Scale-Invariant Noise Fields

Collaborators: Max Gunzburger, John Burkardt (Florida State University), Miroslav Stoyanov (Oak Ridge National Laboratory), Yanzhi Zhang (Missouri University of Science and Technology)

To be realistic, statistical models of parameters should ideally not only be consistent with available measurements, but also incorporate broader, more qualitative information. Scale-invariance is one such property, relating to the frequency content of the random signal. It has been observed widely both in natural time series, as well as spatially varying random fields glaciers, subsurface flow parameters, micro-fracturing in rock, and various growth phenomena. Over the past two decades, a concerted effort was made to extend both the analysis and simulation of solutions of stochastic partial differential equations (SPDEs) to include more realistic statistical models for time-varying parameters that exhibit scale-invariance, such as fractional Brownian motion or \(1/f^\alpha\) processes. Apart from fractional Brownian Surfaces, defined over simple rectangular regions, scale-invariant random parameters over spatial domains have received comparatively little attention. Our research in \cite{vanwyk2014pln} addresses the generalization of scale invariant noises to arbitrary spatial domains, and their numerical synthesis over arbitrary spatial meshes.

By invoking the theory of elliptic Gaussian fields, we model power-law noise fields as solutions to the fractional Poisson problem with a white noise forcing term, conferring on them certain desirable properties, such as Hölder continuity and rotational invariance, while still incorporating the geometry of the underlying region. We were particularly interested in efficiently simulating these fields on arbitrary, non-standard meshes, and therefore chose to discretize the system using finite element methods. The fractional Laplacian is a non-local operator that has been widely studied in fields such as physics, finance, and hydrology, where it is associated with models of anomalous diffusion. In principle, the construction of fractional powers of the Laplacian is based on functional calculus, requiring its complete spectral decomposition. However, more efficient methods exist, that involve estimating a contour integral over the operators' spectrum, using numerical quadrature. My interest in this operator has also lead to collaboration with Yanzhi Zhang (Missouri University of Science and Technology), with whom I am currently working on the analysis of finite difference methods related to the fractional Schrödinger equations.

The theory of Gaussian models for spatially varying power-law noises, also known as elliptic Gaussian processes, or Riesz fields, is fairly recent and there is a need for a more complete understanding of the nature of solutions of SDE’s and SPDE’s and their approximations, when the underlying parameters are spatially varying power-law noises. This includes a convergence theory for approximations based on finite elements, or finite differences. Another direction of future research involves the quantification of more specific features of the field, such as the presence of a grain, as well as the incorporation of these into the random field model.