Center for Statistical Sciences Seminar
In this talk, we consider fitting marginal additive hazards regression models for case-cohort studies with multiple disease outcomes. Most modern analyses of survival data focus on multiplicative models for relative risk using proportional hazards models. However, in many biomedical studies, the proportional hazards assumption might not hold or the investigators are often interested in risk differences. The additive hazards model, which model the risk differences, has often been suggested as an alternative to the proportional hazards model.
We consider a weighted estimating equation approach for the estimation of model parameters. The asymptotic properties of the proposed estimators are derived and their finite sample properties are assessed via simulation studies. The proposed method is applied to the Atherosclerosis Risk in Communities (ARIC) Study for illustration.
Center for Fluid Mechanics, Division of Applied Mathematics Fluids, Thermal and Chemical Processes Group, School of Engineering Joint Seminar Series
Abstract: Droplets impacting on surfaces are common in our everyday life from raindrops splashing on car windows to inkjets printing on paper. Surprisingly, however, many aspects of the dynamics of droplet impact are not well understood; even the initial impact of the drop remains controversial. Here I discuss a new experimental approach and directly measure the interface between the drop and the surface. I show that the drop initially skates along a very thin film of air before contacting the surface, consistent with recent predictions. Moreover, the dynamics of the drop impact are governed by the ultimate rupture of this thin film of air, which itself exhibits a fascinating range of instabilities; these dynamics can be directly observed by refining these new experimental techniques. In this talk I describe these experiments and report the results.
Brown Applied Mathematics Pattern Theory and Vision Seminar
A large number of natural phenomena can be formulated on analytic manifolds. More specifically in computer vision, such underlying notions emerge in multi-factor analysis including feature selection, pose estimation, structure from motion, appearance tracking, and shape embedding. Unlike Euclidean spaces, analytic manifolds does not exhibit global homeomorphism, thus, differential geometry is applicable only locally. This prevents application of conventional inference and learning methods, which require vector norms.
Recently we introduced appearance based object descriptors and motion transformations that exhibit Riemannian manifold structure on positive definite matrices that enable projections of the original problems onto tangent spaces. In this manner, we do not need to flatten the underlying manifold or discover its topology. This talk will demonstrate entertaining results of manifold learning on human detection, regression tracking, unusual event analysis, and affine pose estimation.
Boston/Brown PDE Seminar
Abstract: Ravi Srinivasan and I found a Lax pair that described a kinetic theory for shock statistics in scalar conservation laws with random initial data. More recent work shows that the kinetic equations are completely integrable. Precisely: they have a Hamiltonian structure, can be linearized via a matrix factorization, and solved by an inverse scattering theory.
Boston/Brown PDE Seminar
Abstract: In systems that posses stable limit cycles and a property known as ``shear,'' it is possible to apply time-periodic forcing and create a strange attractor, on which the dynamics can be characterized by an SRB measure. The resulting chaos is sustained (ie is not transient) and is physically observable. This has been previously analyzed in finite dimensions, and in infinite dimensions for limit cycles that are created by Hopf bifurcations. We show how to extend this to more general limit cycles in infinite dimensions. This is join work with Will Ott.
Scientific Computing Group Seminar
Abstract: Some problems in Solid and Structural mechanics require special care when analyzed via finite element approximations. Typical examples are problems that involve kinematic constraints, such as in incompressible elasticity or Reissner-Mindlin plate models, and problems involving moving boundaries, such as evolving cracks, phase transition interfaces or shape optimizations. Nonlinearities in the material behavior only exacerbate these difficulties, and immediately rule out many of the proposed solutions. In this talk I will show how under these circumstances Discontinuous Galerkin methods provide an attractive and advantageous alternative. The overarching idea I will convey is that by relaxing the constraint of having continuous displacements across element boundaries, Discontinuous Galerkin methods are able to impose other kinematic constraints in the problem and still provide accurate solutions. I will demonstrate it by showcasing the performance of a class of Discontinuous Galerkin methods we introduced in a variety of circumstances. First, in nonlinear elasticity problems involving different kinematic constraints. Second, in a class of immersed boundary methods, which sidestep the need for automatic remeshing in problems with evolving boundaries by embedding the boundary in any mesh. And finally, in the accurate solution of the stress and displacement fields around cracks when cracks are "embedded" in the mesh, as in extended finite element methods. In the three cases, I will comment on recent convergence results we obtained. In all cases, a special interpolant taking advantage of the discontinuities in the finite element space had to be constructed, since standard interpolants in conforming finite element spaces can be guaranteed to converge only at suboptimal rates. I will conclude the talk by briefly commenting on an adaptive stabilization technique we created to enhance the robustness of the method in highly nonlinear problems. Some of these ideas carry over to a wider variety of popular methods in Solid Mechanics, such as Enhanced Strain Methods.
Abstract: In the context of nematic liquid crystals, it is of interest to see how Oseen-Frank (OF) energy minimizers can be used to approximate Landau-de Gennes (LdG) energy minimizers, which depends on a small parameter $L$. The simplest approach is to think of a LdG minimizer as the sum of a OF minimizer with a correction term which is ``small'' for small $L$. Because of the failure of OF energy minimizers in predicting ``line defects'' it is possibly desirable to have a better understanding of the correction term. We show that the correction term is of size $O(L)$ in an appropriate sense. Furthermore, we will derive analytic relation between the OF minimizers and the first order term in the above asymptotic expansion. Joint work with Arghir Zarnescu.
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