Variational methods for moving gridsVariational methods for moving gridsThe numerical solution of differential equations on fixed grids is a highly developed branch of numerical analysis. However there are still many problems which cause difficulties owing to problems in resolving the solution. In this talk we discuss the possibilities of using variational methods in one or more dimensions to adapt or move the grid. If a variational principle is available for the problem it can also be used to determine the grid while otherwise a least squares formulation can be employed. A motivation for the latter is given in terms of approximate equidistribution.
The methods described incorporate iterative algorithms possessing a monotonicity property and emphasise the solution of local problems.
Properly picked projections (and their inverses)(Joint work with Michael Kirby, Colorado State University)
Imagine a numerical model having a priori many degrees of freedom, but which---in some parameter regime of interest---is known to have a low-dimensional attractor. Are there techniques which enable us to extract an equivalent low-dimensional model from such system?
This talk will describe an approach to this problem which is based on picking projections of the high-dimensional system which are easy to invert. The talk will cover the basic theory of approach, give an algorithm for finding the projections, discuss the construction of the inverse projection and illustrate the whole with some numerical examples.
Mathematics in the petroleum industryWe present an overview of some of the uses of mathematics, particularly computational mathematics, in the oil industry. We look at well test analysis, in which mathematical models of gas flow in reservoirs are fitted to pressure transient measurements to derive information about the physical characteristics of the reservoir. We also examine flow in pipe networks used to gather oil and gas from wells, and consider how this can be optimized for the best financial return.
MultiMATLAB: MATLAB on multiple processorsMATLAB is a high-level language, and a problem-solving environment, for mathematical and scientific calculations. In many of the scientific and engineering communities it has become the dominant language for desktop numerical computing. MultiMATLAB is a system that enables one to run MATLAB conveniently on multiple processors, high-performance multiprocessors or networks of workstations.
We will describe the system architecture and show applications that have been developed using MultiMATLAB.
Random Fibonacci series and the number 1.13198824...(Joint work with Divakar Viswanath)
The usual Fibonacci series grows asymptotically at the rate 1.61803398..., but what if instead of plus signs at each step, we add up the terms using plus or minus signs chosen at random? Numerical experiments suggest that exponential growth still occurs at a rate about 1.13. My student Divakar Viswanath has been able to prove that this is indeed the case and that the asymptotic growth constant, with probability 1, is C = 1.13198824....
This seemingly specialized problem has surprisingly rich links with numerous subjects including Markov processes, fractal geometry, dynamical systems, Kleinian groups, continued fractions, and random matrices. In particular, C is obtained from an integral of a smooth function against a fractal measure obtained from an infinite Fibonacci or Stern-Brocot tree. We do not know an analytic expression for C.