American Institute of Mathematical Sciences

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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
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show all references

References:
[1]

Springer-Verlag, Berlin, 2007 doi: 10.1007/978-3-540-34514-5.  Google Scholar

[2]

Theory Probab. Appl., 46 (2002), 20-38. doi: 10.1137/S0040585X97978701.  Google Scholar

[3]

Prindle, Weber & Schmidt, 1978.  Google Scholar

[4]

Mem. Amer. Math. Soc., 1951 (1951), 12 pp.  Google Scholar

[5]

John Wiley & Sons, New York, 1953.  Google Scholar

[6]

de Gruyter Studies in Mathematics, 19, Walter de Gruyter & Co., Berlin, 1994. doi: 10.1515/9783110889741.  Google Scholar

[7]

Academic Press, New York-London, 1964.  Google Scholar

[8]

$7^{th}$ edition, Academic Press, 2007.  Google Scholar

[9]

Inverse Problems and Imaging, 4 (2009), 567-597. doi: 10.3934/ipi.2009.3.567.  Google Scholar

[10]

Kluwer Academic Publishers Group, Dordrecht, 1993.  Google Scholar

[11]

Mem. Amer. Math. Soc., 1951 (1951), 51 pp.  Google Scholar

[12]

Proc. Imp. Acad. Tokyo, 20 (1944), 519-524. doi: 10.3792/pia/1195572786.  Google Scholar

[13]

New York: Wiley, p. 11, 1962.  Google Scholar

[14]

Springer, 2005.  Google Scholar

[15]

American Mathematical Monthly, 99 (1992), 403-422. doi: 10.2307/2325085.  Google Scholar

[16]

CRC Press, Boca Raton, FL, 1996.  Google Scholar

[17]

Ph.D thesis, University of Oulu, 2002.  Google Scholar

[18]

Inverse Problems and Imaging, 6 (2012), 215-266. doi: 10.3934/ipi.2012.6.215.  Google Scholar

[19]

Inverse Problems and Imaging, 6 (2012), 267-287. doi: 10.3934/ipi.2012.6.267.  Google Scholar

[20]

Inverse Problems and Imaging, 3 (2009), 87-122. doi: 10.3934/ipi.2009.3.87.  Google Scholar

[21]

Inverse Problems, 20 (2004), 1537-1563. doi: 10.1088/0266-5611/20/5/013.  Google Scholar

[22]

Inverse Problems, 5 (1989), 599-612. doi: 10.1088/0266-5611/5/4/011.  Google Scholar

[23]

Journal of the Royal Statistical Society: Series B, 73 (2011), 423-498. doi: 10.1111/j.1467-9868.2011.00777.x.  Google Scholar

[24]

Inverse Problems and Imaging, 4 (2010), 482-503. doi: 10.3934/ipi.2010.4.485.  Google Scholar

[25]

Ann. Acad. Sci. Fenn. Math. Diss., (2005), 89 pp.  Google Scholar

[26]

Inverse Problems and Imaging, 5 (2011), 167-184. doi: 10.3934/ipi.2011.5.167.  Google Scholar

[27]

Chapman & Hall/CRC, 2005. doi: 10.1201/9780203492024.  Google Scholar

[28]

Comm. Pure Appl. Math., 22 (1969), 345-400. doi: 10.1002/cpa.3160220304.  Google Scholar

[29]

Comm. Pure Appl. Math., 22 (1969), 479-530. doi: 10.1002/cpa.3160220404.  Google Scholar

[30]

Comm. Pure Appl. Math., 24 (1971), 147-225. doi: 10.1002/cpa.3160240206.  Google Scholar

[31]

J. Comp. Phys., 217 (2006), 82-99. doi: 10.1016/j.jcp.2006.02.006.  Google Scholar

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