Mathematical Modeling and Analysis
Mathematical Modeling and Analysis
Pavel Vachal
We suggest a new rezoning technique for ALE simulations, based on decomposition of mesh movement into simple deformation modes, represented by an orthogonal set of base vectors.
Arbitrary Lagrangian-Eulerian (ALE) methods are widely used to simulate problems involving large deformations and volume changes of the computational domain, typically in shock hydrodynamics or plasma physics. To maintain a sufficient precision during the whole computation, it is critical to have a robust general method for mesh rezoning (adaptation), which is able to recognize and remove unphysical mesh distortions. Despite continuous development in the course of last 30 years, none of the methods presented so far is completely satisfactory and sufficiently general.
To address this problem, we construct a physically motivated and mathematically rigorous approach to modal decomposition of discrete fluid flows. This is done by expressing mesh movement as a linear combination of simple, synoptic modes such as translation, rotation, inflation and deviatoric strains. Subsequently, we can filter or damp the components causing high-frequency mesh distortion, while preserving the low-frequency deformation which contains valuable information about physical behavior of the simulated system.
The final objective of this project is to develop a new general mesh rezoning strategy for ALE simulation codes which are currently under development at LANL as well as at CTU. Although testing and tuning of the method in real physical applications is still under way, promising preliminary results will be presented.
Amber Smith
Secondary bacterial infections associated with influenza are a leading cause of death in the United States. These bacterial infections, mainly caused by Streptococcus pneumoniae, capitalize on the environment in the respiratory tract created by the Influenza virus. Experiments suggest a lethal synergism between these pathogens, but the precise mechanisms involved are unknown. However, some hypotheses attribute the interaction to specific viral properties, dysfunctional immune responses, and/or accelerated cell regeneration. In addition, an interesting and surprising observation is the change in viral levels following the bacterial challenge suggesting a truly dual effect. While the kinetics and interactions of these two pathogens are not well understood, we are developing ordinary differential equation models of the following three infections: (i) influenza, (ii) S. pneumoniae, and (iii) influenza followed by S. pneumoniae 7 days later. Using experimental mouse lung data, we are fitting the models, estimating parameters, and investigating synergistic mechanisms. So far, we have good candidates for control models of influenza and S. pneumoniae; however, choosing functional forms for the interaction between the two pathogens presents extreme difficulty. Since the equations are highly coupled, obtaining a fit that accurately represents both viral and bacterial data sets in addition to having biological meaning is not trivial. Therefore, we are continuing to explore possible mechanisms as well as the functional forms involved in these models.
A. Sasha Gutfraind
Countless ex-Soviet nuclear sites are not protected reliably against theft. Consequently, the DOE and the State Department have initiated the Second Line of Defense program for stopping nuclear smugglers and nuclear-armed terrorists. The program employs a network of nuclear material detectors installed at border crossings and transportation junctions of the ex-USSR.
Our research is in developing mathematical algorithms that place those detectors optimally, ie. maximize the probability of detecting the smugglers given a limited budget. Unlike previous work in the area, our approach (1) relaxes important assumptions about smuggler behavior, and (2) uses matrix computations to improve algorithmic performance. Briefly, rather than assuming that smugglers move in the least-cost path between the nuclear site and the target, we allow them to move stochastically on the transportation graph. This transforms a maximin to an expectamin optimization problem, whose goal function we express as a certain product of the transition probability and cost matrices.
Thus far, I have demonstrated a prototype implementation of that approach, and moved towards developing a faster version. A future task is to determine whether the optimal detector locations match any graph-theoretical "centrality" metrics of the transportation links.
Kirill Velizhanin
An ability to model photoinduced energy transport between anharmonicaly coupled vibrational modes in polyatomic molecular systems in condensed phase is crucial for understanding and controlling a verity of photo-induced chemical reactions as well as photo-physical processes, e.g., light harvesting in photosynthetic antenna complexes, T-jump protein folding/unfolding, and optically induced detonation of energetic materials. Such phenomena occur on sub-picosecond to nanosecond time scale, and can be efficiently initiated, manipulated, and probed using ultra-fast time resolved optical and infrared laser techniques. To model the energy transport pathways, we have developed and implemented as a computational tool, the first principles (DFT) based approach for the anharmonic potential energy surfaces reconstruction in polyatomic materials with periodic translational symmetries. As a result the anharmonic couplings between the phonons and/or vibrons can be extracted and the energy transport pathways can be identified. The knowledge of the anharmonic couplings should further allow us to simulate the infrared and Raman frequency-resolved vibrational spectra as well as the time-resolved nonlinear, e.g., photon-echo, responses. The comparison of these spectra with available experimental results should provide method validation, and should help one to gain an insight into the hidden microscopic aspects of the experimentally probed dynamics. The applications to pentaerythritol tetranitrate (PETN) molecular crystal, an important energetic material, will be discussed to illustrate the approach.
Carlos Torres
According to the NIH, 50 to 100 million cases of dengue infection occur each year. This includes 100 to 200 cases in the United States, mostly in people who have recently traveled abroad. Dengue cases range from asymptomatic, clinically non-specific flu like symptoms, dengue fever, dengue hemorrhagic fever, and dengue shock syndrome. We developed a spatial mathematical model that incorporates the epidemiology of dengue fever to study the patterns of transmissibility of dengue in Peru. We used data of the number of weekly dengue cases in Peru at the level of Provinces and departments for the years 1994-2006. We assessed the correlations of transmissibility and final epidemic size with climatological, demographic, and geographic variables. We also studied the distribution of the final epidemic size and the distribution of epidemic duration. We are currently evaluating different ways of coupling 195 provinces to study the global spread of dengue in Peru.
Michael Ham
Progress towards understanding and emulating high level neural functions like memory, visual pattern processing, or cognition is predicated on the quantitative characterization of collective activity patterns. To this end, small neural networks growing on micro-electrode arrays are well established biological models and provide access to a relatively large number of neurons with fine temporal resolution. Activity features found in such network are also likely present \textit{in vivo}.
Here, we work towards characterizing the onset of spontaneously occurring collective bursting patterns in cortical neural networks \textit{in vitro}. This activity results from network excitation by small subgroup of "burst leader neurons," which form a primary circuit and are collectively responsible for igniting and maintaining coordinated network wide bursting activity.
We use an information theoretic approach and a statistical characterization of first spike response delay distributions to provide information about burst leader interaction with the rest of the network. We show that primary circuit neurons are mono-synaptically connected to one another and explore the information structure that exists between these highly connected neurons.
The results provide new views of functional connectivity between neurons in terms of information processing and give quantitative characterizations of the internal dynamics of living cortical networks that should be reproduced in models of spontaneously active neural networks
Vadas Gintautas
Complex networks underpin some of the most important phenomena in nature as well as critical aspects of the infrastructure and social interaction in human society. The structure of these complex networks is usually dictated by non-trivial dynamics associated with functional principles that should be understood in terms of information. In many networks, groups of nodes necessarily form functional units, in that the output of a particular node is a function of one or more inputs from other nodes. Some of the hidden structure in the network may be revealed using a method that targets identification of the functional connectivity of these ensembles.
We present an order-by-order information theoretical expansion of the mutual information contained in a group of nodes in a network, and demonstrate that each term in the expansion represents a measure of functional connectivity of precisely that order. This method is a simple and reliable test of whether a node is fundamentally part of a functional pair, triplet, or higher order group, and extracts the informational content of such a module. We show how this expansion leads naturally to a computational algorithm for the identification of node modules associated with specific information structures, by enabling an optimization strategy in information landscapes analogous to steepest descent. We demonstrate the efficacy of this method by applying it to the action potential time series of a cortical neuronal network grown on a microelectrode array. We find that it is possible to identify groups of neurons with redundant information arranged in chains, as in short-term memory, as well as neurons which combine information from other cells to generate new functional output.
Chad Gonzales
Influenza is a common illness and is a major cause of acute respiratory diseases. It infects millions of people annually and it is one whose complications, usually secondary infection with pneumonia, cause an estimated one million deaths worldwide. Estimating the burden of influenza is difficult due to the transmission mechanism and is an important public health problem.
We have constructed a compartmental model of the transmission dynamics of influenza followed by secondary infection with bacterial pneumonia. The model coupled with data from pneumonia hospitalization cases in Guadalajara, Mexico have allowed us to estimate the annual burden of influenza.
We also estimated the transmissibility of the influenza seasons as measured by the reproduction number (R), defined as the number of secondary cases caused by an infectious individual in a partially immune population. If R is greater than 1 an epidemic can occur. If R is less than 1, the epidemic cannot be sustained. We estimated the reproduction number for each of the years of data and found estimates that range from 1.6 to 1.9, which is in agreement with the estimates obtained using data from France, Australia and the United States.
Vitaliy Gyrya
We develop and analyze a new high-order mimetic finite difference discretization method. A numerical method is called mimetic if it ``mimics'' the fundamental properties of the underlying equations such as conservation laws, symmetries, solution positivity, and fundamental identities of vector and tensor calculus.
As a sample problem we consider a second order diffusion equation written as a system of two linear equations for unknown velocity and pressure. The first equation relates the velocity to the gradient of the pressure (the constitutive equation), while the second equation gives the conservation of mass.
The existing mimetic finite difference (MFD) method produces a locally conservative discretization which is exact for the piecewise linear pressure. This means that the error in pressure is of order h2 and the error in velocity is of order h, where h is the size of mesh elements. Our goal is to develop a method, which will have the error of order h2 for the velocity. The more accurate velocity approximation will advance capabilities of the existing simulation tools for computational fluid dynamics, transport in porous media and other diffusion type problems.
We begin with developing a new method for quadrilateral meshes with possible extension to polygonal meshes. Polygonal meshes allow a more efficient partitioning of the computational domain. The locally refined meshes (where an element can share an edge with more than one element), used frequently in complex simulations, are particular examples of polygonal meshes.
There are a few fundamentally different approaches to increase accuracy of numerical solutions. Finite volume and finite difference methods tend to increase the effective stencils of discrete operators. They impose severe restrictions on mesh smoothness and, usually, result in non-symmetric discretizations. The finite element and spectral element methods increase the number of unknowns inside each element. They impose severe restrictions on the shape of mesh elements. Other approaches use the Pade-type approximations and are not practical for polygonal meshes. The MFD method was originally designed for general polygonal meshes. It always results in a symmetric discretization.
The existing MFD method has (n+1) degrees of freedom for polygon with n vertices: the pressure is approximated by the average over each cell, while the velocity is approximated by an average (0-th moment) of the flux through each of n edges. The main idea for increasing accuracy in our method is to introduce n additional degrees of freedom for the unknown velocity, in the form of the 1-st moments of the flux through each edge.
Valentina Staneva
In our work we show how through solving a nonconvex minimization problem we can exactly reconstruct signals from very few measurements. This is equivalent to solving an underdetermined system of linear equations, which in general has an infinite number of solutions. What helps us find a unique solution is the fact that the signals we are interested in are usually sparse or can have a sparse representation. The importance of this result is that in many settings it is too difficult or too expensive to acquire enough information. Applications can be found in medical and seismic imaging, remote sensing, and sensor networks, among many other disciplines.
The field known as compressed sensing has recently developed a technique for exact signal recovery from fewer measurements than it was believed to be possible. The key to success is having a certain relationship between sparsity and number of measurements and solving an L_1-norm minimization problem. We can obtain the correct solution with even fewer measurements by minimizing the L_p-norm of the signal, where p lies in (0,1). So at the price of losing convexity we can reconstruct signals which are unrecoverable by current techniques. We give a geometric and probabilistic interpretation of why this is true.
Humberto C Godinez
Due to the computational power now available, more realistic mathematical models can be used to describe various physical phenomena. As the models improve new parameters are being introduced whose values are not accurately known, and may lack a strong connection to first principles. Determining the values of all parameters as accurately as possible can be a challenging and expensive endeavor, hence, identifying the most influential parameters is of great importance. Sensitivity analysis enables the identification of the most significant parameters in a model, thus providing valuable insight into which of these should be accurately determined.
The objective of our research is to perform a forward and adjoint sensitivity analysis, as well as to study other aspects of uncertainty quantification, in a new cloud resolving aerosol model. Sensitivity of the solution to the models various parameters, such as activation time scales, fall speed of water droplets as well as aspects of its discretization, such as artificial diffusion coefficients, will be investigated. An MPI-based parallel code is used for the simulation of the cloud resolving aerosol model. The sensitivity of the model with respect to the parameters of interest is computed using the SUNDIALS integration package. It is clear that the robustness of the model, with respect to the parameters, will depend on its sensitivity. A low sensitivity indicates that the solution is not drastically affected by perturbations in the parameters, such as small error on their estimation.
This work contributes to current research on sensitivity analysis and uncertainty quantification in new high-resolution models of cloud aerosols. In addition, it will bring insight into the influence of numerical discretization on the predictability of these models. Future work lies in computing the most sensitive parameters using Data Assimilation methods, such as 3D-Var, 4D-Var or Ensemble Kalman Filter (EKF). Data Assimilation methods use experimental or observational data available, as well as statistical information, to determine the values of the parameters that ``steer" the solution of the model to best fit the data.
Kevin Flores
Understanding the correlation of gene specific mutations and tumor development has important implications in cancer therapy. Recent empirical data have elucidated the candidate cancer genes responsible for carcinogenesis through mutation and expression analysis. This work has revealed the heterogeneities in genotype that encode cancers of the same malignancy grade, providing evidence for the existence of multiple mutational paths that a population of cancer cells can take to manifest itself as a disease. The cell genotypes that are present in a tumor affect the malignancy grade through their effect on the phenotypes of individual cells that the tumor is comprised of. We use a graph theoretical approach to connect the gene expression and mutation data to cell phenotype. We have constructed a gene regulatory network from the KEGG pathway database. This network includes most accurately and completely the relevant pathways that contain the known cancer genes, which in turn encode distinct cell phenotypes. We are analyzing the network to predict the sensitivity of cell signaling pathways that control cell growth and death to alterations caused by gene mutations. The prevalence of gene mutations show no correlation to simple measures of their equivalent representations in the network. Because of the lack of necessary reaction rate data to model any of the interactions, we turn to a network boolean dynamics model in which the state of proteins, represented by nodes in the network, are on or off and are updated in time using functions depending on the network connections. When all the nodes are updated simultaneously at each time step we find that the phenotypic output resulting from the deterministic network dynamics are insensitive to the candidate gene mutations. With nonsimultaneous updating we find that the state space of the dynamics becomes too large to sample using random initial conditions. We employ a specific type of monte carlo algorithm to determine the proportions of inital conditions that have attractors whose protein profiles(on or off) are classified into distinct phenotypes. We consider 4 distinct cell phenotypes: Proliferation, Apoptosis, Survival, and None of the above. With this algorithm we estimate that the entire state space, whose size is on the order of 2^800, can be sampled with less than the equivalent of 30,000 random initial conditions when we want to determine the proportions of the state space that lead to any of the predefined phenotypes. With this type of sampling we can determine whether changes in the network caused by mutations lead to altered proportions of states whose progression will end in the distinct phenotypes.
Lauren O'Malley
Studying the spread of epidemics on scale-free networks is a useful tool in understanding the behavior of real life epidemics, whether they be the spread of human diseases, or computer/email viruses. In these networks, the overlay topology (or connectivity graph) is crucial in determining the behavior of the spread of an epidemic, including defining the epidemic threshold, a value above which the disease spreads and becomes endemic. In scale-free networks with degree exponent <=3, the value of the epidemic threshold vanishes in the limit of infinite (or very large) systems, due to the presence of "hubs". Yet for real-life systems there is a finite limit, not only to the number of nodes, but for the number of connections each node can acquire. It is therefore worthwhile to study systems with an imposed hard cutoff for the number of connections each node can have. We investigate the spread of epidemics on such scale-free topologies with hard cutoffs (i.e. there are not any major hubs) and the effect of these hard cutoffs on the epidemic threshold using an SIS (Susceptible-Infected-Susceptible) model of epidemics. We find that for small values of the cutoff (relative to the network size), the value of the epidemic threshold increases significantly, most likely due to the restriction in the formation of hubs.
Niall Mangan
Fundamental microscopic laws such as Newtons laws and quantum mechanics are reversible; however, most microscopic systems give rise to irreversible behavior. The origin of irreversible behavior is therefore recognized as an important issue and persistent theme in physics. We study a system which undergoes transitions between reversible and irreversible behavior in order to better understand the onset of irreversibly and loss of predictability.
When a magnetic field is applied to a type II superconductor it penetrates the material in the form of quantized magnetic flux lines called vortices. Using numerical simulations, we study vortices shaken by an ac drive in the presence of a random pinning substrate, made up of attractive parabolic wells. Even in the absence of thermal fluctuations, we observe both reversible and irreversible regimes of vortex motion. The irreversible behavior arises as a result of increasing interaction between the vortices. Intermittent transitions between the two regimes occur at a frequency which is determined by the vortex density as well as the amplitude and period of the driving force. By characterizing this system we show how irreversibility emerges statistically from a deterministic many-body dynamical system.
Eben Kenah
One of the most important goals of infectious disease epidemiology is to design efficient interventions to prevent or contain epidemics. In a network-based ``Susceptible-Infectious-Removed" (SIR) epidemic model, infection is transmitted across edges between nodes in a network called the ``contact network". The most effective vaccination strategy for these models is to target nodes with the highest degree (number of neighbors) in the contact network. But can we really have the same optimal vaccination strategy for all diseases spreading on the same network? In this talk, we present an alternative strategy based on a mapping from a stochastic SIR model to a directed random network that we call the ``epidemic percolation network" (EPN). Above the epidemic threshold, the EPN contains a ``giant strongly-connected component" (GSCC), a unique largest group of nodes in which every node can be reached from every other node by following a series of edges. We show that targeting nodes in the GSCC reduces both the probability and final size of an epidemic more rapidly than targeting high-degree nodes in the contact network, particularly in models with substantial heterogeneity in infectiousness and susceptibility. Another important advantage of our approach is that it applies to all time-homogeneous SIR models, including fully-mixed models (which still dominate infectious disease research). The concept of the EPN gives us a unified theoretical framework for SIR epidemic models that may have extremely important practical applications.
Jing Ai
Connectivity is a fundamental problem in wireless ad-hoc and sensor networks, and has been extensively studied. None of these studies, however, has explicitly incorporated traffic knowledge other than just simply aiming to guarantee connectivity between any two nodes. We raise a question that whether the network performance can be improved by providing traffic-aware connectivity. As a preliminary analysis, we study a one-dimensional wireless sensor network for rare event detection. In this scenario, traffic pattern is fixed in that all sensor nodes are responsible to transmit their detected-event back to the sink node. Moreover, it is critical for the sink node to be notified by the detected event as soon as possible (i.e., minimizing end-to-end delay) and it is also desirable that the network can last as long as possible (i.e.,minimizing energy consumption). However, these two objectives are contradictive and thus we characterize the optimal trade-off between those two metrics as a benchmark. First, we present the deterministic transmission-power scheme as the representative scheme widely used in the literature. Then, as a lightweight and dynamic coordination protocol, we present our proposed randomized transmission-power control scheme, where each node's PMFs on controlling transmission power are dependent on each other somehow yet they still behave independently. Through extensive simulations, we surprisingly find that as long as these PMFs are properly calculated, the randomized transmission power scheme can always achieve better energy consumption, end-to-end delay and connectivity all simultaneously and than those of the deterministic one.
G. Alonzo Vera
A sparse representation is an adaptive signal decomposition consisting of a linear combination of atoms from an overcomplete dictionary, where the coefficients of the linear combination are optimized according to some sparsity criterion. Denoising can be perform by allowing a misfit equal to the desired noise variance (between the original and reconstructed signals) when calculating the signal's sparse reprensentation.
On the other hand, Total Variation (TV) regularization is a widely used method to directly calculate a signal's denoised version. TV regularization is known to be exceptionally effective when applied over blocky signals.
Although both techniques solve the same problem, their philosophy is fundamentally different. It have been shown in the related literature that BP (specifically BP denoising) and TV are equivalent in the 1-dimensional case. The 2-D case is non-trivial and the equivalence has not been proven.
We aim to show that Total Variation can be used as an adaptive dictionary for sparse representation (i.e. Basis Pursuit), showing an equivalence between the two methods in the case of 2-D signals. Software routines will be written to test our hypothesis and increase the functionality of the NUMIPAD (numerical methods for inverse problems and adaptive decomposition) software being developed at T-7.