American Institute of Mathematical Sciences

Advanced
May  2013, 7(2): 611-647. doi: 10.3934/ipi.2013.7.611

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

Lassi Roininen 1, , Petteri Piiroinen 2, and  Markku Lehtinen 3,

1. 

University of Oulu, Sodankylä Geophysical Observatory, Tähteläntie 62, FI-99600 Sodankylä
2. 

University of Helsinki, Department of Mathematics and Statistics, Gustaf Hällströmin katu 2b, FI-00014 University of Helsinki, Finland
3. 

University of Oulu, Sodankylä Geophysical Observatory, Sodankylä

Received  November 2011 Revised  August 2012 Published  May 2013

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.
Keywords: covariance convergence, Stochastic difference equation, statistical inversion..
Mathematics Subject Classification: Primary: 65Q10, 60G10; Secondary: 42A3.
Citation: Lassi Roininen, Petteri Piiroinen, Markku Lehtinen. Constructing continuous stationary covariances as limits of the second-order stochastic difference equations. Inverse Problems & Imaging, 2013, 7 (2) : 611-647. doi: 10.3934/ipi.2013.7.611
References:
[1]

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

[2]

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

[3]

R. L. Burden, J. D. Faires and A. C. Reynolds, "Numerical Analysis,", Prindle, (1978). Google Scholar

[4]

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

[5]

J. L. Doob, "Stochastic Processes,", John Wiley & Sons, (1953). Google Scholar

[6]

M. Fukushima, Y. Ōshima and M. Takeda, "Dirichlet Forms and Symmetric Markov Processes,", de Gruyter Studies in Mathematics, (1994). doi: 10.1515/9783110889741. Google Scholar

[7]

I. M. Gel'fand and N. Ya. Vilenkin, "Generalized Functions. Vol. 4. Applications Of Harmonic Analysis,", Academic Press, (1964). Google Scholar

[8]

I. S. Gradshteyn and I. M. Ryzhik, "Table of Integrals, Series and Products,", $7^{th}$ edition, (2007). Google Scholar

[9]

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

[10]

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

[11]

K. Itō, On stochastic differential equations,, Mem. Amer. Math. Soc., 1951 (1951). Google Scholar

[12]

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

[13]

K. E. Iverson, "A Programming Language,", New York: Wiley, (1962). Google Scholar

[14]

J. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems,", Springer, (2005). Google Scholar

[15]

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

[16]

H-H. Kuo, "White Noise Distribution Theory,", CRC Press, (1996). Google Scholar

[17]

S. Lasanen, "Discretizations of Generalized Random Variables with Applications to Inverse Problems,", Ph.D thesis, (2002). Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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. doi: 10.1111/j.1467-9868.2011.00777.x. Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

show all references

References:
[1]

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

[2]

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

[3]

R. L. Burden, J. D. Faires and A. C. Reynolds, "Numerical Analysis,", Prindle, (1978). Google Scholar

[4]

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

[5]

J. L. Doob, "Stochastic Processes,", John Wiley & Sons, (1953). Google Scholar

[6]

M. Fukushima, Y. Ōshima and M. Takeda, "Dirichlet Forms and Symmetric Markov Processes,", de Gruyter Studies in Mathematics, (1994). doi: 10.1515/9783110889741. Google Scholar

[7]

I. M. Gel'fand and N. Ya. Vilenkin, "Generalized Functions. Vol. 4. Applications Of Harmonic Analysis,", Academic Press, (1964). Google Scholar

[8]

I. S. Gradshteyn and I. M. Ryzhik, "Table of Integrals, Series and Products,", $7^{th}$ edition, (2007). Google Scholar

[9]

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

[10]

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

[11]

K. Itō, On stochastic differential equations,, Mem. Amer. Math. Soc., 1951 (1951). Google Scholar

[12]

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

[13]

K. E. Iverson, "A Programming Language,", New York: Wiley, (1962). Google Scholar

[14]

J. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems,", Springer, (2005). Google Scholar

[15]

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

[16]

H-H. Kuo, "White Noise Distribution Theory,", CRC Press, (1996). Google Scholar

[17]

S. Lasanen, "Discretizations of Generalized Random Variables with Applications to Inverse Problems,", Ph.D thesis, (2002). Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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. doi: 10.1111/j.1467-9868.2011.00777.x. Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[1]

John A. D. Appleby, Xuerong Mao, Alexandra Rodkina. On stochastic stabilization of difference equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 843-857. doi: 10.3934/dcds.2006.15.843
[2]

Stefan Sommer, Anne Marie Svane. Modelling anisotropic covariance using stochastic development and sub-Riemannian frame bundle geometry. Journal of Geometric Mechanics, 2017, 9 (3) : 391-410. doi: 10.3934/jgm.2017015
[3]

Lassi Roininen, Janne M. J. Huttunen, Sari Lasanen. Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography. Inverse Problems & Imaging, 2014, 8 (2) : 561-586. doi: 10.3934/ipi.2014.8.561
[4]

Elena Braverman, Alexandra Rodkina. Stochastic difference equations with the Allee effect. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5929-5949. doi: 10.3934/dcds.2016060
[5]

Timoteo Carletti. The lagrange inversion formula on non--Archimedean fields, non--analytical form of differential and finite difference equations. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 835-858. doi: 10.3934/dcds.2003.9.835
[6]

Arnaud Debussche, Jacques Printems. Convergence of a semi-discrete scheme for the stochastic Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 761-781. doi: 10.3934/dcdsb.2006.6.761
[7]

Liu Liu. Uniform spectral convergence of the stochastic Galerkin method for the linear semiconductor Boltzmann equation with random inputs and diffusive scaling. Kinetic & Related Models, 2018, 11 (5) : 1139-1156. doi: 10.3934/krm.2018044
[8]

Esther S. Daus, Shi Jin, Liu Liu. Spectral convergence of the stochastic galerkin approximation to the boltzmann equation with multiple scales and large random perturbation in the collision kernel. Kinetic & Related Models, 2019, 12 (4) : 909-922. doi: 10.3934/krm.2019034
[9]

Jialin Hong, Lijun Miao, Liying Zhang. Convergence analysis of a symplectic semi-discretization for stochastic nls equation with quadratic potential. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4295-4315. doi: 10.3934/dcdsb.2019120
[10]

Yan Zheng, Jianhua Huang. Exponential convergence for the 3D stochastic cubic Ginzburg-Landau equation with degenerate noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5621-5632. doi: 10.3934/dcdsb.2019075
[11]

Alexandra Rodkina, Henri Schurz. On positivity and boundedness of solutions of nonlinear stochastic difference equations. Conference Publications, 2009, 2009 (Special) : 640-649. doi: 10.3934/proc.2009.2009.640
[12]

Sari Lasanen. Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns. Inverse Problems & Imaging, 2012, 6 (2) : 267-287. doi: 10.3934/ipi.2012.6.267
[13]

Anne Bronzi, Ricardo Rosa. On the convergence of statistical solutions of the 3D Navier-Stokes-$\alpha$ model as $\alpha$ vanishes. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 19-49. doi: 10.3934/dcds.2014.34.19
[14]

Esha Chatterjee, Sk. Sarif Hassan. On the asymptotic character of a generalized rational difference equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1707-1718. doi: 10.3934/dcds.2018070
[15]

Anatoli F. Ivanov, Sergei Trofimchuk. Periodic solutions and their stability of a differential-difference equation. Conference Publications, 2009, 2009 (Special) : 385-393. doi: 10.3934/proc.2009.2009.385
[16]

Seiji Ukai, Tong Yang, Huijiang Zhao. Exterior Problem of Boltzmann Equation with Temperature Difference. Communications on Pure & Applied Analysis, 2009, 8 (1) : 473-491. doi: 10.3934/cpaa.2009.8.473
[17]

Mou-Hsiung Chang, Tao Pang, Moustapha Pemy. Finite difference approximation for stochastic optimal stopping problems with delays. Journal of Industrial & Management Optimization, 2008, 4 (2) : 227-246. doi: 10.3934/jimo.2008.4.227
[18]

J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008
[19]

Hakima Bessaih, Benedetta Ferrario. Statistical properties of stochastic 2D Navier-Stokes equations from linear models. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 2927-2947. doi: 10.3934/dcdsb.2016080
[20]

Giuseppe Maria Coclite, Lorenzo di Ruvo. A note on the convergence of the solution of the Novikov equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2865-2899. doi: 10.3934/dcdsb.2018290

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences