Center for Statistical Sciences Seminar
Abstract: Dynamic treatment regimes (aka adaptive treatment strategies) are individually tailored treatments, with treatment type and/or dosage changing according to patient’s intermediate response at different stages of therapy. Availability of multiple treatment options at each stage and possibility of variable intermediate responses to these treatments may result in many different dynamic treatment regimes. Sequential multiple assignment randomization trials (SMARTs) are often used to study different dynamic treatment regimes in the treatment of cancer, leukemia, depression, and AIDS. Frequently the goal is to compare multiple treatment regimes based on time-to-event outcome such as overall survival. In this talk we will discuss some parametric and semi-parametric methods to compare two-stage dynamic treatment regimes based on time-to-event data collected from SMART design. Application of these methods will be demonstrated by applying to a leukemia dataset.
Lefschetz Center for Dynamical Systems Seminar
Abstract: We study the problem of finding the instability index of certain non-selfadjoint fourth order differential operators that appear as linearizations of coating and rimming flows, where a thin layer of fluid coats a horizontal rotating cylinder. The main result reduces the computation of the instability index to a finite-dimensional space of trigonometric polynomials. The proof uses Lyapunov's method to associate the differential operator with a quadratic form, whose maximal positive subspace has dimension equal to the instability index. The quadratic form is given by a solution of Lyapunov's equation, which here takes the form of a fourth order linear PDE in two variables. Elliptic estimates for the solution of this PDE play a key role.
Probability Seminar
Abstract: Consider the so-called Lauricella problem with an integer $m\geq 1$ $$ \begin{array}{lcll} (\frac{1}{2}\Delta+V)^m u &=& 0,&\quad x\in D,\smallskip\\ (\frac{1}{2}\Delta+V)^{k} u&=&(-1)^{k}g_{k},&\quad x\in\partial D. \end{array} \eqno (1) $$ where $k=0,\dots, m-1$, $(\frac{1}{2}\Delta+V)^{0}:=id$, $D\subset {\bf R}^d$ is a bounded domain with a regular boundary $\partial D$, $V\in C(\overline{D})$ and $g_k\in C(\partial {D})$. Let $\lambda_1=\lambda_1(D,V)$ denote the principle Dirichlet eigenvalue of $-\frac{1}{2}\Delta-V$ in $D$. Under condition $\lambda_1>0$ we prove existence, uniqueness and different bounds for solution of problem (1). The main tool is a probabilistic representation of the solution in terms of iterated stochastic integral.
Brown Applied Mathematics Pattern Theory and Vision Seminar
Abstract: In this talk we discuss a computational framework for learning complex high dimensional data. The proposed methods rely on estimating regularized version of data-dependent kernel matrices, where regularization is achieved filtering out the unstable components corresponding to small eigenvalues. In many cases these matrices can be interpreted as empirical versions of underlying covariance matrices, integral operators or closely related objects, such as diffusion operators. We will describe different strategies to implement regularization and discuss the corresponding computational properties. The sample complexity of the methods is analyzed and shown to be optimal for a large class of problems. Experimental results in the context of supervised learning classification and vector fields estimation will be presented.
Boston University/Brown University PDE Seminar
Abstract: Our ability to perceive and respond to transient external stimuli arises from the brain's ability to generate sustained, spatially localized activity in response to transient inputs. Mathematical models designed to represent this process in networks of neurons must provide a mechanism for recruitment of neurons into an active group as well as a mechanism to limit the spread of activity and hence maintain its localization. Analysis of such models often focuses on how different localization features, such as long- range inhibition, contribute to the existence and stability of sustained, localized activity patterns. In this talk, I will discuss work on the recruitment of neuronal oscillators, in contexts where desynchronization of inputs or competition complicate outcomes. I will consider discrete models, most of which are not specific to neurons, as well as a continuum limit. This work involves a geometric perspective, and the talk assumes no background knowledge of neuroscience.
Department of Mathematics Colloquium
Boston University/Brown University PDE Seminar
Abstract: KdV equation is a standard model of weakly nonlinear long waves on the surface of shallow water. It will be shown that in KdV with periodic boundary conditions, high frequency solutions evolve almost as the linear ones for large time. For KdV (or some other dispersive equations) on the real line such behavior could be expected due to the dispersive decay. While on the circle (i.e. periodic boundary conditions) such dispersive decay is not possible, the dispersion manifests itself in averaging out nonlinearity over high frequency solutions. This result is obtained by the application of normal form transformations in the appropriate spaces. The integrability properties of KdV are not used, so similar results could be obtained for other KdV like equations. The interaction of these high frequency solutions with a cnoidal wave will be discussed, too. This work has been motivated by an attempt to explain some phenomena in nonlinear optics and fluid dynamics. This is a joint work with M.B. Erdogan and N. Tzirakis (also University of Illinois).
Applied Mathematics Colloquium (Turkey Talk)
Abstract: Can the Physical/Mathematical Notions of Entropy be Usefully Imported into the Social Sphere ?
Scientific Computing Seminar
Abstract: We consider the finite element method for stochastic partial differential equations with white noise perturbations in the forcing terms. Our main interest is the error estimates of the finite element approximations. The main difficulty for the error estimates is the lack of regularity of the exact solutions. We first discretize the white noise based on the partition of the domain used for the construction of finite element subspaces. Then we construction the finite element solutions for the stochastic PDEs with discretized white noise perturbations and derive the error estimates. Both semi-linear elliptic SPDEs and Stochastic Stokes equations will be discussed. Some numerical experiments will also be presented.
Scientific Computing Seminar
Abstract: Coupled flow problems which depend on an accurate mass balance are presented, namely the coupling of free and porous media flow and the mixing layer problem in oceanography. An inf-sup stable, mimetic finite element method of arbitrary order for these problems is introduced and analyzed. Finally, application of these methods to parameter estimation in hurricane models is discussed.
PDE Seminar
Applied Mathematics / Center for Statistical Sciences Seminar
Abstract: Multiple hypothesis testing is challenging when signals in data are weak, the distributions of noise and signals are not completely known, or there is statistical dependency in data. I will present some theoretical results on these issues. I will show that self-normalized large deviations can be useful in dealing with experiment designs for weak signals, linear programming can be used to evaluate p-values when data distributions are not completely known, and techniques similar to nonlinear filtering can be used to analyze the performance of multiple testing on hidden states of Markov models that may be nonstationary.