Brown University Center for Statistical Sciences Seminar
Candidate for Assistant Professor, Department of Community Health | |
Abstract: In biomedical studies, survival time may be censored by many events. Some censoring events, such as patient voluntary withdrawal or changing treatment, may be related to potential failure. However, due to the identifiability problem, these events are often assumed to be independent of failure, even though this false assumption can lead to biased inference. It is also common that subjects in the study are naturally clustered, for example, they are patients from multiple medical centers, or they are family members, or they are litter mates. Subjects in the same cluster share some common factors. Therefore, their survival outcomes are likely to be correlated, instead of independent of each other. In my talk, I will consider these two problems (dependent censoring and correlated data) simultaneously. I developed a test to check if dependent censoring is present. I also developed a model to analyze correlated survival data with dependent censoring. The EM algorithm is used to fit the model. In the E-steps, Markov Chain Monte Carlo method is used to evaluate integrals without closed form. Simulation studies and analysis of a real data set of kidney disease patients are provided.
Special Brown University, Division of Applied Mathematics Seminar
Abstract: 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 and 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 system.
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 of 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.
Special Brown Analysis Seminar
Applied Mathematics Colloquium
Abstract: One of the most commonly used boundary conditions, when modelling the relation between a voltage potential and the corresponding current at a corroding surface, is the (simplified) Butler-Volmer condition
$$ \frac{\partial u}{\partial n}= \lambda(e^{u/2}- e^{-u/2}$$
In conjunction with the equation $\delta u = 0 $ this boundary condition leads to a surprisingly rich solution structure for positive values of $\lambda$. I shall carefully analyse the qualitative as well as the quantitative behaviour of solutions, with special emphasis on multiplicity and the development of singularities (blow up). Numerical computations have played a significant role in this work, both as a means to develop intuition and (in one case) as a means to derive rather surprising explicit formulas for the solutions. I shall present some of these numerical results, and time permitting, I shall also discuss the general case
$$\frac{\partial u}{\partial n}= \lambda f(u} $$.
Scientific Computing Seminar
Abstract: Finite element method is widely used for solving different kind of partial differential equations. The obtained finite element solution has superconvergence at some special points for the solution or the gradient if the mesh has special structures. By doing a proper post-processing, supertconvergence can also be obtained.
For practical engineering problems, the mesh may be generated by different kind of requirements and the conditions needed for the superconvergence are often not easy to be satisfied. We shall propose a gradient recovery technique and analyze it for finite element solutions which provides new gradient approximations with high order of accuracy on nearly arbitray meshes. The recovery technique is based on the method of least-squares surface fitting in a finite dimensional space corresponding to a coarse mesh. It is proved that the recovered gradient has a high order of superconvergence for appropriately-chosen surface fitting spaces. The recovery technique is robust, efficient, and applicable to a wide class of problems such as the Stokes and elasticity equations.
Center for Statistical Sciences Seminar
Department of Community Health, Purdue University, Department of Statistics | |
Abstract In order to understand the role of microorganisms in an environment, the identification and characterization of the relevant microbial community is necessary. Characteristic profiles of microbial communities are obtained by denaturing gradient gel electrophorei (DGGE) of polymerase chain reaction (PCR) amplified 16S rDNA from soil extracted DNA. These characteristic profiles, commonly called community DNA fingerprints, can be represented in the form of high-dimensional binary vectors. The corresponding statistical problem of modeling and variable selection for high-dimensional multivariate binary data can be addressed from both a frequentist and a Bayesian perspective. Permutation-based approaches are employed to select variables which vary significantly with respect to a treatment effect and the properties of these methods are explored via simulation. Bayesian methods for model selection were also employed using an Emperical Bayes model for multivariate binary response data, but these results will only be discussed as time permits. in conclusion, an application of the proposed methodology is presented in the context of a controlled agricultural experiment.
Brown University Mathematics Colloquium
PDE Seminar
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