Brown University Center for Statistical Sciences Seminar
Abstract: Modern day approaches toward causal inference for noncompliance in placebo controled clinical trials draw heavily on missing data theory. Indeed, besides the observed outcome on experimental treatment one explicityly considers the potential latent outcome on placebo.
When structural (mean) models hold for the effect of actually received treatment, randomization based asymptotically unbiased estimators for the treatment effect can be found without further assumptions on the compliance selection mechanism. In a second stage one can then also estimate the mean selection effect. As with other occurrences of missing data, typically very little is known a priori about the missingness mechanism and assumptions about this mechanism are thus hard to justify.
Molenberghs, Kenward and Goetghebeur (1999) proposed a general approach towards sensitivity analysis when categorical data suffer from unintentionally missing observations. Rather than imposing untestable assumptions on the missing data mechanism, it is acknowledged that even with infinite sample size there will remain a level of ignorance on certain parameters. Hence, for a parameter set of interest a region of ignorance becomes the focus of inference. In a data set with finite sample size, imprecision is accompanying the estimated region of ignorance and the recognition of both sources of uncertainty leads to the construction of (l-alpha) 100% regions of uncertainty.
Using this approach, we reanalyze a placebo controlled trial on cholesterol reduction, previously analyzed by Efron and Feldman (1991) and Goetghebeur and Molenberghs (1996) and investigate to what extent previous conclusions were driven by the assumptions rather than the data...
Brown Univeristy Graduate School, Dissertation Defense Information
Special PDE Seminar
Abstract: We consider here the problem of deriving rigorously, globally in time and for general initial conditions, fluid mechanics equations such as Navier-Stokes, Euler or Stokes equations from the Boltzmann's equations. Our results may be viewed as extensions of the important series of works by C. Bardos, F. Golse and D. Levermore. The methods used here are very much related to those used for the study of low Much number limits (i.e., the convergence of solutions of compressible, isentropic, Navier-Stokes equations to those of incompressible equations.
Brown Applied Mathematics Pattern Theory and Vision Seminar
Abstract: We describe some exciting mathematical advances in our work as part of a collective effort to construct a probabilistic atlas of the human brain. Extreme variations in brain structure in human populations complicate the design of statistical approaches (1) to automatically identify structures in brain images; (2) to detect abnormal brain structure in an individual or clinical group; (3) to relate these abnormalities to genetic, clinical and demographic factors; and (4) to compare the dynamics of 4-dimensional growth or degenerative processes in the human brain. Probabilistic brain atlases address these challenges by encoding variations in brain structure and function in large human populations. We describe our development of atlases that synthesize data across age, gender, time, and multiple brain imaging modalities, to represent normal populations and diseased subpopulations with Alzheimer's Disease and schizophrenia. To identify disease-specific patterns of brain structure and function, we combine mathematical approaches from Riemannian geometry, the theory of Gaussian random fields, and covariant partial differential equations, with supercomputing algorithms for image warping and analysis. Current challenges will be described that will be of interest to applied mathematicians and statisticians in general, as well as researchers in the image analysis and brain mapping fields.
Brown Analysis Seminar
Applied Mathematics Colloquium
Mathematics Department, Distinguished Lecture Series
Scientific Computing Seminar
Abstract: Time-domain modeling of acoustic or electromagnetic scattering in free space requires a reduction of the infinite physical domain to a finite computational domain. The exact nonreflecting boundary conditions - well-known to be nonlocal in both space and time - can be expressed as a convolution of the solution at the boundary from the time of quiescence to the present. The kernel of this intergral operator, which depends on the computational domain, is presented for several boundary geometries. An efficient implementation of this formulation is discussed.
We also present a new time-symmetric evolution formula for the scalar wave equation. It is simply related to the classical D'Alembert or spherical means representations, and can be used to develop stable, robust numerical schemes on irregular meshes.
PDE Seminar
Mathematics Department, Distinguished Lecture Series
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