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Multiparameter regularization for Volterra kernel identification via multiscale collocation methods

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Abstract

Identification of the Volterra system is an ill-posed problem. We propose a regularization method for solving this ill-posed problem via a multiscale collocation method with multiple regularization parameters corresponding to the multiple scales. Many highly nonlinear problems such as flight data analysis demand identifying the system of a high order. This task requires huge computational costs due to processing a dense matrix of a large order. To overcome this difficulty a compression strategy is introduced to approximate the full matrix resulted in collocation of the Volterra kernel by an appropriate sparse matrix. A numerical quadrature strategy is designed to efficiently compute the entries of the compressed matrix. Finally, numerical results of three simulation experiments are presented to demonstrate the accuracy and efficiency of the proposed method.

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References

  1. Aiordachioaiea, D., Ceangaa, E., Keyserb, R., Naka, Y.: Detection and classification of non-linearities based on Volterra kernels processing. Eng. Appl. Artif. Intell. 14, 497–503 (2001)

    Article  Google Scholar 

  2. Atkinson, K., Graham, I., Sloan, I.: Piecewise continuous collocation for integral equations. SIAM J. Numer. Anal. 20, 172–186 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  3. Belge, M., Kilmer, M.E., Miller, E.L.: Efficient determination of multiple regularization parameters in a generalized L-curve framework. Inverse Problems 18, 1161–1183 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Brenner, M., Xu, Y.: A factorization method for identification of Volterra systems. J. Comput. Appl. Math. 144, 105–117 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Brezinski, C., Redivo-Zaglia, M., Rodriguez, G., Seatzu, S.: Multi-parameter regularization techniques for ill-conditioned linear system. Numer. Math. 94, 203–228 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Differential Equations. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  7. Ciarlet, P.: Introduction to Linear Algebra and Optimization. Cambridge University Press, Cambridge (1989)

    Google Scholar 

  8. Chen, L., Narendra, K.: Identification and control of a nonlinear discrete-time system based on its linearization: a unified framework. IEEE Trans. Neural Netw. 15, 663–673 (2004)

    Article  Google Scholar 

  9. Chen, Z., Lu, Y., Xu, Y., Yang, H.: Multi-parameter Tikhonov regularization for ill-posed operator equations. J. Comput. Math. 26, 37–55 (2008)

    MATH  MathSciNet  Google Scholar 

  10. Chen, Z., Micchelli, C., Xu, Y.: A construction of interpolating wavelets on invariant sets. Comput. Math. 68, 1560–1587 (1999)

    MathSciNet  Google Scholar 

  11. Chen, Z., Micchelli, C., Xu, Y.: Fast collocation methods for second kind integral equations. SIAM J. Numer. Anal. 40, 344–375 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  12. Chen, Z., Wu, B., Xu, Y.: Multilevel augmentation methods for solving operator equations. Numer. Math. J. Chinese Univ. 14, 31–55 (2005)

    MATH  MathSciNet  Google Scholar 

  13. Chen, Z., Wu, B., Xu, Y.: Fast numerical collocation solutions of integral equations. Comm. Pure Appl. Anal. 6, 643–666 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Chen, Z., Wu, B., Xu, Y.: Fast collocation methods for high dimensional weakly singular integral equations. J. Integral Equations Appl. 20, 49–92 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Chen, Z., Xu, Y., Yang, H.: A multilevel augmentation method for solving ill-posed operator equations. Inverse Problems 22, 155–174 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Chon, K., Chen, Y., Holstein-Rathlou, N., Marmarelis, V.: Nonlinear system analysis of renal autoregulation in normotensive and hypertensive rats. IEEE Trans. Biomed. Eng. 45, 342–353 (1998)

    Article  Google Scholar 

  17. Doicu, A., Schreier, F., Hilgers, S., Hess, M.: Multi-parameter regularization method for atmospheric remote sensing. Comput. Phys. Comm. 165, 1–9 (2005)

    Article  Google Scholar 

  18. Doyle, F., Ogunnaike, B., Pearson, R.: Nonlinear model control using second-order volterra models. Automatica 31, 697–714 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  19. Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996)

    MATH  Google Scholar 

  20. French, A., Sekizawa, S., Hoger, U., Torkkeli, P.: Predicting the responses of mechanoreceptor neurons to physiological inputs by nonlinear system identification. Ann. Biomed. Eng. 29, 187–194 (2001)

    Article  Google Scholar 

  21. Ford, W.F., Xu, Y., Zhao, Y.: Derivative correction for quadrature formulas. Adv. Comput. Math. 6, 139–157 (1996)

    Article  MathSciNet  Google Scholar 

  22. Groetsch, C.W.: The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Research Notes in Mathematics, 105. Pitman, Boston (1984)

    Google Scholar 

  23. Glentis, G., Koukoulas, P., Kalouptsidis, N.: Efficient algorithms for volterra system identification. IEEE Trans. Signal Process. 47, 3042–3057 (1999)

    Article  MATH  Google Scholar 

  24. Khouaja, A., Kibangou, A., Favier, G.: Third-order Volterra kernels complexity reduction using PARAFAC. In: Proc. 1st IEEE International Symposium on Control, Communications and Signal Processing (ISCCSP 04), pp. 857–860. IEEE, Piscataway (2004)

    Chapter  Google Scholar 

  25. Kaneko, H., Xu, Y.: Gauss-type quadratures for weakly singular integrals and their application to Fredholm integral equations of the second kind. Math. Comp. 62, 739–753 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  26. Lu, Y., Shen, L., Xu, Y.: Shadow block iteration for solving linear systems obtained from wavelet transforms. Appl. Comput. Harmon. Anal. 19, 359–385 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  27. Lu, Y., Shen, L., Xu, Y.: Multi-parameter regularization methods for high-resolution image reconstruction with displacement errors. IEEE Trans. Circuits Systems I 54, 1788–1799 (2007)

    Article  MathSciNet  Google Scholar 

  28. Maner, B., Doyle, F., Ogunnaike, B., Pearson, R.: Nonlinear model predictvie control of a simulated multivariable polymerization reactor using second-order volterra models. Automatica 32, 1285–1301 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  29. Marmarelis, V., Juusola, M., French, A.: Principal dynamic mode analysis of nonlinear transduction in a spider mechanoreceptor. Ann. Biomed. Eng. 27, 391–402 (1999)

    Article  Google Scholar 

  30. Micchelli, C.A., Sauer, T., Xu, Y.: A construction of refinable sets for interpolating wavelets. Results Math. 34, 359–372 (1998)

    MATH  MathSciNet  Google Scholar 

  31. Micchelli, C.A., Xu, Y.: Using the matrix refinement equation for the construction of wavelets on invariant sets. Appl. Comput. Harmon. Anal. 1, 391–401 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  32. Micchelli, C.A., Xu, Y.: Reconstruction and decomposition algorithms for biorthogonal wavelets. Multidimens. Systems Signal Process. 8, 31–69 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  33. Micchelli, C.A., Xu, Y., Zhao, Y.: Wavelet Galerkin methods for second-kind integral equations. J. Comput. Appl. Math. 86, 251–270 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  34. Nemeth, J., Kollar, I., Schoukens, J.: Identification of volterra kernels using interpolation. IEEE Trans. Instrum. Meas. 51, 770–775 (2002)

    Article  Google Scholar 

  35. Parsa, V., Parker, P., Scott, R.: Adaptive stimulus artifact reduction in noncortical somatosensory evoked potential studies. IEEE Trans. Biomed. Eng. 45, 165–179 (1998)

    Article  Google Scholar 

  36. Phillips, D.L.: A technique for the numerical solution of certain integral equations of the first kind. J. Assoc. Comput. Mach. 9, 84–97 (1962)

    MATH  MathSciNet  Google Scholar 

  37. Prazenica, R., Kurdila, A.: Volterra kernel identification using triangular wavelet. J. Vib. Control 10, 597–622 (2004)

    Article  MathSciNet  Google Scholar 

  38. Richter, G.R.: Supercovergence of piecewise polynomial galerkin approximations, for Fredholm integral equation of the second kind. Numer. Math. 31, 63–70 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  39. Renardy, M., Rogers, R.C.: An Introduction to Partial Differential Equations, 2nd edn. Springer, New York (2004)

    MATH  Google Scholar 

  40. Schwab, C.: Variable order composite quadrature of singular and nearly singular integrals. Computing 53, 173–194 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  41. Strauch, P., Mulgrew, B.: Active control of nonlinear noise processes in a linear duct. IEEE Trans. Signal Process. 46, 2404–2412 (1998)

    Article  Google Scholar 

  42. Tikhonov, A.N.: On the solution of ill posed problems and the method of regularization (Russian). Dokl. Akad. Nauk SSSR 151, 501–504 (1963)

    MathSciNet  Google Scholar 

  43. Tikhonov, A.N., Arsenin, V.Y.: Solutions of Ill-posed Problems. Wiley, New York (1977)

    MATH  Google Scholar 

  44. Taylor, A.E., Lay, D.C.: Intruduction to Functional Analysis, 2nd edn. Wiley, New York (1980)

    Google Scholar 

  45. Tsoulkas, V., Koukoulas, P., Kalouptsidis, N.: Identification of input-output bilinear systems using cumulants. IEEE Trans. Signal Process. 49, 2753–2761 (2001)

    Article  Google Scholar 

  46. Wang, Y., Xu, Y.: A fast wavelet collocation method for integral equations on polygons. J. Integral Equations Appl. 17, 227–330 (2005)

    Article  Google Scholar 

  47. Xu, Y., Chen, H.L., Zou, Q.: Limit values of derivatives of the Cauchy integrals and computation of the logarithmic potential. Computing 73, 295–237 (2004)

    Article  MathSciNet  Google Scholar 

  48. Zhang, Q., Suki, B., Westwick, D., Lutchen, K.: Factors affecting Volterra kernel estimation: emphasis on lung tissue viscoelasticity. Ann. Biomed. Eng. 26, 103–116 (1998)

    Article  Google Scholar 

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Authors and Affiliations

  1. NASA Dryden Flight Research Center, MS 4830D/RS, Edwards, CA, 93523-0273, USA

    Marty Brenner

  2. Department of Mathematics, Syracuse University, Syracuse, NY, 13244-1150, USA

    Ying Jiang & Yuesheng Xu

Authors
  1. Marty Brenner
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  2. Ying Jiang
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  3. Yuesheng Xu
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Corresponding author

Correspondence to Yuesheng Xu.

Additional information

Communicated by Juan Manuel Peña.

Supported in part by the US National Aeronautics and Space Administration under Cooperative Agreement NNX07AC37A, by the US National Science Foundation under grants CCR-0407476 and DMS-0712827, by the Natural Science Foundation of China under grants 10371122 and 10631080, and by the Education Ministry of the People’s Republic of China under the Changjiang Scholar Chair Professorship Program through Sun Yat-sen University.

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Brenner, M., Jiang, Y. & Xu, Y. Multiparameter regularization for Volterra kernel identification via multiscale collocation methods. Adv Comput Math 31, 421 (2009). https://doi.org/10.1007/s10444-008-9077-4

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  • DOI: https://doi.org/10.1007/s10444-008-9077-4

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