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Local singularity reconstruction from integrals over curves in $\mathbb{R}^3$
Constructing continuous stationary covariances as limits of the second-order stochastic difference equations
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 |
References:
[1] |
V. I. Bogachev, "Measure Theory Vol I, II,", Springer-Verlag, (2007).
doi: 10.1007/978-3-540-34514-5. |
[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. |
[3] |
R. L. Burden, J. D. Faires and A. C. Reynolds, "Numerical Analysis,", Prindle, (1978).
|
[4] |
M. D. Donsker, An invariance principle for certain probability limit theorems,, Mem. Amer. Math. Soc., 1951 (1951).
|
[5] |
J. L. Doob, "Stochastic Processes,", John Wiley & Sons, (1953).
|
[6] |
M. Fukushima, Y. Ōshima and M. Takeda, "Dirichlet Forms and Symmetric Markov Processes,", de Gruyter Studies in Mathematics, (1994).
doi: 10.1515/9783110889741. |
[7] |
I. M. Gel'fand and N. Ya. Vilenkin, "Generalized Functions. Vol. 4. Applications Of Harmonic Analysis,", Academic Press, (1964).
|
[8] |
I. S. Gradshteyn and I. M. Ryzhik, "Table of Integrals, Series and Products,", $7^{th}$ edition, (2007).
|
[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. |
[10] |
T. Hida, H-H. Kuo, J. Potthoff and L. Streit, "White Noise. An Infinite-Dimensional Calculus,", Kluwer Academic Publishers Group, (1993).
|
[11] |
K. Itō, On stochastic differential equations,, Mem. Amer. Math. Soc., 1951 (1951).
|
[12] |
K. Itō, Stochastic integral,, Proc. Imp. Acad. Tokyo, 20 (1944), 519.
doi: 10.3792/pia/1195572786. |
[13] |
K. E. Iverson, "A Programming Language,", New York: Wiley, (1962).
|
[14] |
J. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems,", Springer, (2005).
|
[15] |
D. Knuth, Two notes on notation,, American Mathematical Monthly, 99 (1992), 403.
doi: 10.2307/2325085. |
[16] |
H-H. Kuo, "White Noise Distribution Theory,", CRC Press, (1996).
|
[17] |
S. Lasanen, "Discretizations of Generalized Random Variables with Applications to Inverse Problems,", Ph.D thesis, (2002).
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[25] |
P. Piiroinen, Statistical measurements, experiments and applications,, Ann. Acad. Sci. Fenn. Math. Diss., (2005).
|
[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. |
[27] |
H. Rue and L. Held, "Gaussian Markov Random Fields: Theory and Applications,", Chapman & Hall/CRC, (2005).
doi: 10.1201/9780203492024. |
[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. |
[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. |
[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. |
[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. |
show all references
References:
[1] |
V. I. Bogachev, "Measure Theory Vol I, II,", Springer-Verlag, (2007).
doi: 10.1007/978-3-540-34514-5. |
[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. |
[3] |
R. L. Burden, J. D. Faires and A. C. Reynolds, "Numerical Analysis,", Prindle, (1978).
|
[4] |
M. D. Donsker, An invariance principle for certain probability limit theorems,, Mem. Amer. Math. Soc., 1951 (1951).
|
[5] |
J. L. Doob, "Stochastic Processes,", John Wiley & Sons, (1953).
|
[6] |
M. Fukushima, Y. Ōshima and M. Takeda, "Dirichlet Forms and Symmetric Markov Processes,", de Gruyter Studies in Mathematics, (1994).
doi: 10.1515/9783110889741. |
[7] |
I. M. Gel'fand and N. Ya. Vilenkin, "Generalized Functions. Vol. 4. Applications Of Harmonic Analysis,", Academic Press, (1964).
|
[8] |
I. S. Gradshteyn and I. M. Ryzhik, "Table of Integrals, Series and Products,", $7^{th}$ edition, (2007).
|
[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. |
[10] |
T. Hida, H-H. Kuo, J. Potthoff and L. Streit, "White Noise. An Infinite-Dimensional Calculus,", Kluwer Academic Publishers Group, (1993).
|
[11] |
K. Itō, On stochastic differential equations,, Mem. Amer. Math. Soc., 1951 (1951).
|
[12] |
K. Itō, Stochastic integral,, Proc. Imp. Acad. Tokyo, 20 (1944), 519.
doi: 10.3792/pia/1195572786. |
[13] |
K. E. Iverson, "A Programming Language,", New York: Wiley, (1962).
|
[14] |
J. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems,", Springer, (2005).
|
[15] |
D. Knuth, Two notes on notation,, American Mathematical Monthly, 99 (1992), 403.
doi: 10.2307/2325085. |
[16] |
H-H. Kuo, "White Noise Distribution Theory,", CRC Press, (1996).
|
[17] |
S. Lasanen, "Discretizations of Generalized Random Variables with Applications to Inverse Problems,", Ph.D thesis, (2002).
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[25] |
P. Piiroinen, Statistical measurements, experiments and applications,, Ann. Acad. Sci. Fenn. Math. Diss., (2005).
|
[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. |
[27] |
H. Rue and L. Held, "Gaussian Markov Random Fields: Theory and Applications,", Chapman & Hall/CRC, (2005).
doi: 10.1201/9780203492024. |
[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. |
[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. |
[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. |
[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. |
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