Abstract
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.
Highlights
Gaussian random fields over functions, sometimes referred to as continuously indexed Gaussian random fields, are important in spatial statistical modeling, geostatistics and in inverse problems. They are described through a mean and a covariance operator, usually defined over a Hilbert space
PDE operators require the definition of boundary conditions, which has implications for the resulting covariance operators
The second method we propose amounts to a rescaling of the covariance operator
Summary
Gaussian random fields over functions, sometimes referred to as continuously indexed Gaussian random fields, are important in spatial statistical modeling, geostatistics and in inverse problems They are described through a mean and a covariance operator, usually defined over a Hilbert space. Constructing covariance operators from elliptic PDE operators, which has recently gained popularity [1,2,3,4,5], allows one to build on available fast PDE solvers for the required manipulations This leads to a correspondence between domain Green’s functions of PDE operators and covariance functions of the Gaussian random fields. PDE operators require the definition of boundary conditions, which has implications for the resulting covariance operators This can lead to increased/decreased correlation and pointwise variance close to the boundary, which is usually undesirable from a statistical perspective. The first method combines the PDE operator with a homogeneous Robin boundary condition βu
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
