Center for Statistical Sciences Seminar
Abstract: It is crucial to understand immune responses to infectious agents in order to develop antiviral treatments and prevention vaccines to control pandemics of infectious diseases which pose a great threat to public health. A cutting-edge approach for quantitatively understanding pathogenesis of immune responses and viral infections is to integrate mathematical modeling approaches with well-designed experiments in a systematical way. Dynamics models such as ordinary differential equations (ODE) and state-space models are usually used to describe dynamic immune responses to viral infections. It is quite challenging to perform statistical estimation and inference for dynamic models from experimental data. For instance, the standard least squares (LS) estimation approach is widely used to estimate the unknown parameters in a non-linear ODE model. However, it requires repeated evaluation of the ODEs in order to obtain the LS estimate using numerical methods, and there are many theoretical and practical issues that need to be resolved for the LS method for ODE models. We propose some new estimation methods for ODE models, which will be applied to study immune responses to HIV and influenza infections. Based on the ODE models, we are able to quantify cell kinetics and viral dynamics as well as immune cells trafficking among different compartments for viral infections. A user-friendly software, DEDiscover developed by our group, for differential equation model simulations and parameter estimation will be introduced.
Probability Seminar
Abstract: Consider the following problems: 1. Given an undamped harmonic oscillator driven by additive Gaussian white noise, estimate oscillator's frequency from the observations of the oscillations; 2. Given an undamped wave equation driven by additive space-time white noise, estimate the propagation speed from the observations of the solution. It turns out that the the first (one-dimensional) problem is not necessarily easier to study than the second (infinite-dimensional) problem. The objective of the talk is to study the asymptotic properties of the maximum likelihood estimator in both problems and to discuss various generalizations.
Computational Molecular Biology (CCMB) Seminar
Hosted by: Ben Raphael (Refreshments will be served at 11:45 am) |
Abstract: Recent studies show that along with single nucleotide polymorphisms and small indels, larger structural differences contribute significantly to human genetic diversity. The realization of new ultra-high-throughput sequencing platforms has made it feasible to detect the full spectrum of genomic variation among many individual genomes, including those between healthy tissues and those susceptible to disease with genomic origin. Conventional algorithms for identifying structural variation (SV) have not been designed to handle the short read lengths and the errors implied by the available and future high throughput sequencing technologies. In this talk we will provide combinatorial formulations for the SV detection between a reference genome and a high throughput paired-end sequenced individual genome. We will provide efficient algorithms for each of the formulations we give, which all turn out to be fast and quite reliable; they are also applicable to all currently available sequencing methods and traditional capillary sequencing technology.
Applied Math Dissertation Defense
Applied Math Dissertation Defense
Brown Applied Mathematics Pattern Theory and Vision Seminar
Abstract:
The images acquired via medical imaging modalities are frequently subject to low signal-to-noise ratio, bias field and partial volume effects. These artifacts, together with the naturally low contrast between image intensities of some neighboring structures, make the extraction of regions of interest (ROIs) in clinical images a challenging problem. Probabilistic atlases, typically generated from comprehensive sets of manually labeled examples, facilitate the analysis by providing statistical priors for tissue classification and structure segmentation. However, the limited availability of training examples that are compatible with the images to be segmented renders the atlas-based approaches impractical in many cases. In the talk I will present a generative model for joint segmentation of corresponding regions of interest in a collection of aligned images that does not require labeled training data. Instead, the evolving segmentation of the entire image set supports each of the individual segmentations. This is made possible by iteratively inferring a subset of the model parameters, called the spatial parameters, as part of the joint segmentation processes. These spatial parameters are defined in the image domain and can be viewed as a latent atlas. Our latent atlas formulation is based on probabilistic principles, but we solve it using partial differential equations and energy minimization criteria. We evaluate the method successfully for the segmentation of cortical and subcortical structures within different populations and of brain tumors in a single-subject multi-modal longitudinal experiment.
[pizza will be provided]
Applied Math Dissertation Defense
Joint LCDS-Probability Seminar
Abstract: H-infinity control is a robust control theory where problems can be formulated as zero-sum dynamic games between the controller and the antagonistic deterministic disturbance. It is known that partially observed H-infinity control with an integral running cost can be reduced to a perfectly observed differential game with the information state, which is an infinite-dimensional sufficient statistics for the state of the system. In this talk, we consider an H-infinity control problem with a maximum running cost under partial observations. We first reduce the problem to a perfectly observed infinite-dimensional differential game with a maximum running cost of the information state. Then, by using dynamic programming on the information state and introducing a viscosity notion in an infinite-dimensional space, we show that the value function of the differential game is a viscosity solution of the dynamic programming partial differential equation of a quasivariational type in the infinite-dimensional space.
Probability Seminar
Abstract: The celebrated Gittins index, its generalizations and related techniques play an important role in applied probability models, resource allocation problems, optimal portfolio management problems as well as other problems of nancial mathematics. It is well known that 1) a connection exists between the Ratio (cycle) maximization problem, the Kathehakis-Veinot (KV) Restart Problem and the Whittle family of Retirement Problems, and 2) that their key characteristics, the classical Gittins index, the KV index, and the Whittle index are equal. These indices were generalized by the author (Statistics and Probability Letters, 2008) in such a way that it is possible to use the so called State Elimination algorithm, developed earlier to solve the problem of Optimal Stopping of Markov Chains to calculate this common index: The main goal of this talk is to demonstrate that the equality of these indices is a special case of a similar equality for three simple abstract optimization problems. By an abstract optimization problem we mean a problem with maximization over an abstract set of indices U without any species about this set.
Scientific Computing Seminar
(joint work with Jay Golapalakrishnan and Antti Niemi) | |
Abstract:
The hp-adaptive finite elements combine elements of varying size h and polynomial order p to deliver approximation properties superior to any other discretization methods. The best approximation error converges exponentially fast to zero as a function of number of degrees-of-freedom. The hp methods are thus a natural candidate for singularly perturbed problems experiencing internal or boundary layers like in compressible gas dynamics.
This is the good news. The bad news is that only a small number of variational formulations is stable for hp-discretizations. By the hp-stability we mean a situation where the discretization error can be bounded by the best approximation error times a constant that is independent of both h and p. To this class belong classical elliptic problems (linear and non-linear), and a large class of wave propagation problems whose discretization is based on hp spaces reproducing the classical exact grad-curl-div sequence. Examples include acoustics, Maxwell, elastodynamics, poroelasticity and various coupled and multiphysics problems.
We will present a new paradigm for constructing discretization schemes for virtually arbitrary systems of linear PDE's that remain stable for arbitrary hp meshes, extending thus dramatically the applicability of hp approximations. For a start, we focus on a challenging model problem - convection dominated diffusion.
The presented methodology incorporates the following features:
The problem of interest is formulated as a system of first order PDE's in the distributional (weak) form, i.e. all derivatives are moved to test functions. We use the DG setting, i.e. the integration by parts is done over individual elements.
As a consequence, the unknowns include not only field variables within elements but also fluxes on interelement boundaries. We do not use the concept of a numerical flux but, instead, treat the fluxes as independent, additional unknowns.
For each trial function corresponding to either field or flux variable, we determine a corresponding optimal test function by solving an auxiliary local problem on one element. The use of optimal test functions guarantees attaining the supremum in the famous inf-sup condition from Babuska-Brezzi theory.
The local problems for determining optimal test functions are solved approximately with an enhanced approximation (a locally hp-refined mesh).
By selecting right norms for test functions, we can obtain amazing stability properties uniform not only with respect to discretization parameters but also with respect to the diffusion constant (the resulting discretization is robust).
The presentation will consist of three parts.
In the first part, we will present a general abstract variational framework for the optimal test functions.
The second part will focus on the analysis of 1D convection-dominated diffusion problem. We will present a 1D stability analysis.
The third part will focus on adaptivity. We will present numerous numerical examples for 1D and 2D ``confusion'' problems. We have been able to solve in a fully automatic mode problems with diffusion constant eps =10^{-11} in 1D and eps = 10^{-7} in 2D using hp-adaptivity.
The presentations will be accompanied with ``live'' demonstrations of 1D and 2D codes.
For a detailed presentation on the subject, see [1,2,3].
[1] L. Demkowicz and J. Gopalakrishnan.
A Class of Discontinuous Petrov-Galerkin Methods.
Part I: The Transport Equation.
Comput. Methods Appl. Mech. Engrg., accepted.
see also ICES Report 2009-12.
[2] L. Demkowicz and J. Gopalakrishnan.
A Class of Discontinuous Petrov-Galerkin Methods
Part II: Optimal Test Functions.
Numer. Mth. Partt. D.E., in review
ICES Report 2009-16.
[3] L. Demkowicz, J. Gopoalakrishnan and A. Niemi.
A Class of Discontinuous Petrov-Galerkin Methods.
Part III: Adaptivity.
PDE Seminar
Abstract: The two-phase Stefan problem is expected to exhibit waiting time phenomena, in other words, an initial Lipschitz free boundary might remain just Lipschitz for some positive time. On the other hand, adding a Gibbs-Thomson correction term to the model is believed to stabilize the interface and make it more regular. I will present a partial result in this direction, namely, I will show that weak solutions whose free boundaries are Lipschitz in space and time are actually C^{2,\alpha} in space and become so instantaneously. This will follow first from observing that the De Giorgi-Moser-Nash theory for parabolic equations works even when one has a singular right hand side and secondly from the regularity theorem of almost-minimal boundaries of Almgren, De Giorgi and Tamanini.
Pattern Theory Seminar
<--- 2010 Index