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

Yair Daon, Georg Stadler

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)1083-1102
Number of pages20
JournalInverse Problems and Imaging
Volume12
Issue number5
DOIs
StatePublished - Jan 1 2018

Fingerprint

Covariance Operator
Mathematical operators
Elliptic PDE
Gaussian Random Field
Operator
Boundary conditions
Hilbert spaces
Boundary Effect
Robin Boundary Conditions
Normalize
Computational complexity
Rescaling
Large-scale Problems
Random Field
Manipulation
Computational Complexity
Hilbert space
Optimization Problem
Three-dimensional
Influence

Keywords

  • Bayesian statistics
  • Boundary conditions
  • Fast PDE solvers
  • Gaussian random fields
  • Inverse problems
  • Matérn kernels

ASJC Scopus subject areas

  • Analysis
  • Modeling and Simulation
  • Discrete Mathematics and Combinatorics
  • Control and Optimization

Cite this

Mitigating the influence of the boundary on PDE-based covariance operators. / Daon, Yair; Stadler, Georg.

In: Inverse Problems and Imaging, Vol. 12, No. 5, 01.01.2018, p. 1083-1102.

Research output: Contribution to journalArticle

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