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October 2018, 12(5): 1083-1102. doi: 10.3934/ipi.2018045

## Mitigating the influence of the boundary on PDE-based covariance operators

 Courant Institute, New York University, 251 Mercer street, New York, NY 10012, USA

* Corresponding author: Yair Daon

Received  October 2016 Revised  May 2018 Published  July 2018

Fund Project: Supported in part by the National Science Foundation under grants #1507009 and #1522736, and by the U.S. Department of Energy Office of Science, Advanced Scientific Computing Research (ASCR), Scientific Discovery through Advanced Computing (SciDAC) program

Gaussian random fields over infinite-dimensional Hilbert spaces require the definition of appropriate covariance operators. The use of elliptic PDE operators to construct covariance operators allows to build on fast PDE solvers for manipulations with the resulting covariance and precision operators. However, PDE operators require a choice of boundary conditions, and this choice can have a strong and usually undesired influence on the Gaussian random field. We propose two techniques that allow to ameliorate these boundary effects for large-scale problems. The first approach combines the elliptic PDE operator with a Robin boundary condition, where a varying Robin coefficient is computed from an optimization problem. The second approach normalizes the pointwise variance by rescaling the covariance operator. These approaches can be used individually or can be combined. We study properties of these approaches, and discuss their computational complexity. The performance of our approaches is studied for random fields defined over simple and complex two- and three-dimensional domains.

Citation: Yair Daon, Georg Stadler. Mitigating the influence of the boundary on PDE-based covariance operators. Inverse Problems & Imaging, 2018, 12 (5) : 1083-1102. doi: 10.3934/ipi.2018045
##### References:
 [1] C. Bekas, A. Curioni and I. Fedulova, Low cost high performance uncertainty quantification in Proceedings of the 2nd Workshop on High Performance Computational Finance, WHPCF '09, ACM, New York, NY, USA, 2009, Article No. 8. doi: 10.1145/1645413.1645421. [2] C. Bekas, E. Kokiopoulou and Y. Saad, An estimator for the diagonal of a matrix, Applied Numerical Mathematics, 57 (2007), 1214-1229. doi: 10.1016/j.apnum.2007.01.003. [3] J. Besag, On a system of two-dimensional recurrence equations, Journal of the Royal Statistical Society. Series B (Methodological), 43 (1981), 302-309. [4] T. Bui-Thanh, O. Ghattas, J. Martin and G. Stadler, A computational framework for infinite-dimensional Bayesian inverse problems Part Ⅰ: The linearized case, with application to global seismic inversion, SIAM Journal on Scientific Computing, 35 (2013), A2494-A2523. doi: 10.1137/12089586X. [5] D. Calvetti, J. P. Kaipio and E. Somersalo, Aristotelian prior boundary conditions, International Journal of Mathematics and Computer Science, 1 (2006), 63-81. [6] G. Da Prato, An Introduction to Infinite-dimensional Analysis, Universitext, Springer, 2006. doi: 10.1007/3-540-29021-4. [7] NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1.0.10 of 2015-08-07, Online companion to [16]. [8] L. C. Evans, Partial Differential Equations, 2nd edition, Graduate studies in mathematics, American Mathematical Society, 2010, URL http://books.google.com/books?id=Xnu0o_EJrCQC. doi: 10.1090/gsm/019. [9] M. Hairer, Introduction to Stochastic PDEs, Lecture Notes, 2009. [10] T. Isaac, N. Petra, G. Stadler and O. Ghattas, Scalable and efficient algorithms for the propagation of uncertainty from data through inference to prediction for large-scale problems, with application to flow of the Antarctic ice sheet, Journal of Computational Physics, 296 (2015), 348-368. doi: 10.1016/j.jcp.2015.04.047. [11] S. G. Johnson, Cubature—Adaptive Multi-dimension Integration, http://ab-initio.mit.edu/wiki/index.php/Cubature. [12] Lin, Lu, Ying, Car and W. E, Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems, Communications in Mathematical Sciences, 7 (2009), 755-777. doi: 10.4310/CMS.2009.v7.n3.a12. [13] F. Lindgren, H. Rue and J. Lindström, An explicit link between Gaussian fields and Gaussian Markov random fields: The stochastic partial differential equation approach, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73 (2011), 423-498, URL http://dx.doi.org/10.1111/j.1467-9868.2011.00777.x. doi: 10.1111/j.1467-9868.2011.00777.x. [14] A. Logg, K.-A. Mardal and G. N. Wells (eds.), Automated Solution of Differential Equations by the Finite Element Method, vol. 84 of Lecture Notes in Computational Science and Engineering, Springer, 2012. doi: 10.1007/978-3-642-23099-8. [15] B. Øksendal, Stochastic Differential Equations, Springer, 2003. doi: 10.1007/978-3-642-14394-6. [16] F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, New York, NY, 2010, Print companion to [7]. [17] L. Roininen, J. M. J. Huttunen and S. Lasanen, Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography, Inverse Problems Imaging, 8 (2014), 561-586. doi: 10.3934/ipi.2014.8.561. [18] H. Rue and S. Martino, Approximate Bayesian inference for hierarchical Gaussian Markov random field models, Journal of Statistical Planning and Inference, 137 (2007), 3177-3192. doi: 10.1016/j.jspi.2006.07.016. [19] D. Simpson, F. Lindgren and H. Rue, In order to make spatial statistics computationally feasible, we need to forget about the covariance function, Environmetrics, 23 (2012), 65-74. doi: 10.1002/env.1137. [20] D. Simpson, F. Lindgren and H. Rue, Think continuous: Markovian Gaussian models in spatial statistics, Spatial Statistics, 1 (2012), 16-29. doi: 10.1016/j.spasta.2012.02.003. [21] A. Singer, Z. Schuss, A. Osipov and D. Holcman, Partially reflected diffusion, SIAM Journal on Applied Mathematics, 68 (2008), 844-868. doi: 10.1137/060663258. [22] A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica, 19 (2010), 451-559. doi: 10.1017/S0962492910000061. [23] J. M. Tang and Y. Saad, A probing method for computing the diagonal of a matrix inverse, Numerical Linear Algebra with Applications, 19 (2012), 485-501. doi: 10.1002/nla.779. [24] S. R. Varadhan, Probability Theory, Courant Lecture Notes in Mathematics, 7. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2001. doi: 10.1090/cln/007. [25] P. Whittle, Stochastic-processes in several dimensions, Bulletin of the International Statistical Institute, 40 (1963), 974-994.

show all references

##### References:
 [1] C. Bekas, A. Curioni and I. Fedulova, Low cost high performance uncertainty quantification in Proceedings of the 2nd Workshop on High Performance Computational Finance, WHPCF '09, ACM, New York, NY, USA, 2009, Article No. 8. doi: 10.1145/1645413.1645421. [2] C. Bekas, E. Kokiopoulou and Y. Saad, An estimator for the diagonal of a matrix, Applied Numerical Mathematics, 57 (2007), 1214-1229. doi: 10.1016/j.apnum.2007.01.003. [3] J. Besag, On a system of two-dimensional recurrence equations, Journal of the Royal Statistical Society. Series B (Methodological), 43 (1981), 302-309. [4] T. Bui-Thanh, O. Ghattas, J. Martin and G. Stadler, A computational framework for infinite-dimensional Bayesian inverse problems Part Ⅰ: The linearized case, with application to global seismic inversion, SIAM Journal on Scientific Computing, 35 (2013), A2494-A2523. doi: 10.1137/12089586X. [5] D. Calvetti, J. P. Kaipio and E. Somersalo, Aristotelian prior boundary conditions, International Journal of Mathematics and Computer Science, 1 (2006), 63-81. [6] G. Da Prato, An Introduction to Infinite-dimensional Analysis, Universitext, Springer, 2006. doi: 10.1007/3-540-29021-4. [7] NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1.0.10 of 2015-08-07, Online companion to [16]. [8] L. C. Evans, Partial Differential Equations, 2nd edition, Graduate studies in mathematics, American Mathematical Society, 2010, URL http://books.google.com/books?id=Xnu0o_EJrCQC. doi: 10.1090/gsm/019. [9] M. Hairer, Introduction to Stochastic PDEs, Lecture Notes, 2009. [10] T. Isaac, N. Petra, G. Stadler and O. Ghattas, Scalable and efficient algorithms for the propagation of uncertainty from data through inference to prediction for large-scale problems, with application to flow of the Antarctic ice sheet, Journal of Computational Physics, 296 (2015), 348-368. doi: 10.1016/j.jcp.2015.04.047. [11] S. G. Johnson, Cubature—Adaptive Multi-dimension Integration, http://ab-initio.mit.edu/wiki/index.php/Cubature. [12] Lin, Lu, Ying, Car and W. E, Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems, Communications in Mathematical Sciences, 7 (2009), 755-777. doi: 10.4310/CMS.2009.v7.n3.a12. [13] F. Lindgren, H. Rue and J. Lindström, An explicit link between Gaussian fields and Gaussian Markov random fields: The stochastic partial differential equation approach, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73 (2011), 423-498, URL http://dx.doi.org/10.1111/j.1467-9868.2011.00777.x. doi: 10.1111/j.1467-9868.2011.00777.x. [14] A. Logg, K.-A. Mardal and G. N. Wells (eds.), Automated Solution of Differential Equations by the Finite Element Method, vol. 84 of Lecture Notes in Computational Science and Engineering, Springer, 2012. doi: 10.1007/978-3-642-23099-8. [15] B. Øksendal, Stochastic Differential Equations, Springer, 2003. doi: 10.1007/978-3-642-14394-6. [16] F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, New York, NY, 2010, Print companion to [7]. [17] L. Roininen, J. M. J. Huttunen and S. Lasanen, Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography, Inverse Problems Imaging, 8 (2014), 561-586. doi: 10.3934/ipi.2014.8.561. [18] H. Rue and S. Martino, Approximate Bayesian inference for hierarchical Gaussian Markov random field models, Journal of Statistical Planning and Inference, 137 (2007), 3177-3192. doi: 10.1016/j.jspi.2006.07.016. [19] D. Simpson, F. Lindgren and H. Rue, In order to make spatial statistics computationally feasible, we need to forget about the covariance function, Environmetrics, 23 (2012), 65-74. doi: 10.1002/env.1137. [20] D. Simpson, F. Lindgren and H. Rue, Think continuous: Markovian Gaussian models in spatial statistics, Spatial Statistics, 1 (2012), 16-29. doi: 10.1016/j.spasta.2012.02.003. [21] A. Singer, Z. Schuss, A. Osipov and D. Holcman, Partially reflected diffusion, SIAM Journal on Applied Mathematics, 68 (2008), 844-868. doi: 10.1137/060663258. [22] A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica, 19 (2010), 451-559. doi: 10.1017/S0962492910000061. [23] J. M. Tang and Y. Saad, A probing method for computing the diagonal of a matrix inverse, Numerical Linear Algebra with Applications, 19 (2012), 485-501. doi: 10.1002/nla.779. [24] S. R. Varadhan, Probability Theory, Courant Lecture Notes in Mathematics, 7. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2001. doi: 10.1090/cln/007. [25] P. Whittle, Stochastic-processes in several dimensions, Bulletin of the International Statistical Institute, 40 (1963), 974-994.
Left: Cross sections through covariance functions induced by elliptic PDE operators with different boundary conditions. Shown is also a sketch of the domain $\Omega = [0, 1]^2$ and the cross section $\boldsymbol{x} (s) = (s, 0.5)^T$. The center is located at $\boldsymbol{x} ^\star = \boldsymbol{x} (0.05) = (0.05, 0.5)^T$. Right: Two covariance functions on the Antarctica domain (see Sec. 6.2). The magnitude of the left covariance function exceeds the gray scale used to show the covariance between the centers and the points of the domain. The discrepancy between the covariance is due to the use of Neumann boundary conditions for the differential operator
Optimal Robin boundary coefficients $\beta$ for an edge of a square using $\mathcal{A} = -\Delta + 121$ (a), (c) and a line on a face of a cube using $\mathcal{A} = -\Delta + 25$ (b), (d). Shown are coefficients computed by adaptive quadrature, and their discrete approximations on regular meshes obtained by dividing $n^2$ squares into $4n^2$ triangles in two dimensions, and $n^3$ cubes into $6n^3$ tetrahedra in three dimensions. The approximations are either based on approximate $L_2$-projections followed by finite element quadrature (a), (b) or on direct finite element quadrature (c), (d) as discussed in section 4.3
The left plot shows covariance functions derived from PDE operators with different boundary conditions for the parallelogram domain example (section 6.1). Shown are slices of the Green's function along a cross section. The right plot shows part of the parallelogram domain $\Omega$. The black dot is $\boldsymbol{x} ^{\star} = (0.025, 0.025)^T$—the center of the Green's functions. The red line indicates the cross section $\boldsymbol{x} (s) = (s, 0.6s + 0.01 )$, which is used in the left plot
Green's functions for the Antarctica domain detailed in section 6.2. Results for optimal Robin boundary conditions combined with variance normalization are shown in (a). These results should be compared with figure 1, which uses homogeneous Neumann boundary conditions. Magnifications are shown for Neumann conditions with normalized variance (b), varying Robin boundary condition from section 4 (c), Robin condition with constant coefficient taken from [17] (d), and Neumann boundary condition (e)
Pointwise standard deviation fields for Antarctica with different boundary conditions for the underlying PDE operator: Dirichlet conditions (a), Neumann conditions (b), Robin conditions with constant coefficient following [17] (c), and Robin conditions with varying coefficient computed as in section 4.2 (d)
Two-dimensional slices through Green's functions for the unit cube example from section 6.3. The center of the green's function is located at $\boldsymbol{x} ^{\star} = (0.05, 0.5, 0.5)^{T}$, and the slice shown is $\{ \boldsymbol{x} ^{\star} + (s, 0, 0)^{T} + (0, t, 0)^{T}, s, t \in \mathbb{R} \} \cap [0, 1]^3$. Shown are the free-space Green's function (a), the Green's function computed with Neumann boundary with normalized variance (b), with Robin boundary conditions with variable coefficient $\beta$ (c), and with Robin boundary conditions with variable coefficient and normalized variance (d)
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