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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
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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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