

  • [14]

    G. Guiochon, A. Felinger and D. G. G. Shirazi, Fundamentals of Preparative and Nonlinear Chromatography, Academic Press, New York, 2006. Google Scholar

  • [15]

    G. Guiochon and B. Lin, Modeling for Preparative Chromatography, Academic Press, New York, 2003. Google Scholar

  • [16]

    C. Horvath, High-performance Liquid Chromatography: Advances and Perspectives. Vol. 5, Academic Press, New York, 1988. Google Scholar

  • [17]

    S. Jacobson, S. Golshan-Shirazi and G. Guiochon, Chromatographic band profiles and band separation of enantiomers at high concentration, J. Amer. Chem. Soc. 112 (1990), no. 18, 6492–6498. CrossrefGoogle Scholar

  • [18]

    F. James and M. Sepulveda, Parameter identification for a model of chromatographic column, Inverse Problems 10 (1994), Article ID 1299. Google Scholar

  • [19]

    S. Javeed, S. Qamar, A. Seidel-Morgenstern and G. Warnecke, Efficient and accurate numerical simulation of nonlinear chromatographic processes, Comput. Chem. Eng. 35 (2011), 2294–2305. Web of ScienceCrossrefGoogle Scholar

  • [20]

    S. Kim, K. J. Lee, M. C. Kim and K. Y. Kim, Estimation of temperature-dependent thermal conductivity with a simple integral approach, Int. Commun. Heat Mass Transf. 30 (2003), no. 4, 485–494. CrossrefGoogle Scholar

  • [21]

    P. Kügler and H. W. Engl, Identification of a temperature dependent heat conductivity by Tikhonov regularization, J. Inverse Ill-Posed Probl. 10 (2002), no. 1, 67–90. Google Scholar

  • [22]

    K. Kunisch and W. Ring, Regularization of nonlinear ill-posed problems with closed operators, Numer. Funct. Anal. Optim. 14 (1993), no. 3–4, 389–404. CrossrefGoogle Scholar

  • [23]

    O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Society, London, 1991. Google Scholar

  • [24]

    G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, London, 2005. Google Scholar

  • [25]

    G. Lin, Y. Zhang, X. Cheng, M. Gulliksson, P. Forssén and T. Fornstedt, A regularizing Kohn–Vogelius formulation for the model-free adsorption isotherm estimation problem in chromatography, Appl. Anal. (2017), 10.1080/00036811.2017.1284311. Web of ScienceGoogle Scholar

  • [26]

    J. Lindholm, P. Forssen and T. Fornstedt, Validation of the accuracy of the perturba-tion peak method for determination of single and binary adsorption isothermparameters in lc, Anal. Chem. 76 (2004), 4856–4865. CrossrefPubMedGoogle Scholar

  • [27]

    O. Lisec, P. Hugo and A. Seidel-Morgenstern, Frontal analysis method to determine competitive adsorption isotherms, J. Chromatogr. A 908 (2001), no. 1–2, 19–34. PubMedCrossrefGoogle Scholar

  • [28]

    M. Mierzwiczak and J. A. Kołodziej, The determination temperature-dependent thermal conductivity as inverse steady heat conduction problem, Int. Commun. Heat Mass Transf. 54 (2011), no. 4, 790–796. CrossrefGoogle Scholar

  • [29]

    J. Nocedal and S. Wright, Numerical Optimization, Springer, New York, 2006. Google Scholar

  • [30]

    D. M. Ruthven, Principles of Adsorption and Adsorption Processes, Wiley, New York, 1984. Google Scholar

  • [31]

    J. Samuelsson, R. Arnell and T. Fornstedt, Potential of adsorption isotherm measurements for closer elucidating of binding in chiral liquid chromatographic phase systems, J. Sep. Sci. 32 (2009), no. 10, 1491–1506. CrossrefPubMedWeb of ScienceGoogle Scholar

  • [32]

    E. Schock, Arbitrarily slow convergence, uniform convergence and superconvergence of galerkin-like methods, IMA J. Numer. Anal. 5 (1985), no. 2, 153–160. CrossrefGoogle Scholar

  • [33]

    A. Seidel-Morgenstern, Experimental determination of single solute and competitive adsorption isotherms, J. Chromatogr. A 1037 (2004), 255–272. CrossrefPubMedGoogle Scholar

  • [34]

    A. Vergnaud, G. Beaugrand, O. Gaye, L. Perez, P. Lucidarme and L. Autrique, On-line identification of temperature-dependent thermal conductivity, 2014 European Control Conference (ECC), IEEE Press, Piscataway (2014), 2139–2144. Google Scholar

  • [35]

    Y. Zhang, G. Lin, P. Forssén, M. Gulliksson, T. Fornstedt and X. Cheng, A regularization method for the reconstruction of adsorption isotherms in liquid chromatography, Inverse Problems 32 (2016), no. 10, Article ID 105005. Web of ScienceGoogle Scholar

  • [36]

    Y. Zhang, G. Lin, M. Gulliksson, P. Forssén, T. Fornstedt and X. Cheng, An adjoint method in inverse problems of chromatography, Inverse Probl. Sci. Eng. (2016), 10.1080/17415977.2016.1222528. Web of ScienceGoogle Scholar

About the article

Received: 2016-09-10

Revised: 2017-03-03

Accepted: 2017-03-07

Published Online: 2017-05-05

Published in Print: 2018-02-01

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11571311

Award identifier / Grant number: 11401304

Funding Source: Swedish Foundation for International Cooperation in Research and Higher Education

Award identifier / Grant number: IB2015-5989

Funding Source: Knowledge Foundation

Award identifier / Grant number: 20150233

Funding Source: Vetenskapsrådet

Award identifier / Grant number: 2015-04627

The work was partially supported by NSFC (grant numbers 11571311 and 11401304), STINT (grant number IB2015-5989), KK HÖG (grant number 20150233), ÅForsk (grant number 15/497), and VR (grant number 2015-04627).

Citation Information: Journal of Inverse and Ill-posed Problems, Volume 26, Issue 1, Pages 33–49, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip-2016-0057.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Ye Zhang, Rongfang Gong, Mårten Gulliksson, and Xiaoliang Cheng
Journal of Inverse and Ill-posed Problems, 2018, Volume 0, Number 0

Comments (0)

Please log in or register to comment.
Log in