Abstract
We consider the problem of constructing a vector-valued linear Markov process in continuous time, such that its first coordinate is in good agreement with given samples of the scalar autocorrelation function of an otherwise unknown stationary Gaussian process. This problem has intimate connections to the computation of a passive reduced model of a deterministic time-invariant linear system from given output data in the time domain. We construct the stochastic model in two steps. First, we employ the AAA algorithm to determine a rational function which interpolates the z-transform of the discrete data on the unit circle and use this function to assign the poles of the transfer function of the reduced model. Second, we choose the associated residues as the minimizers of a linear inequality constrained least squares problem which ensures the positivity of the transfer function’s real part for large frequencies. We apply this method to compute extended Markov models for stochastic processes obtained from generalized Langevin dynamics in statistical physics. Numerical examples demonstrate that the algorithm succeeds in determining passive reduced models and that the associated Markov processes provide an excellent match of the given data.
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matlab code and data for the numerical examples in Section 6 are available at https://mathematik.uni-mainz.de/hanke-aaa.
References
Anderson, B.D.O.: An algebraic solution to the spectral factorization problem. IEEE Trans. Automat. Control 12, 410–414 (1967)
Anderson, B.D.O.: The inverse problem of stationary covariance generation. J. Stat. Phys. 1, 133–147 (1969)
Anderson, B.D.O., Vongpanitlerd, S.: Network analysis and synthesis: a modern systems theory approach. Prentice-Hall, Englewood Cliffs, NJ (1973)
Bai, Z., Freund, R.W.: Eigenvalue-based characterization and test for positive realness of scalar transfer functions. IEEE Trans. Automat. Control 45, 2396–2402 (2000)
Bai, Z., Freund, R.W.: A partial Padé-via-Lanczos method for reduced-order modeling. Linear Algebra Appl. 332–334, 139–164 (2001)
Bini, D.A., Iannazzo, B., Meini, B.: Numerical solution of algebraic Riccati equations. SIAM, Philadelphia (2012)
Björck, Å.: Numerical methods for least squares problems. SIAM, Philadelphia (1996)
Bockius, N., Shea, J., Jung, G., Schmid, F., Hanke, M.: Model reduction techniques for the computation of extended Markov parameterizations for generalized Langevin equations. J. Phys.: Condens. Matter 33, 214003 (2021)
Brüll, T., Schröder, C.: Dissipativity enforcement via perturbation of para-Hermitian pencils. IEEE Trans. Circuits Syst. I. Regul. Pap. 60, 164–177 (2012)
Cherifi, K., Goyal, P., Benner, P.: A non-intrusive method to inferring linear port-Hamiltonian realizations using time-domain data. Electron. Trans. Numer. Anal. 56, 102–116 (2022)
Coelho, C.P., Phillips, J., Silveira, L.M.: Convex programming approach for generating guaranteed passive approximations to tabulated frequency-data. IEEE Trans. Computer-Aided Design 23, 293–301 (2004)
Derevianko, N., Plonka, G., Petz, M.: From ESPRIT to ESPIRA: estimation of signal parameters by iterative rational approximation. IMA J. Numer. Anal. 43, 789–827 (2023)
Fazzi, A., Guglielmi, N., Lubich, C.: Finding the nearest passive or nonpassive system via Hamitonian eigenvalue optimization. SIAM J. Matrix Anal. Appl. 42, 1553–1580 (2021)
Freund, R.W., Jarre, F., Vogelbusch, C.H.: Nonlinear semidefinite programming: sensitivity, convergence, and an application in passive reduced-order modeling. Math. Program. 109, 581–611 (2007)
Gillis, N., Sharma, P.: Finding the nearest positive-real system. SIAM J. Numer. Anal. 56, 1022–1047 (2018)
Grivet-Talocia, S.: Passivity enforcement via perturbation of Hamiltonian matrices. IEEE Trans. Circuits Syst. I. Regul. Pap. 51, 1755–1749 (2004)
Hanke, M.: Mathematical analysis of some iterative methods for the reconstruction of memory kernels. Electron. Trans. Numer. Anal. 54, 483–498 (2021)
Jung, G., Hanke, M., Schmid, F.: Iterative reconstruction of memory kernels. J. Chem. Theory Comput. 13, 2481–2488 (2017)
Jung, G., Schmid, F.: Fluctuation-dissipation relations far from equilibrium: a case study. Soft Matter 17, 6413–6425 (2021)
Kalman, R.E.: Linear stochastic filtering theory–reappraisal and outlook. In: Proceedings of the Symposium on System Theory, New York, 1965, Polytechnic Press, Brooklyn, pp. 197–205 (1965)
Lindquist, A., Picci, G.: Linear stochastic systems. Springer, Heidelberg (2015)
Nakatsukasa, Y., Sète, O., Trefethen, L.N.: The AAA algorithm for rational approximation. SIAM J. Sci. Comput. 40, A1494–A1522 (2018)
Pavliotis, G.A.: Stochastic processes and applications. Diffusion Processes, the Fokker-Planck and Langevin Equations. Springer, New York (2014)
Plonka, G., Potts, D., Steidl, G., Tasche, M.: Numerical Fourier analysis. Springer, Cham (2018)
Saraswat, D., Achar, R., Nakhla, M.: Enforcing passivity for rational function based macromodels of tabulated data. In: Electrical Performance of Electrical Packaging (IEEE Cat. No. 03TH8710), IEEE, pp. 295-298 (2003)
Schneider, C., Werner, W.: Some new aspects of rational interpolation. Math. Comp. 47, 285–299 (1986)
Varah, J.M.: On fitting exponentials by nonlinear least squares. SIAM J. Sci. Comput. 6, 30–44 (1985)
Weiss, L., McDonough, R.N.: Prony’s method, \(z\)-transforms, and Padé approximation. SIAM Rev. 5, 145–149 (1963)
Zwanzig, R.: Nonequilibrium statistical mechanics. Oxford University Press, Oxford (2001)
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The author likes to thank Peter Benner and Volker Mehrmann for valuable pointers to the literature.
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Open Access funding enabled and organized by Projekt DEAL. The research leading to this work has been done within the Collaborative Research Center TRR 146; corresponding funding by the DFG is gratefully acknowledged.
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Communicated by: Helge Holden
Dedicated to the memory of Claus Schneider.
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Hanke, M. Stochastic modeling of stationary scalar Gaussian processes in continuous time from autocorrelation data. Adv Comput Math 50, 60 (2024). https://doi.org/10.1007/s10444-024-10150-7
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DOI: https://doi.org/10.1007/s10444-024-10150-7