Possible Ph.D. Thesis Research

My research is centered on addressing mathematical issues that arise in the practical application of differential equation models to the study of the physical environment.

Differential equations are widely used in science and engineering to describe complicated physical situations in mathematical terms. The motivation is to understand the underlying processes. Simple observation does not often reveal the "why" and "how" of processes. Along with observation, we need to analyze, and this is where mathematics can help. A valid mathematical description can explain the relative importance of different components of a physical system, can indicate what happens when parts of the system are changed, and can lead to predictions about what a system will do. Derivatives crop up in mathematical models because it is often the case that it easier to measure a change in quantity as opposed to an value on an absolute scale, and derivatives are computed by taking limits of ratios of differences in measured quantities.

While it is true that differential equations are used to help in understanding complicated phenomena, it is also true that such models are themselves very complicated. In fact, such models are not determined completely or are not complete descriptions, depend on parameters that need to be determined from experiment and therefore subject to uncertainty, and cannot be solved exactly and approximate solutions have to be computed using the computer. All of these issues give rise to hard mathematical problems and provide a critical need for the involvement of mathematicians.

My work has been motivated by seeking answers to two fundamental scientific questions:

• What is the uncertainty in information computed from an approximate solution of a differential equation? This includes not only the effects of discretization, but error arising from uncertainty in the model, data, and parameters. We pursue deterministic and statistical approaches to quantify uncertainty.
• How should we compute specified information from a solution of a differential equation? This involves understanding what information is required in an application and whether or not a particular model can yield that information.
  

My approach to these questions is driven entirely by the needs of application. My research involves development of new numerical techniques, new and interesting mathematical analysis, computational investigations of specific problems, and software development. I publish in mathematics, engineering, and science journals, and likewise my texts are used both by mathematicians and engineers. I work with mathematicians specializing in the theory of partial differential equations, engineers and scientists on practical applications, and computer scientists on algorithmic development.

I advise Ph.D. thesis projects on both mathematical and computational aspects of numerical solution of differential equations. In general, however, all of my Ph.D. students can expect to have significant amounts of analysis and computation in their theses.

Depending on the project, my students can have the opportunity to work with ecologists, engineers and scientists at national laboratories such as Idaho National Laboratory, Livermore National Laboratory and Sandia National Laboratory, or with my colleagues in Itally, Sweden, and the Netherlands.

General research areas for which there are good t hesis projects include:

• Computational error estimation for numerical solutions of multi-physics reacting flows: Reacting flows have both the physics of fluids and of chemical reactions. One way such problems are solved are to use method specialized to each kind of physics, which are highly evolved and efficient, and then to patch together the two solutions to solve the overall problem. This raises all kinds of interesting mathematical issues in stability and estimating the error of the overall process.
• Data assimilation/data-model fusion/inverse problems. Data assimilation is the problem of incorporating real field/experimental data into a differential equation in order to make accurate predictions about the behavior of a physical system. The standard approach treat sthis like as a least-squares-like problem for fitting the parameters in a differential equation to produce the closest fit to values measured in field experiments. This involves the numerical solution of complicated multi-physics, multi-scale differential equations as well as nonlinear optimization techniques. We are developing a new approach that constrains the fit process in such a way that insures the model has the same stability and sensitivity properties as the physical system
• Solution of reaction-diffusion equations on domains with uncertain or changing geometry. Reaction-diffusion equations have long been used to model the formation of patterns in nature, such as found in animal coloration. But, realistic models should allow for the domains of the model to grow, e.g., the way an animal grows as it becomes older. This possibility opens up numerous interesting mathematical and numerical issues
• A posteriori analysis of numerical methods for partial differential equations with constraints. These kinds of problems arise in geometric PDEs and models of black holes. The solutions of PDEs are constrained to lie on a specified manifold determined by, e.g., an algebraic constraint. Numerical approximations typically do not satisfy the constraint precisely, and we seek estimates to determine the effects of this inaccuracy on the computed information.
• A new approach to domain decomposition. Recently, we devised a new approach to the decomposition of the solution of a differential equation based on using the Green's function. This approach could be used as an alternative to the standard domain decomposition technique, which is based on a decomposition of the spatial domain. But, there is a lot of interesting mathematics to be done in order to develop the technique.
• A posteriori detection of stochasticity in a differential equation. Differential equation models of complicated physical systems, such as ecological systems like the Carbon Cycle in the terrestrial system, often have the interesting property that they mix stochastic and deterministic components. Parts of such models are determined precisely by balance and conservation principles. But parts of the models depend on quantities that cannot be precisely determined by field measurements, and any measurements are subject to error, and may include components for which no model is known. Since the kinds of information that can be computed from solutions of deterministic and stochastic models are vastly different, this raises a natural question. Given a model, can we estimate the effect of the stochastic elements and determine, in some sense, the degree of stochasticity of the model, and hence say what kinds of information can be computed accurately from the model.
• A posteriori analysis of stochastic and deterministic coupling methods. The close coupling of stochastic and deterministic continuum models has emerged as a critical component in several fields of engineering and science, ranging from fusion to computational materials science. However, there remain significant mathematical issues regarding the understanding of how to solve such problems, such as how information is exchanged between the stochastic and deterministic models; how are errors propagated, the derivation of error estimates, and the identification of numerical artifacts; the phenomenological compatibility of the mathematics developed; and assessing the effects of uncertainty. In this project, we are developing a general mathematical framework that can provide a unified theoretical foundation for the formulation, analysis, and implementation of stochastic/deterministic coupling methods based on an operator based mathematical formalism. The operator characterization of the coupling problem allows us to draw upon powerful results, tools, and ideas from approximation theory, probability, functional analysis, calculus of variations, and numerical analysis.
• Continuum modeling of extremely large networks of simple receiver/transmitters. The goal of simulating large networks is to determine rough characteristics (throughput) that can help guide the design and implementation. SImulation of large network behavior is notoriously difficult however. We are attempting to replace direct simulation by numerical solution of continuum models that determine the characteristic behavior directly. This is joint work with statisticans and engineers at CSU.
• Various topics in the fundamental foundations of computational error estimation and adaptive error control. I always have several projects in this area.