Several models have recently been introduced in various applied settings, from chemotaxis to pedestrian transport. Those equations include local non linear effects and can be seen as hybrid models between linear transport equations and scalar conservation laws. We introduce new quantitative, explicit estimates that imply well posedness.
First-passage problems involving multiple diffusing particles will be discussed. In general, there is a family of nontrivial first-passage exponents that quantify how the original ordering of the random walks unravels with time. Remarkably, the scaling exponents themselves obey universal scaling laws, when the number of random walks is large, and moreover, the scaling function is obtained analytically. The theoretical analysis maps the multiple particle trajectories onto a single trajectory in high dimensions and uses the kinetics of first-passage inside spherical cones.
E. Ben-Naim, Phys. Rev. E 82, 061103 (2010); E. Ben-Naim and P.L. Krapivsky, J. Phys. A 43, 495007 & 495008 (2010).
Recent discoveries about various capabilities of small RNAs to affect gene expression have led to some biotechnological advances and medical applications. After a brief overview of these discoveries, we will focus on the phenomenon of RNA conformational switching and how it relates to function. The computational methods that are involved in RNA folding prediction, RNA mutational analysis, and the design and search for novel RNA switches will be described. A representation of the RNA secondary structure by a Laplacian matrix and how its spectral decomposition can assist in detecting conformational rearrangements will also be discussed.
Prof. David Levermore, Department of Mathematics, IPST, AMSC and the Burgers Program, University of Maryland
Hierarchies in Simulation-Based Science and Engineering
Simulation has emerged as the third pillar of science
and engineering, complementing observation and theory. It plays
a major role when the system being studied is either too remote,
too complicated, or too large to allow a thorough observational
interrogation. The accompanying theoretical framework is usually
multi-description, multi-physics, and multi-scale. This demands
an interdisciplinary hierarchical approach to models, algorithms,
and data. We show how this approach applies to model validation,
algorithm verification, model and algorithm building, calibration,
and uncertainty quantification. It gives importance to different
mathematical questions than a classical non-hierarchical approach.
It demands a larger role for statistics. It requires scientists
and mathematicians to again be good engineers in order to do good
science. Examples will be drawn from climate, weather, plasmas,
astrophysics, engineering, biochemistry, fisheries, and other
Prof. Benedetto Piccoli, Department of Mathematical Sciences and the Center for Computational and Integrative Biology, Rutgers University - Camden
Mixed models for nonlinear flows on networks
Continuous and discrete models for traffic flow on networks are commonly used.
Applications ranges from vehicular traffic to supply chains and data networks.
We focus on recent mixed models, involving continuous-discrete spaces and ode-pde systems, and
models coupling. Finally, we will discuss a measure theoretical framework,
particularly efficient for dynamics of large groups.
Application of the Implicit Particle Filter to a Model of Nearshore Circulation
In the implicit particle filter the trajectory of each particle is informed by observations. The implicit particle filter typically requires fewer particles than other particle filtering schemes, but more computational effort per particle. In its simplest form, the implicit particle filter reduces to the method of optimal importance sampling.
In other work, we have applied an implicit particle smoother to the problem of parameter estimation, and found it to produce reliable parameter estimates even if the initial estimate is very poor.
Today the implementation of the implicit particle filter to the problem of state estimation in a shallow water model with 30,000 state variables will be described. This system produces reliable state estimates with O(10) particles, and runs conveniently on a workstation.
Dr. Cory Hauck, Computer Science and Mathematics Division, Oak Ridge National Laboratory
Optimization-Based Methods for Discretization of Partial Differential Equations.
In extremely large computational problems, the lack of sufficient memory resources can place limitations on both the communication between problem unknowns and the resolution of fine dynamical scales. In such cases, many numerical methods which are otherwise stable and convergent may lack important properties of the continuum solution. One way to enforce such properties in a discretization is through the use of constrained optimization. Optimization-based discretizations typically involves a significant increase in computational cost. However, if the optimization is local to a computational cell, then the additional floating point operations can be handled by local processing units. In this context, the optimization-based approach fits squarely into the emerging paradigm of data parallel computing, where the bottleneck to fast simulation is not floating-point operations, but rather data transfer. In this talk, we present discretizations and preliminary numerical results for several hyperbolic problems based on the optimization approach.
River Ecology: How Branching Geometry Influences Connectivity, Extinction Risk and Biogeography
Riverine landscapes differ in fundamental ways from terrestrial ones. In particular, the branching hierarchical geometry and downstream flow of river systems lead to a suite of network properties rarely considered in 2-dimensional systems that have long been the mainstay of landscape ecology.
Intrinsic effects of configuration, directional biases, transient connectivity, and opportunities for 'out of network' movement may all lead to inherently asymmetrical opportunities for connections among parts of a riverine landscape thereby influencing ecological processes and biogeographic patterns. This 'alternative geometry' of riverine networks provides excellent opportunities for scientists to explore how network connectivity shapes habitat occupancy, metacommunity dynamics, and biogeographic patterns. Using examples involving fish communities that inhabit the river networks of North America and India, I will discuss here how human modifications to the spatial characteristics of river systems, such as habitat fragmentation and interbasin water transfer projects, influence ecological dynamics and biogeographic patterns. Taken together these research projects illustrate the important contributions that riverine geometry makes to our understanding of interspecific variation in extinction risks and the potentially broad relevance of the neutral theory of biodiversity.
The FENE dumbbell model consists of the incompressible Navier-Stokes equation and the Fokker-Planck equation for the polymer distribution. In such a model, the polymer elongation cannot exceed a limit, yielding all interesting features near the boundary. In this talk I’ll recount progress toward understanding the sharpness of boundary requirement dictated by the elongation parameter, with a resolution based on a well-posedness theorem for the coupled Fokker-Planck-Navier-Stokes system. I’ll also present an entropy satisfying conservative method to compute the underlying probability density distribution.
Eigen problems in renewal equations: the effect of time dependence
In order to model the effect of circadian rhythms, it has been necessary to developp linear renewal type models for the cell cycle and cell divsion with periodic coefficients. We are interested here in specific properties of those models in term of growth rate. Particularly, we compare the to constant coefficients models and exhibit unexpected behaviors due to time dependence.
PDE models for nanowire bio- and gas sensors including stochastic effects
Recently, we have developed PDE models for the quantitative
understanding of nanowire bio- and gas sensors. Homogenization of the
boundary layer leads to interface conditions and we have shown
existence and (local) uniqueness for the resulting self-consistent
model for bio- and gas sensors. We have also developed a new parallel
algorithm for the numerical solution of the drift-diffusion-Poisson
system including interface conditions based on the FETI method. It is
especially advantageous for structures with high aspect ratios such as
nanowire sensors and transistors and/or for structures consisting of
different materials (equations).
In both bio- and gas sensors, several stochastic effects occur due to
Brownian motion, binding and unbinding of molecules, chemical
reactions, etc. Recently, we have obtained an existence and
uniqueness result for a diffusion equation coupled with a stochastic
process for molecule binding and unbinding. Numerical results are
compared with measurements as well. Furthermore, a nonlinear elliptic
stochastic equation is discussed.
Using these results, we have performed simulations of realistic
nanowire DNA sensors and we have found the optimal operating regime
(maximum sensor response) by optimizing device parameters. Different
surface models are used for gas sensors and results from their
simulation are presented as well.
Existence, Stability and Dynamics of Some Single- and Multi-Component Solitary Waves: From Theory to Experiments
In this talk, we will present an overview of recent theoretical, numerical and experimental work concerning the static, stability, bifurcation and dynamic properties of coherent structures that can emerge in one- and higher-dimensional settings within Bose-Einstein condensates at the coldest temperatures in the universe (i.e., at the nanoKelvin scale). We will discuss how this ultracold quantum mechanical setting can be approximated at a mean-field level by a deterministic PDE of the nonlinear Schrodinger type and what the fundamental nonlinear waves of the latter are, such as dark solitons and vortices. Then, we will try to go to a further layer of simplified description via nonlinear ODEs encompassing the dynamics of the waves within the traps that confine them, and the interactions between them. Finally, we will attempt to compare the analytical and numerical implementation of these reduced descriptions to recent experimental results and speculate towards a number of interesting future directions within this field.