We describe a general Bayesian approach for using computer codes for a complex physical system to assist in forecasting actual system outcomes. Our approach is based on expert judgements and experiments on fast versions of the computer code. These are combined to construct models for the relationships between the code's inputs and outputs, respecting the natural space/time features of the physical system. The resulting beliefs are systematically updated as we make evaluations of the code for varying input sets and calibrate the input space against past data on the system. The updated beliefs are then used to construct forecasts for future system outcomes. While the approach is quite general, it has been developed particularly to handle problems with high-dimensional input and output spaces, for which each run of the computer code is expensive. The methodology will be applied to problems in uncertainty analysis for hydrocarbon reservoirs.

This lecture, which assumes only a basic abstract algebra background, provides an introduction to the Galois theory of linear ordinary differential equations. At present, no algorithm exists for computing the Galois group of an arbitrary equation. We will give a survey of recent work on some special classes of equations, including results to be included in the speaker's PhD dissertation.

The p-power torsion of a family of elliptic curves in characteristic p defines a local system on the base space. Igusa shows that the associated monodromy representation is maximal, in the sense that the image of the fundamental group is the full automorphism group of a fiber. I will discuss variants of this situation which can make the group in question bigger (higher-dimensional abelian varieties), smaller (with large endomorphism rings), or simply different (and no physical p-torsion). I hope to interpret these results as statements about certain differential equations.

The combinatorial structure of a d-dimensional simple convex polytope -- as given, for example, by the set of the (d-1)-regular subgraphs of facets -- can be reconstructed from its abstract graph [Blind & Mani 1988, Kalai 1988]. However, no polynomial/efficient algorithm is known for this task, although a polynomially checkable certificate for the correct reconstruction exists [Kaibel & Koerner 2000]. A much stronger certificate would be given by the following characterization of the facet subgraphs, conjectured by M. Perles: ``The facet subgraphs of a simple d-polytope are exactly all the (d-1)-regular, connected, induced, non-separating subgraphs'' [Perles 1970,1984]. We give examples for the validity of Perles conjecture: In particular, it holds for the duals of cyclic polytopes, and for the duals of stacked polytopes. On the other hand, we observe that for any counterexample, the boundary of the (simplicial) dual polytope P^* contains a 2-complex without a free edge, and without 2-dimensional homology. Examples of such complexes are known; we use a simple modification of ``Bing's house'' (two walls removed) to construct explicit 4-dimensional counterexamples to Perles' conjecture.

Associated with a dynamical system, which evolves single initial conditions, the Frobenius-Perron operator evolves ensemble densities of initial conditions. Including a brief tutorial on the topic, we will present our new applications of this global and statistical point of view:

1. The inverse Frobenius-Perron problem (IFPP) is a global open-loop strategy to control chaos by constructing a "nearby" dynamical system with desirable invariant density. We reduce the question of stabilizing an arbitrary invariant density to the question of a hyperplane intersecting a unit hyperbox; several controllability theorems follow. Applications will be described.
2. Well-known models have been found to exhibit new and interesting dynamics under the addition of stochastic perturbations. Using the Frobenius-Perron operator for stochastic dynamical systems, we develop new tools designed to predict the effects of noise and to pinpoint stochastic transport regions in phase space. As an example, we study a model from population dynamics for which chaos-like behavior can be induced, as the standard deviation of the noise is increased. We identify how stochastic perturbations destabilize two attracting orbits, effectively completing a heteroclinic orbit, to create chaos-like behavior. Other physical applications will also be discussed.

I will explain some basic concepts for fluid flow in porous media, go through some details about the 3D streamline simulator and SGSIM (Sequential Gaussian SIMulation) for generating random permeability fields under multiGaussian assumption.

A major problem of Petroleum engineering si the prediction of future oil and water production from a reservoir whose properties are inferred from measurements along well paths, and from observations of pressure, production, and fluid saturations at well locations. If the properties of the porous material were known at all locations, and all boundary conditions were specified, the production rates of fluids would be computed from the numerical solution of a set of partial differential equations governing mass conservation and flow. Rock properties are known to be heterogeneous on many scales, however, and the measurements are always insufficient to determine the properties throughout the reservoir. In the petroleum and groundwater fields, rock properties (permeability and porosity) are modeled as spatial random fields, whose auto-covariance and cross-covariances are known from ovservations of outcrops and cores. Uncertainty in future production is characterized by the empirical distribution from the suite of realizations of rock properties. The problem is assessing uncertainty in reservoir production or groundwater remediation predictions is that while valid prodecures for sampling the posterior pdf are available, the computational cost of generating the necessary number of samples from such procedures is prohibitive. An increase in computer speed is unlikely to solve this problem as the trend has been to build more complex numerical models of the reservoir as computer capability increases. Most recent effort has gone in to approximate methods of sampling. In this talk, I will describe our experience with the use of Markov Chain Monte Carlo methods and with approximate sampling methods.

The combinatorial laplacian is a method for computing the rational homology of an algebraic or simplicial complex. Although the method is quite old, it has recently been applied with success to a number of problems in algebraic combinatorics. We will survey the method and describe some of these recent applications.