Alexey Miroshnikov

Parallel Markov chain Monte Carlo for non‐Gaussian posterior distributions

Recent developments in big data and analytics research have produced an abundance of large data sets that are too big to be analyzed in their entirety, due to limits on computer memory or storage capacity. To address these issues, communication-free parallel Markov chain Monte Carlo (MCMC) methods have been developed for Bayesian analysis of big data. These methods partition data into manageable subsets, perform independent Bayesian MCMC analysis on each subset, and combine the subset posterior samples to estimate the full data posterior. Current approaches to combining subset posterior samples include sample averaging, weighted averaging, and kernel smoothing techniques. Although these methods work well for Gaussian posteriors, they are not well-suited to non-Gaussian posterior distributions. Here, we develop a new direct density product method for combining subset marginal posterior samples to estimate full data marginal posterior densities. Using a commonly-implemented distance metric, we show in simulation studies of Bayesian models with non-Gaussian posteriors that our method outperforms the existing methods in approximating the full data marginal posteriors. Since our method estimates only marginal densities, there is no limitation on the number of model parameters analyzed. Our procedure is suitable for Bayesian models with unknown parameters with fixed dimension in continuous parameter spaces.

parallelMCMCcombine: an R package for bayesian methods for big data and analytics.

Recent advances in big data and analytics research have provided a wealth of large data sets that are too big to be analyzed in their entirety, due to restrictions on computer memory or storage size. New Bayesian methods have been developed for data sets that are large only due to large sample sizes. These methods partition big data sets into subsets and perform independent Bayesian Markov chain Monte Carlo analyses on the subsets. The methods then combine the independent subset posterior samples to estimate a posterior density given the full data set. These approaches were shown to be effective for Bayesian models including logistic regression models, Gaussian mixture models and hierarchical models. Here, we introduce the R package parallelMCMCcombine which carries out four of these techniques for combining independent subset posterior samples. We illustrate each of the methods using a Bayesian logistic regression model for simulation data and a Bayesian Gamma model for real data; we also demonstrate features and capabilities of the R package. The package assumes the user has carried out the Bayesian analysis and has produced the independent subposterior samples outside of the package. The methods are primarily suited to models with unknown parameters of fixed dimension that exist in continuous parameter spaces. We envision this tool will allow researchers to explore the various methods for their specific applications and will assist future progress in this rapidly developing field.

Computing the joint distribution of the total tree length across loci in populations with variable size

In recent years, a number of methods have been developed to infer complex demographic histories, especially historical population size changes, from genomic sequence data. Coalescent Hidden Markov Models have proven to be particularly useful for this type of inference. Due to the Markovian structure of these models, an essential building block is the joint distribution of local genealogical trees, or statistics of these genealogies, at two neighboring loci in populations of variable size. Here, we present a novel method to compute the marginal and the joint distribution of the total length of the genealogical trees at two loci separated by at most one recombination event for samples of arbitrary size. To our knowledge, no method to compute these distributions has been presented in the literature to date. We show that they can be obtained from the solution of certain hyperbolic systems of partial differential equations. We present a numerical algorithm, based on the method of characteristics, that can be used to efficiently and accurately solve these systems and compute the marginal and the joint distributions. We demonstrate its utility to study the properties of the joint distribution. Our flexible method can be straightforwardly extended to handle an arbitrary fixed number of recombination events, to include the distributions of other statistics of the genealogies as well, and can also be applied in structured populations.

On the Construction and Properties of Weak Solutions Describing Dynamic Cavitation

We consider the problem of dynamic cavity formation in isotropic compressible nonlinear elastic media. For the equations of radial elasticity we construct self-similar weak solutions that describe a cavity emanating from a state of uniform deformation. For dimensions d=2,3 we show that cavity formation is necessarily associated with a unique precursor shock. We also study the bifurcation diagram and do a detailed analysis of the singular asymptotics associated to cavity initiation as a function of the cavity speed of the self-similar profiles. We show that for stress free cavities the critical stretching associated with dynamically cavitating solutions coincides with the critical stretching in the bifurcation diagram of equilibrium elasticity. Our analysis treats both stress-free cavities and cavities with contents.

Relative entropy in hyperbolic relaxation for balance laws

We present a general framework for the approximation of systems of hyperbolic balance laws. The novelty of the analysis lies in the construction of suitable relaxation systems and the derivation of a delicate estimate on the relative entropy. We provide a direct proof of convergence in the smooth regime for a wide class of physical systems. We present results for systems arising in materials science, where the presence of source terms presents a number of additional challenges and requires delicate treatment. Our analysis is in the spirit of the framework introduced by Tzavaras [A. Tzavaras, Commun. Math. Sci., 3-2, 2005] for systems of hyperbolic conservation laws.

CONVERGENCE OF VARIATIONAL APPROXIMATION SCHEMES FOR ELASTODYNAMICS WITH POLYCONVEX ENERGY

We consider a variational scheme developed by S. Demoulini, D. M. A. Stuart and A. E. Tzavaras [Arch. Rat. Mech. Anal. 157 (2001)] that approximates the equations of three dimensional elastodynamics with polyconvex stored energy. We establish the convergence of the time-continuous interpolates constructed in the scheme to a solution of polyconvex elastodynamics before shock formation. The proof is based on a relative entropy estimation for the time-discrete approximants in an environment of L p -theory bounds, and provides an error estimate for the approximation before the formation of shocks.

Weak* Solutions II: The Vacuum in Lagrangian Gas Dynamics

We develop a framework in which to make sense of solutions containing the vacuum in Lagrangian gas dynamics. At and near a vacuum, the specific volume becomes infinite, and enclosed vacuums are represented by Dirac masses, so they cannot be treated in the usual weak sense. However, the weak* solutions recently introduced by the authors can be extended to include solutions containing vacuums. We present a definition of these natural vacuum solutions and provide explicit examples which demonstrate some of their features. Our examples are isentropic for clarity, and we briefly discuss the extension to the full

Motile Geobacter dechlorinators migrate into a model source zone of trichloroethene dense non-aqueous phase liquid: Experimental evaluation and modeling

3\times3

Asymptotic properties of parallel Bayesian kernel density estimators

system of gas dynamics. We also extend our methods to one-dimensional dynamic elasticity to show that fractures cannot form in an entropy solution.

Cellulose biodegradation models; an example of cooperative interactions in structured populations

Microbial migration towards a trichloroethene (TCE) dense non-aqueous phase liquid (DNAPL) could facilitate the bioaugmentation of TCE DNAPL source zones. This study characterized the motility of the Geobacter dechlorinators in a TCE to cis-dichloroethene dechlorinating KB-1(™) subculture. No chemotaxis towards or away from TCE was found using an agarose in-plug bridge method. A second experiment placed an inoculated aqueous layer on top of a sterile sand layer and showed that Geobacter migrated several centimeters in the sand layer in just 7days. A random motility coefficient for Geobacter in water of 0.24±0.02cm(2)·day(-1) was fitted. A third experiment used a diffusion-cell setup with a 5.5cm central sand layer separating a DNAPL from an aqueous top layer as a model source zone to examine the effect of random motility on TCE DNAPL dissolution. With top layer inoculation, Geobacter quickly colonized the sand layer, thereby enhancing the initial TCE DNAPL dissolution flux. After 19days, the DNAPL dissolution enhancement was only 24% lower than with an homogenous inoculation of the sand layer. A diffusion-motility model was developed to describe dechlorination and migration in the diffusion-cells. This model suggested that the fast colonization of the sand layer by Geobacter was due to the combination of random motility and growth on TCE.