Advanced Search
Article Contents
Article Contents

Constructing continuous stationary covariances as limits of the second-order stochastic difference equations

Abstract Related Papers Cited by
  • In Bayesian statistical inverse problems the a priori probability distributions are often given as stochastic difference equations. We derive a certain class of stochastic partial difference equations by starting from second-order stochastic partial differential equations in one and two dimensions. We discuss discretisation schemes on uniform lattices of these stationary continuous-time stochastic processes and convergence of the discrete-time processes to the continuous-time processes. A special emphasis is given to an analytical calculation of the covariance kernels of the processes. We find a representation for the covariance kernels in a simple parametric form with controllable parameters: correlation length and variance. In the discrete-time processes the discretisation step is also given as a parameter. Therefore, the discrete-time covariances can be considered as discretisation-invariant. In the two-dimensional cases we find rotation-invariant and anisotropic representations of the difference equations and the corresponding continuous-time covariance kernels.
    Mathematics Subject Classification: Primary: 65Q10, 60G10; Secondary: 42A38.


    \begin{equation} \\ \end{equation}
  • [1]

    V. I. Bogachev, "Measure Theory Vol I, II," Springer-Verlag, Berlin, 2007doi: 10.1007/978-3-540-34514-5.


    V. I. Bogachev and A. V. Kolesnikov, Open mappings of probability measures and Skorokhod's representation theorem, Theory Probab. Appl., 46 (2002), 20-38.doi: 10.1137/S0040585X97978701.


    R. L. Burden, J. D. Faires and A. C. Reynolds, "Numerical Analysis," Prindle, Weber & Schmidt, 1978.


    M. D. Donsker, An invariance principle for certain probability limit theorems, Mem. Amer. Math. Soc., 1951 (1951), 12 pp.


    J. L. Doob, "Stochastic Processes," John Wiley & Sons, New York, 1953.


    M. Fukushima, Y. Ōshima and M. Takeda, "Dirichlet Forms and Symmetric Markov Processes," de Gruyter Studies in Mathematics, 19, Walter de Gruyter & Co., Berlin, 1994.doi: 10.1515/9783110889741.


    I. M. Gel'fand and N. Ya. Vilenkin, "Generalized Functions. Vol. 4. Applications Of Harmonic Analysis," Academic Press, New York-London, 1964.


    I. S. Gradshteyn and I. M. Ryzhik, "Table of Integrals, Series and Products," $7^{th}$ edition, Academic Press, 2007.


    T. Helin, On infinite-dimensional hierarchical probability models in statistical inverse problems, Inverse Problems and Imaging, 4 (2009), 567-597.doi: 10.3934/ipi.2009.3.567.


    T. Hida, H-H. Kuo, J. Potthoff and L. Streit, "White Noise. An Infinite-Dimensional Calculus," Kluwer Academic Publishers Group, Dordrecht, 1993.


    K. Itō, On stochastic differential equations, Mem. Amer. Math. Soc., 1951 (1951), 51 pp.


    K. Itō, Stochastic integral, Proc. Imp. Acad. Tokyo, 20 (1944), 519-524.doi: 10.3792/pia/1195572786.


    K. E. Iverson, "A Programming Language," New York: Wiley, p. 11, 1962.


    J. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems," Springer, 2005.


    D. Knuth, Two notes on notation, American Mathematical Monthly, 99 (1992), 403-422.doi: 10.2307/2325085.


    H-H. Kuo, "White Noise Distribution Theory," CRC Press, Boca Raton, FL, 1996.


    S. Lasanen, "Discretizations of Generalized Random Variables with Applications to Inverse Problems," Ph.D thesis, University of Oulu, 2002.


    S. Lasanen, Non-Gaussian statistical inverse problems. Part I: Posterior distributions, Inverse Problems and Imaging, 6 (2012), 215-266.doi: 10.3934/ipi.2012.6.215.


    S. Lasanen, Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns, Inverse Problems and Imaging, 6 (2012), 267-287.doi: 10.3934/ipi.2012.6.267.


    M. Lassas, E. Saksman and S. Siltanen, Discretization invariant Bayesian inversion and Besov space priors, Inverse Problems and Imaging, 3 (2009), 87-122.doi: 10.3934/ipi.2009.3.87.


    M. Lassas and S. Siltanen, Can one use total variation prior for edge-preserving Bayesian inversion?, Inverse Problems, 20 (2004), 1537-1563.doi: 10.1088/0266-5611/20/5/013.


    M. S. Lehtinen, L. Päivärinta and E. Somersalo, Linear inverse problems for generalised random variables, Inverse Problems, 5 (1989), 599-612.doi: 10.1088/0266-5611/5/4/011.


    F. Lindgren, H. Rue and J. Lindström, An explicit link between Gaussian Markov random fields: The stochastic partial differential equation approach, Journal of the Royal Statistical Society: Series B, 73 (2011), 423-498.doi: 10.1111/j.1467-9868.2011.00777.x.


    M. Orispää and M. Lehtinen, Fortran linear inverse problem solver, Inverse Problems and Imaging, 4 (2010), 482-503.doi: 10.3934/ipi.2010.4.485.


    P. Piiroinen, Statistical measurements, experiments and applications, Ann. Acad. Sci. Fenn. Math. Diss., (2005), 89 pp.


    L. Roininen, M. Lehtinen, S. Lasanen, M. Orispää and M. Markkanen, Correlation priors, Inverse Problems and Imaging, 5 (2011), 167-184.doi: 10.3934/ipi.2011.5.167.


    H. Rue and L. Held, "Gaussian Markov Random Fields: Theory and Applications," Chapman & Hall/CRC, 2005.doi: 10.1201/9780203492024.


    D. W. Stroock and S. R. S. Varadhan, Diffusion processes with continuous coefficients. I, Comm. Pure Appl. Math., 22 (1969), 345-400.doi: 10.1002/cpa.3160220304.


    D. W. Stroock and S. R. S. Varadhan, Diffusion processes with continuous coefficients. II, Comm. Pure Appl. Math., 22 (1969), 479-530.doi: 10.1002/cpa.3160220404.


    D. W. Stroock and S. R. S. Varadhan, Diffusion processes with boundary conditions, Comm. Pure Appl. Math., 24 (1971), 147-225.doi: 10.1002/cpa.3160240206.


    C. H. Su and D. Lucor, Covariance kernel representations of multidimensional second-order stochastic processes, J. Comp. Phys., 217 (2006), 82-99.doi: 10.1016/j.jcp.2006.02.006.

  • 加载中

Article Metrics

HTML views() PDF downloads(79) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint