Brown Analysis Seminar
Abstract: The flat forms of Whitney have been useful in solving many problems in geometric analysis and elsewhere. A classical theorem of Wolfe states that the space of flat forms is in fact the dual to the space of flat chains in Euclidean space. Recent work by Adams generalizes flat chains to Banach spaces. We will define a flat differential form in a Banach space and discuss the generalization of Wolfe's theorem to this setting.
Brown University Center for Statistical Sciences Seminar
Abstract: We consider modeling a deterministic computer response as a realization from a stochastic heteroskedastic process (SHP), which incorporates a spatially-correlated volatility process into the traditional spatially-correlated Gaussian process (GP) model. Unconditionally, the SHP is a stationary non-Gaussian process, with stationary GP as a special case. Conditional on a latent process, the SHP is a non-stationary GP. The sample paths of this process offer more modeling flexibility than those produced by a traditional GP, and can better reflect prediction uncertainty. GP prediction error variances depend only on the locations of inputs, while SHP can reflect local inhomogeneities in a response surface through prediction error variances that depend on both input locations and output responses. We use maximum likelihood for inference, which is complicated by the high dimensionality of the latent process. Accordingly, we develop an importance sampling method for likelihood computation and use a low-rank kriging approximation to reconstruct the latent process. Responses at unobserved locations can be predicted using empirical best predictors or by empirical best linear unbiased predictors. Prediction error variances are also obtained. In examples with simulated and real computer experiment data, the SHP model is superior to traditional GP models. (This is joint work with Jay Breidt, Wenying Huang, and Ke Wang.)
Brown Applied Mathematics Pattern Theory and Vision Seminar
Abstract:
We consider modeling a deterministic computer response as a realization from a stochastic heteroskedastic process (SHP), which incorporates a spatially-correlated volatility process into the traditional spatially-correlated Gaussian process (GP) model. Unconditionally, the SHP is a stationary non-Gaussian process, with stationary GP as a special case. Conditional on a latent process, the SHP is a non-stationary GP. The sample paths of this process offer more modeling flexibility than those produced by a traditional GP, and can better reflect prediction uncertainty. GP prediction error variances depend only on the locations of inputs, while SHP can reflect local inhomogeneities in a response surface through prediction error variances that depend on both input locations and output responses.
We use maximum likelihood for inference, which is complicated by the high dimensionality of the latent process. Accordingly, we develop an importance sampling method for likelihood computation and use a low-rank kriging approximation to reconstruct the latent process. Responses at unobserved locations can be predicted using empirical best predictors or by empirical best linear unbiased predictors. Prediction error variances are also obtained. In examples with simulated and real computer experiment data, the SHP model is superior to traditional GP models. (This is joint work with Jay Breidt, Wenying Huang, and Ke Wang.)
Special PDE and LCDS eminar
Department of Mathematics Colloquium Talk
Abstract: Floor diagrams are a class of weighted oriented graphs introduced by E. Brugalle and G. Mikhalkin. Tropical geometry arguments yield combinatorial descriptions of (ordinary and relative) Gromov-Witten invariants of projective spaces in terms of floor diagrams and their generalizations. In the case of the projective plane, these descriptions can be used to obtain new formulas for the corresponding enumerative invariants. In particular, we give a proof of Goettsche's polynomiality conjecture for plane curves, and enumerate plane rational curves of given degree passing through given points and having maximal tangency to a given line. On the combinatorial side, we show that labeled floor diagrams of genus 0 are equinumerous to labeled trees, and therefore counted by the celebrated Cayley's formula. The corresponding bijections lead to interpretations of the Kontsevich numbers (the genus-0 Gromov-Witten invariants of the projective plane) in terms of certain statistics on trees. This is joint work with Grisha Mikhalkin.
Scientific Computing Seminar
Abstract:
The nonlinear Helmholtz equation models the propagation of intense laser beams in Kerr media such as water,
silica and air. It is a semilinear elliptic equation which requires nonselfadjoint
radiation boundary-conditions, and remains unsolved in many
cases.
Its commonly-used parabolic approximation, the nonlinear Schrodinger
equation (NLS), is known to possess singular solutions. We therefore consider
the question, which has been open since the 1960s: do nonlinear Helmholtz
solutions exists, under conditions for which the NLS solution becomes singular? In other words, is the singularity
removed in the elliptic model?
In this work we develop a numerical method which produces such solutions,
thereby showing that the singularity is indeed removed in the elliptic
equation.
We also consider the subcritical case, wherein the NLS has stable solitons.
For beams whose width is comparable to the optical wavelength, the NLS
model becomes invalid, and so the existence of such ``nonparaxial solitons''
requires solution of the Helmholtz model.
Numerically, we consider the case of grated material, that has material
discontinuities in the direction of propagation. We develop a fourth-order
discretization which is compact only in this direction, that is optimal for this
case.
Joint work with Gadi Fibich and Semyon Tsynkov.
PDE Seminar
Brown Applied Mathematics Pattern Theory and Vision Seminar
Abstract: Human visual scene understanding is remarkable: with only a brief glance at an image, an abundance of information is available - spatial structure, scene category and the identity of main objects in the scene. In traditional computer vision, scene and object recognition are two visual tasks generally studied separately. However, it is unclear whether it is possible to build robust systems for scene and object recognition, matching human performance, based only on local representations. Another key component of machine vision algorithms is the access to data that describe the content of images. As the field moves into integrated systems that try to recognize many object classes and learn about contextual relationships between objects, the lack of large annotated datasets hinders the fast development of robust solutions. In the early days, the first challenge a computer vision researcher would encounter would be the difficult task of digitizing a photograph. Even once a picture was in digital form, storing a large number of pictures (say six) consumed most of the available computational resources. In addition to the algorithmic advances required to solve object recognition, a key component to progress is access to data in order to train computational models for the different object classes. This situation has dramatically changed in the last decade, especially via the internet, which has given computer vision researchers access to billions of images and videos. In this talk I will describe recent work on visual scene understanding that try to build integrated models for scene and object recognition, emphasizing the power of large database of annotated images in computer vision. Antonio Torralba is associate Professor of Electrical Engineering and Computer Science at the Computer Science and Artificial Intelligence Laboratory (CSAIL) at MIT. Following his degree in telecommunications engineering, obtained at the Universidad Politecnica de Catalunya, Spain, he was awarded a Ph.D. in Signal, Image, and Speech processing from the Institut National Polytechnique de Grenoble, France. Thereafter, he spent post-doctoral training at the Brain and Cognitive Science Department and the Computer Science and Artificial Intelligence Laboratory at MIT.
LCDS Workshop
Abstract: The schedule can be found at http://www.dam.brown.edu/people/sandsted/lcds-lattice-conference.php