Brown University Center for Statistical Sciences Seminar
Abstract: We have been quantitatively investigating whether and how various health-related phenomena might spread from person to person. For example, we explored the nature and extent of person-to-person spread of obesity. We developed a densely interconnected network of 12,067 people assessed repeatedly from 1971 to 2003. We used longitudinal statistical models and network-scientific methods to examine whether weight gain in one person was associated with weight gain in friends, siblings, spouses, and neighbors. Discernible clusters of obese persons were present in the network at all time points, and the clusters extended three people deep. These clusters were not solely due to selective formation of social ties. A friend becoming obese in a given time interval increased a person's chances of becoming obese by 57% (95% CI: 6%-123%). Among pairs of adult siblings, one becoming obese increased the chance that the other became obese by 40% (21%-60%). Among spouses, one becoming obese increased the likelihood that the other became obese by 37% (7%-73%). Immediate neighbors did not exhibit these effects. We have also conducted similar investigations of other health behaviors, such as smoking, drinking, exercising, and screening, and of other health phenomena, such as depression. Various aspects of our findings suggest that the spread of social norms may partly underlie inter-personal health effects. Our findings have implications for clinical and public health interventions, and for cost-effectiveness assessments of preventive and therapeutic interventions. They also lay a new foundation for public health by providing a rationale for the claim that health is not just an individual, but also a collective, phenomenon.
Probability Seminar
Abstract: Freidlin-Wentzell theory, one of the classical areas in large deviations, deals with path probability asymptotics for small noise stochastic dynamical systems. For finite dimensional stochastic differential equations (SDE) there has been an extensive study of this problem. In this work we are interested in infinite dimensional models, i.e. the setting where the driving Brownian motion is infinite dimensional. In recent years there has been lot of work on the study of large deviations principle (LDP) for small noise infinite dimensional SDEs, much of which is based on the ideas of Azencott (1980). A key in this approach is obtaining suitable exponential tightness and continuity estimates for certain approximations of the stochastic processes. This becomes particularly hard in infinite dimensional setting where such estimates are needed with metrics on exotic function spaces (e.g. Hölder spaces, spaces of diffeomorphisms etc). Our approach to the large deviation analysis is quite different and is based on certain variational representation for infinite dimensional Brownian motions. It bypasses all discretizations and finite dimensional approximations and thus no exponential probability estimates are needed. Proofs of LDP are reduced to demonstrating basic qualitative properties (existence, uniqueness and tightness) of certain perturbations of the original process. The approach has now been adopted by several authors in recent works to study various infinite dimensional models such as stochastic Navier-Stokes equations, stochastic flows of diffeomorphisms, SPDEs with random boundary conditions. As a first example of this approach, we consider a class of stochastic reaction-diffusion equations, which have been studied by various authors. We establish a large deviation principle under conditions that are substantially weaker than those available in the literature. We next study a family of stochastic flows of diffeomorphisms that arise in certain image analysis problems. Large deviations for the case where the driving noise is finite dimensional has been studied by Ben Arous and Castell (1995). We extend these results to an infinite dimensional setting and apply them to a problem of image analysis.
Scientific Computing Seminar
Abstract: There are many physical phenomena develop singular, or nearly singular, behavior in localized regions, e.g., shock waves, boundary layers or blowup solutions. Using uniform grids for these problems becomes computationally prohibitive as the solution approaches singularity. We develop a semi-static adaptive grid method for the solution of these problems. This is the first method of its class that allows control of the distribution of grid points both in the `singular' and outer regions. In addition, this method is more efficient and more robust than other adaptive methods. We demonstrate the efficiency of this method with a numerical example of a solution localized by more than 10 orders of magnitude. Joint work with Adi Ditkowski
PDE Seminar
Abstract: The study of singular solutions of the NLS goes back to the 1960s, with applications in nonlinear optics and, more recently, in BEC. Until recently, the only known singular solutions had a self-similar "peak-type" profile that approaches a delta function near the singularity. In this talk I will present new families of singular solutions of the NLS that collapse with a self-similar ring profile, and whose blowup rate is different from the one of the "old" singular solutions. I will also show, both theoretically and experimentally, that these new blowup profiles are attractors for large super-Gaussian initial conditions. This is joint work with Gadi Fibich and Xiao-Ping Wang
<--- 2008 Index