Purpose: For potential graduate students, what are my research interests?
Short answer: Mathematical modeling and scientific computation, especially applied to real problems
Other people's problems, especially problems someone cares about
I try very hard to be more interested in solving the problems of others than my own. That may be why I am asked to serve on many committees. Applications on which I have worked include: energy forecasting, energy upgrades to historical buildings, chemical reactions, steel structures, robotics, electrocardiograms, control of an artificial heart, data compression, stock valuation, currency value at risk, oil reservoir simulation, geology of crack propagation, remote sensing of crop moisture content, bobsled runner shape, yacht sail design and manufacture, and probably several I've forgotten.
If your application is not in this list, it only means no one has proposed it to me. Yet.
Some currently active problems
How much natural gas will customers of WE Energies need each day for the next week?
Residential, industrial, and commercial customers use natural gas for space heating, cooking and water heating, as feedstock in industrial processes, as fuel for industrial and commercial operations, and hundreds of other purposes.
Theoretically, heat replaced is lost through convection and conduction, with delays, so one could do first-principles architectural and thermodynamic modeling of buildings.
Viewed as a function of temperature (adjusted for wind, humidity, and other weather conditions), consumption is a very noisy signal.
Data availability is critical for any modeling activity. GasDay has over 500,000 days worth of data on weather and consumption for operating areas across the US and over 7 M days worth of weather and consumption data for individual customers. Current models are multiple regression and neural networks, combined using ensemble forecasting techniques. Outlier detection and removal is critical. The application seems appropriate for Bayesian techniques.
GasDay licenses software to 23 local distribution companies.
Each day, we help forecast about 20% of the natural gas delivered to residential, industrial, and commercial customers in the US. We use modern software architecture and tools, including databases, multi-tiered systems, distributed computing, automated testing, and user interface design.
GasDay is an entrepreneurial technology transfer initiative housed in the College of Engineering, Director: Dr. Ron Brown. We support several graduate students, and several more participate, along with about 12-15 undergraduate students.
Students see first-hand how important it is to understand customers and their needs. They participate in business strategy decisions.
Milwaukee County Research Park
How can we improve the energy performance of historically significant (that means old) buildings?
Existing buildings have a profound impact on our natural environment, economy, health and productivity. In the United States alone, buildings account for: 72% of electricity consumption, 39% of energy use, 38% of all carbon dioxide (CO2) emissions, 40% of raw materials use, 30% of waste output (136 million tons annually), and 14% of potable water consumption. Wisconsin spent almost $21.5 billion on energy in 2007. In Milwaukee, 40% of typical building energy is for heating, 25% for lighting, and 20% for cooling. Reducing energy in commercial buildings is a high priority for local, state, and national governments.
Project Goals, Phase I:
- Work with Milwaukee County Research Park to make their Technology Innovation Center a green laboratory for existing building renovations.
- Develop an advanced monitoring system using sensors and Building Information Modeling.
- Convene an Industry Advisory Committee to explore best industry practices in sustainable construction.
This is a joint project with several faculty members from MSOE and UWM, with participation of about 12 industrial partners.
With Dr. Mark Stadtherr and his students at Notre Dame, we are exploring the behavior of certain chemical reactions modeled by differential equations with uncertain parameters.
I hold that, outside of pure mathematics, no number is known to 10 digits, and few are known to four digits. In many cases of life-critical importance, we are lucky to have the first digit right. How can that uncertainty be captured in our models and carried through out numerical computations without leading to estimates that are so pessimistic as to be worthless?
Our analysis relies on Taylor series, differential equations, numerical analysis, probability and statistics, and visualization.
Tools and techniques of mathematical modeling and scientific computation
While I like to make other people's problems go away, my contributions usually are in the application of tools from modeling and computation. I have contributed to the literature of scientific computation of solving linear and non-linear systems; local and global optimization; quadrature; curve fitting and approximation; solving initial and boundary value problems in ordinary, partial, and differential algebraic equations.
Most of those contributions involved interval analysis, automatic differentiation, or both.
Interval analysis: I prefer scientific computations that are robust/reliable/verified/guaranteed, in some sense. Most scientific computation is done with floating point numbers, using approximate algorithms. Usually, they do quite well, but once in a while, approximate algorithms can give wrong answers with serious consequences. In contrast, algorithms based on interval analysis compute with intervals that are guaranteed (up to software correctness) to enclose the correct answer to the problem posed. Interval algorithms can account for uncertain input, round-off errors during computation, and truncation errors in approximate algorithms. Interval algorithms can provide mathematically rigorous proofs that problems have an answer (or that they don't).
Automatic differentiation: Many numerical algorithms (e.g., Newton's method, optimization, stiff ODEs, sensitivity analysis) wish they had true derivatives. Conventional wisdom has held you cannot expect that, so there is a huge literature and practice of approximate, finite-difference differentiation. However, differentiation is so easy first semester calculus students can do it for simple functions. Automatic differentiation tools accept a computer program for a function f and return a computer program for requested derivative objects such as gradient, Jacobian, Hessian, or Taylor series. Automatic differentiation is semi-symbolic. It propagates values, not expressions. Its execution time is usually competitive with finite differences (but more accurate), and it is sometimes much faster (and still more accurate).
A few projects I'd love to undertake include
- Mapping locations of singularities of solutions to differential equations: A singularity (pole) in an analytic function is what determines the radius of convergence of its Taylor series. We can draw beautiful animated movies of singularity locations. If we give that vocabulary to people who understand the equations, there might be profound insights.
- Interval quadrature: I wrote what is still the world's best software for validated (interval) quadrature 25 years ago. It is way over-due for an update.
- Global optimization: I've worked for years with Baker Earshot, University of Louisiana at Lafayette, on global optimization. There remain many unanswered questions, for example, what can you do with an infinite set of "best" points?
- Fuzzy arithmetic: Fuzzy sets are an alternative to probability or intervals for expressing uncertainty. (If you are interested in more theoretical aspects of fuzzy sets, you probably want to work with Prof. Feng.)
I'd entertain projects in computer science/engineering operating systems, data bases, architecture, graphics, user interface design, data mining, networking, applications architecture, etc., but they are not my forte.
Additional philosophical observations
Writing: I hold that good English writing is very important. If you work with me, you will become a better writer, but that process will not be always "fun."
Programming: I enjoy programming (in quite a variety of languages). You should, too. Programming is not an end in itself, but if you think programming is for lower level people, you do not want to work with me.
Choose advisor: Beginning graduate students often over-emphasize the importance of the research topic. I know of no practicing scientist/engineer who has ever worked on a project where the technical part was the hard part. The hard part of every project is the people (or budget, but that's really a people problem, too). Choose and advisor you can tolerate for the duration of your study; you'll change research areas several times in your career anyway.
Colleague and friend: You may enter as a student, but when you graduate, you are supposed to be employable in a position as a colleague. Your time as a graduate student should move you from student to colleague. I do my best to treat my students as colleagues and (eventually) as friends. Evidence? I exchange Christmas cards with 40-50 former students, some from as long as 30 years ago.
Also see Why take a Corliss class?
References: References from students with whom I am working and have worked are available on request.