(Tenative Schedule)

Time & Location: All talks are on Thursdays in Gibson Hall 325 at 3:30 pm unless otherwise noted. Refreshments in Gibson 426 after the talk.

Comments indicating vacations, special lectures, or change in location or time are in

Organizer: Gustavo Didier

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We study (experimentally and theoretically) random curves (in many senses) in space and in the plane, including such of their properties as the knot type.

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In data assimilation that attempts to predict nonlinear evolution of the system by combining complex computational model, observations, and uncertainties associated with them, it is useful to be able to quantify the amount of information provided by an observation or by an observing system. Measures of the observational influence are useful for the understanding of performance of the data assimilation system. The Forecast sensitivity to observation provides practical and useful metric for the assessment of observations. Quite often complex data assimilation systems use a simplified version of the forecast sensitivity formulation based on ensembles. In this talk, we first present the comparison of forecast sensitivity for 4DVar, Hybrid-4DEnVar, and 4DEnKF with or without such simplifications using a highly nonlinear model. We then present the results of ensemble forecast sensitivity to satellite radiance observations for Hybrid-4DEnVart using a global data assimilation system.

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Many microorganisms and cells function in complex (non-Newtonian) fluids, which are mixtures of different materials and exhibit both viscous and elastic stresses. For example, mammalian sperm swim through cervical mucus on their journey through the female reproductive tract, and they must penetrate the viscoelastic gel outside the ovum to fertilize. A swimming stroke emerges from the coupled interactions between the complex rheology of the surrounding media and the passive and active body dynamics of the swimmer. We use computational models of swimmers in viscoelastic fluids to understand these interactions. I will show results from several recent investigations, and give mechanistic explanations for some different experimental observations. In particular I will discuss how flexible filaments (such as flagella) can store energy from the fluid to obtain speed enhancements from fluid elasticity.

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Applications such as electrical-impedance tomography, nanoelectrode sensors, and nanowire sensors lead to deterministic and stochastic partial differential equations that model electrostatics and charge transport. The main model equations are the nonlinear Poisson-Boltzmann equation and the stochastic drift-diffusion-Poisson-Boltzmann system. After a discussion of the model equations, theoretic results as well as a numerical method, namely optimal multi-level Monte Carlo, are presented.

Knowing these model equations, the question how as much information as possible can be extracted from measurements arises next. We use computational Bayesian PDE inversion to reconstruct physical and geometric parameters of the body interior in electrical-impedance tomography and of target molecules in the two nanoscale sensors considered here. Computational Bayesian estimation provides us with the ability not only to estimate unknown parameter values but also their probability distributions and hence the uncertainties in reconstructions, which is important in the case of ill-posed inverse problems. In addition to showing the well-posedness of the Bayesian inversion problem for the nonlinear Poisson-Boltzmann equation, numerical results for the three applications such as multifrequency reconstruction for nanoelectrode sensors are shown.

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This talk presents two Bayesian approaches to making efficient statistical inferences for genome-wide association studies (GWAS) using regression models. The first method concerns the marginal analysis of each covariate, while the second is for the joint analysis of the entire dataset.

In the first work, motivated by the "Bayes/non-Bayes compromise", we characterize the asymptotic distribution of the Bayes factor in linear regression with conjugate priors. We show that, under the null, the log(Bayes factor) is distributed as a weighted sum of independent chi-squared random variables with a shifted mean. This enables us to analytically evaluate the p-value associated with a Bayes factor. By implementing a recent algorithm of Bausch (2013), we are able to compute extremely small p-values to arbitrary precision. Moreover, our result helps explain Bartlett’s paradox and the prior-dependence nature of the Bayes factor.

The second work concerns Bayesian variable selection for linear regression models using Markov chain Monte Carlo methods (MCMC). The key innovation is a novel iterative algorithm for solving a special type of linear systems, which is the most time-consuming step in each MCMC iteration. We call our method iterative complex factorization (ICF) and prove that ICF always converges. In the context of Bayesian variable selection with MCMC, ICF is much faster and more accurate than other iterative methods such as Gauss-Seidel iterations. Our algorithm is particularly useful for the heritability estimation with massive GWAS datasets, which can be prohibitively slow via traditional MCMC methods.

This talk presents two Bayesian approaches to making efficient statistical inferences for genome-wide association studies (GWAS) using regression models. The first method concerns the marginal analysis of each covariate, while the second is for the joint analysis of the entire dataset.

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When organisms locomote and interact in nature, they must navigate through complex habitats that vary on many spatial scales, and they are buffeted by turbulent wind or water currents and waves that also vary on a range of spatial and temporal scales. We have been using the microscopic larvae of bottom-dwelling marine animals to study how the interaction between the swimming by a microorganism and the turbulent water flow around them determines how they move through the environment. Many bottom-dwelling marine animals release tiny larvae that are dispersed to new sites by ambient water currents. To recruit to new sites on the sea floor, these larvae must leave the water column and land on surfaces in suitable habitats. We are studying the mechanisms larvae use to move through turbulent flow and land on surfaces. Field and laboratory measurements enabled us to quantify the fine-scale, rapidly-changing patterns of water velocity vectors and of chemical cue concentrations near coral reefs and along fouling communities (organisms growing on docks and ships). We also measured the locomotory performance of larvae of reef-dwelling and fouling community animals, and their responses to chemical and mechanical cues. We used these data to design agent-based models of larval behavior. By putting model larvae into our real-world flow and chemical data, which varied on spatial and temporal scales experienced by microscopic larvae, we could explore how different responses by larvae affected their transport into reefs or fouling communities. The most effective strategy for recruitment depends on habitat.

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The last two decades have brought swift change in the area of human genetics. For less than $100 per person, genotypes for over 2 million genetic markers can be obtained, and with large publicly available genetic reference panels, a person's genetic data can be further imputed to well over 7 million genetic markers. This data revolution has opened up many scientific and statistical questions, though deciphering such a large amount of information can be daunting and requires an understanding of both genetics and statistics. Here, I provide motivation and context to the scientific question and detail the statistical methods and results in three areas: the use of familial matching in forensics, discovery of disease sub-types, and the combination of genetic findings over studies of individuals with differing ancestry. Finally, I end with some open research questions in statistical genetics, with a particular motivation towards translating the wealth of genetic data and results into public health and medicine.

Mathematics Department, 424 Gibson Hall, New Orleans, LA 70118 504-865-5727 math@math.tulane.edu