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public 01:34:43

Tyler Bongers : Teaching Sample

  -   Other Meetings and Events ( 78 Views )

public 01:05:00
public 01:03:10

Jonathan Mattingly : Ergodicity of Stochastically Forced PDEs

  -   Other Meetings and Events ( 9 Views )

Stochastic PDEs have become important models for many phenomenon. Nonetheless, many fundamental questions about their behavior remain poorly understood. Often such SPDE contain different processes active at different scales. Not only does such structure give rise to beautiful mathematics and phenomenon, but I submit that it also contains the key to answering many seemingly unrelated questions. Questions such as ergodicity and Mixing. Given a stochastically forced dissipative PDE such as the 2D Navier Stokes equations, the Ginzburg-Landau equations, or a reaction diffusion equation; is the system Ergodic ? If so, at what rate does the system equilibrate ? Is the convergence qualitatively different at different physical scales ? Answers to these an similar questions are basic assumptions of many physical theories such as theories of turbulence. I will try both to convince you why these questions are interesting and explain how to address them. The analysis will suggest strategies to explore other properties of these SPDEs as well as numerical methods. In particular, I will show that the stochastically forced 2D Navier Stokes equations converges exponentially to a unique invariant measure. I will discuss under what minimal conditions one should expect ergodic behavior. The central ideas will be illustrated with a simple model systems. Along the way I will explain how to exploit the different scales in the problem and how to overcome the fact that the problem is an extremely degenerate diffusion on an infinite dimensional function space. The analysis points to a class of operators in between STRICTLY ELLIPTIC and HYPOELLIPTIC operators which I call EFFECTIVELY ELLIPTIC. The techniques use a representation of the process on a finite dimensional space with memory. I will also touch on a novel coupling construction used to prove exponential convergence to equilibrium.

public 01:06:54

Lev Rozansky : Beyond the Alexander polynomial

  -   Other Meetings and Events ( 9 Views )

public 56:39

Dean Oliver : Sampling the Posterior Distribution for Reservoir Properties Conditional to Production Data

  -   Other Meetings and Events ( 7 Views )

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.