Hans-Werner van Wyk

Department of Scientific Computing, Florida State

Projects

My current research interests lie at the interface between statistics and applied mathematics and concern the study of uncertainty in dynamical systems governed by partial differential equations. Simulation plays an increasingly central role in modern science and engineering research, by supplementing experiments, aiding in the prototyping of engineering systems or informing decisions on safety and reliability. While physical systems involving sound, heat, electrostatics, electrodynamics, fluid flow, elasticity, or quantum mechanics are generally modeled and understood in the deterministic language of partial differential equations (PDEs), they often operate within complex environments that cannot be described- or even observed in full deterministic detail. The ready availability of modern sensor technology and computational resources, on the other hand, facilitates the detailed statistical analysis of related observable quantities, from which more realistic statistical models can potentially be derived. I am involved in a variety of projects related to both the identification and statistical characterization of parametric uncertainty in PDEs as well as the quantification of its effect on the model response. This highly interdisciplinary subject encompasses fields such as mathematical modeling, statistical data analysis, optimization theory, functional- and numerical analysis as well as scientific computing. Below, I list some of the projects I am involved in.


Multilevel Sampling Methods

Collaborators: Max Gunzburger (FSU)

Stochastic sampling methods, such as Monte Carlo sampling or sparse grid stochastic collocation, are arguably the most direct and least intrusive means of estimating statistical quantities of interest related to solutions of partial differential equations with random parameters. However, they may also carry a considerable computational cost, especially if reliable statistics are sought, requiring a potentially large number of samples computed at high levels of fidelity. Multilevel sampling methods improve upon the efficiency of traditional sampling schemes without compromising on accuracy and parallelizability, by dynamically incorporating the model's physical discretization into the sampling procedure through the use of a hierarchy of discretization models. For more information, click here.


Variational Approximation of Uncertain Parameters

Collaborators: Jeff Borggaard (Virgnia Tech)

The precise statistical characterization of the random parameters alluded to above is rarely known in practice and must often be inferred from measurements of the related model output. In van Wyk 2014 (see also van Wyk 2012 we propose a novel variational approach to the statistical identification of spatially varying, uncertain parameters in second order elliptic partial differential equations and develop an efficient augmented Lagrangian algorithm to solve the problem numerically. For more information, click here.


Scale-Invariant Noise Fields and Anomolous Diffusion

Collaborators: Max Gunzburger, John Burkardt (FSU), Miroslav Stoyanov (ORNL), Yanzhi Zhang (Missouri University of Science and Technology)

For systems governed by differential equations, random influences are commonly modeled as white noise, especially for cases in which not much is known about the precise nature of the noise. However, statistical analyses of a variety of physical-, biological-, and social phenomena reveal instead the presence of scale-invariant noise, also known as \(1/f^\alpha\) noise, due to the decay rate of the signal's power spectral density. Scale invariance has been observed in parameters relating to various growth phenomena, glacial-/as well as various industrial surfaces, micro-fracturing in rock, and subsurface flow to name but a few. Our research van Wyk 2014 addresses the generalization of scale invariant noises, commonly modeled in the time domain as fractional Brownian- or \(1/f^\alpha\) processes, to arbitrary spatial domains. 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. We were particularly interested in efficiently simulating these fields on arbitrary, non-standard meshes. For more information, click here.


Sensitivity Analysis and Uncertainty Quantification

Collaborators: Jeff Borggaard (Virginia Tech), Vitor Nunes (UT Dallas)

When the quantification of uncertainty in distributed systems is important for making design decisions and managing uncertainty under operation, one may want to address questions such as: what is the "worst" distributed perturbation of this parameter?, and how likely is that perturbation? In Borggaard 2013 , we propose a methodology based on Fréchet sensitivity analysis to interrogate stochastic descriptions of parameters. The spectral decomposition of the Fréchet operator yields a hierarchical ordering of the local parametric variations to which the model output is most sensitive. By combining a statistical description of the model with this spectral decomposition we are then able to study, as well as quantify, the interplay between parameter uncertainty and system sensitivity and hence obtain a more holistic understanding of the underlying mechanisms that cause a variation in the system response. For more information, click here.