Rheologica Acta

, Volume 47, Issue 2, pp 169–178 | Cite as

Estimation of the relaxation spectrum from dynamic experiments using Bayesian analysis and a new regularization constraint

Original Contribution


The relaxation spectrum is estimated from dynamic experiments using Bayesian analysis and a new regularization constraint. In the Bayesian framework, a probability can be calculated for each estimate of the spectrum. This offers several advantages; (1) an optimal estimate of the relaxation spectrum may be calculated as the mean of a large number of estimates, and (2) reliable errors for the optimal estimate can be provided using the deviation of all estimates from the mean. Furthermore, the Bayesian approach (3) gives an estimate of the overall noise level of the experiment, which is usually an important but unknown parameter for the calculation of relaxation spectra from dynamic experiments by indirect methods (determining the regularization parameter), and finally, (4) the information content in a given set of experimental data can be quantified. The validity of the Bayesian approach is demonstrated using simulated data.


Rheology Relaxation spectrum Bayesian analysis 


  1. Baumgaertel M, Schausberger A, Winter HH (1990) The relaxation of polymers with linear flexible chains of uniform length. Rheol Acta 29:400–408CrossRefGoogle Scholar
  2. Davies AR, Anderssen RS (1997) Sampling localization in determining the relaxation spectrum. J Non-Newton Fluid Mech 73:163–179CrossRefGoogle Scholar
  3. Dose V (2003) Bayesian inference in physics: case studies. Rep Prog Phys 66:1421–1461CrossRefGoogle Scholar
  4. Emri I, Tschoegl NW (1993) Generating line spectra from experimental responses. I. Relaxation modulus and creep compliance. Rheol Acta 32:311–321CrossRefGoogle Scholar
  5. Elster C, Honerkamp J (1991) Modified maximum entropy method and its application to creep data. Macromolecules 24:310–314CrossRefGoogle Scholar
  6. Ferry JD (1980) Visoelastic properties of polymers. Wiley, New YorkGoogle Scholar
  7. Gull SF (1989) Maximum-entropy and Bayesian methods. In: Skilling O (ed). Kluwer, Dordrecht, pp 53–71Google Scholar
  8. Hansen S, Wilkins SW (1994) On uncertainty in maximum-entropy maps and the generalization of classic MaxEnt. Acta Crystallogr A50:547–550Google Scholar
  9. Hansen S (2000) Bayesian estimation of hyperparameters for indirect Fourier transformation in small-angle scattering. J Appl Crystallogr 33:1415–1421CrossRefGoogle Scholar
  10. Honerkamp J, Weese J (1989) Determination of the relaxation spectrum by a regularization method. Macromolecules 22:4372–4377CrossRefGoogle Scholar
  11. Honerkamp J, Weese J (1990) Tikhonov’s regularization method for ill-posed problems: a comparison of different methods for the determination of the regularization parameter. Contin Mech Thermodyn 2:17–30CrossRefGoogle Scholar
  12. Honerkamp J, Weese J (1993) A nonlinear regularization method for the calculation of relaxation spectra. Rheol Acta 32:65–73CrossRefGoogle Scholar
  13. Jackson JK, de Rosa ME, Winter HH (1994) Molecular weight dependence of relaxation time spectra for the entanglement and flow behavior of monodisperse linear flexible polymers. Macromolecules 27:2426–2431CrossRefGoogle Scholar
  14. Jensen EA (2002) Determination of discrete relaxation spectra using Simulated Annealing. J Non-Newton Fluid Mech 107:1–11CrossRefGoogle Scholar
  15. Mao R, Tang J, Swanson BG (2000) Relaxation time spectrum of hydrogels by contin. Analysis J Food Sci 65:374–381CrossRefGoogle Scholar
  16. MacKay DJC (1992) Maximum entropy and Bayesian methods. In: Smith CR et al. (ed), Seattle, 1991. Kluwer, The Netherlands, pp 39–66Google Scholar
  17. Mead DW (1994) Numerical interconversion of linear viscoelastic material functions. J Rheol 38:1769–1795CrossRefGoogle Scholar
  18. Provencher SW (1982a) A constrained regularization method for inverting data represented by linear algebraic or integral equations. Comput Phys Commun 27:213–227CrossRefGoogle Scholar
  19. Provencher SW (1982b) CONTIN a general purpose constrained regularization program for inverting linear algebraic or integral equations. Comput Phys Commun 27:229–242CrossRefGoogle Scholar
  20. Skilling J (1988) Maximum-entropy and Bayesian methods in science and engineering, vol 1. In: Erickson GJ, Ray Smith C (eds). Kluwer, Dordrecht, pp 173–187Google Scholar
  21. Steenstrup S, Hansen S (1994) The maximum-entropy method without the positivity constraint—applications to determination of the distance distribution function in small-angle scattering. J Appl Crystallogr 27:574–580CrossRefGoogle Scholar
  22. Tikhonov AN, Arsenin VYa (1977) Solution of ill-posed problems. Wiley, New YorkGoogle Scholar
  23. Tschoegl NW (1989) The phenomenological theory of linear viscoelastic behavior. Springer, BerlinGoogle Scholar
  24. Tschoegl NW, Emri I (1993) Generating line spectra from experimental responses. II. Storage and loss functions. Rheol Acta 32:332–327CrossRefGoogle Scholar
  25. Vestergaard B, Hansen S (2006) Application of Bayesian analysis to indirect Fourier transformation in small-angle scattering. J Appl Crystallogr 39:797–804CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of Natural Sciences, Faculty of Life SciencesUniversity of CopenhagenFrederiksberg CDenmark

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