Mathematical Modeling and Analysis
Mathematical Modeling and Analysis
CNLS conference room, 4pm-5pm. Each speaker has 20 minutes plus 10 minutes for questions.
Kevin Flores
We model the competition in a tumor between cells with and without an important gene mutation, the p53 gene inactivation. The primary function of the p53 gene is to stop cell proliferation by inducing cell death. Here, we only consider the p53 gene regulation of cell death under low oxygen conditions. We use a multiscale model of avascular tumor growth: a lattice Monte Carlo model to describe cellular dynamics, and reaction-diffusion equations to describe the chemically dynamic tumor environment. With this model we can determine whether mutated p53 cells are upregulated by the low oxygen environment caused by the tumor formation itself. We find that the low oxygen environment increases the growth rate of the p53 mutated cells compared to their unaltered counterparts. The simulations also predict a change in the maximum growth rate of the tumor and final tumor saturation size due to the transition from unaltered p53 cell domination to p53-mutated cell domination. The p53 mutation also suppresses cell death induced by radiation and chemotherapy. By predicting the proportion of p53 mutated cells in a solid tumor, we can quantify how much resistance a solid tumor may have to cancer therapy.
Ignacio Rozada
We use a multiscale approach for the problem of simulating dynamical processes on large complex networks. We first consider the problem of efficiently partitioning networks in two equally sized pieces while minimizing the cut. Network partitioning is very important when performing simulations on very large networks in computer clusters. We present a fast multiscale algorithm that computes the eigenvectors associated with the smallest eigenvalues of the Laplacian matrix of large complex networks. These eigenvectors are used to obtain the minimum cut. We use a coarsening method specifically designed for networks with irregular degree distributions. The coarsening method preserves the degree distribution, and simple dynamical processes behave very similarly in both the coarse and the original networks. We compare multiscale algorithms using different coarsening methods, and point to some interesting applications.
Karima Nigmatulina
This talk will focus on evaluating how well prepared Southern California hospitals are for pandemic influenza. We apply the lessons learned from this specific scenario to develop nationally applicable results. I will evaluate the effectiveness of several intervention strategies and behavioral changes in decreasing the cumulative number of bed shortages. Our results show that behavioral changes can reduce the potential shortfall of hospitals during pandemic influenza and that aggregation to the metropolitan level underestimates the severity of bed shortages at individual hospitals.
In order to assess the preparedness of 220 hospitals in six Southern California counties with a population of over 18 million individuals, we used the output of an agent based simulation model, EpiSims. We examined different hospitalization rates and varying hospitalization durations throughout our computations. Our observations led to the development of a simple model of hospital stay duration that captures the behavior of more complex models.
The capacity of health-care systems to handle patient visits during normal seasonal influenza outbreaks has been diminishing. Pandemic influenza likely will have much higher morbidity and attack rates than seasonal flu. Preparing ahead of time for an influenza pandemic is essential for public health officials and hospital administrators.
Valentina Staneva
any current models for image restoration are based on the assumption that the noise in images is Gaussian. In our work we extend existing algorithms to account for different kinds of noise, by incorporating some information about the noise distribution. We consider an optimization-based approach that denoises an image by reducing its total variation. While such an approach preserves edges in an image, the computation turns out to be more challenging. For the implementation of our model, we use a fast numerical scheme related to Newton's method, but simpler. The results we obtain for images with Poisson noise show that the algorithm correctly and efficiently reconstructs the original data. Poisson noise is typical for X-ray radiographs, so our reconstruction technique should find many applications throughout the lab, as well as in other settings employing computed tomography.
Diogo Bolster
Several industrial toxins are highly volatile and as such often require a full multiphase modelling approach when studying their flow. In this talk we will focus our attention on the spill of such a volatile compound over a porous medium.
While many analytical solutions for the infiltration of a fluid into a porous medium exist, few account for volatility since in many practical cases, such as water irrigation, it is insignificant. However in the case of many toxic organic compounds, often referred to as DNAPLs (Dense Non Aqueous Phase Liquids) or VOCs (Volatile Organic Compounds), this will not hold true. By accounting for volatility we hope to achieve a better understanding of the flow of such toxins through the ground, which would to aid in the prevention of groundwater contamination and cleanup of toxic spills.
After making several simplifying assumptions we try to find the simplest possible model that still accounts for the fluid's volatility. Then, using asymptotic perturbation techniques, we find analytical solutions to a one dimensional problem. For small levels of volatility our solutions reduce to previously published solutions for nonvolatile fluids. However, when volatility is accounted for some interesting further dynamics are observed. Our solutions are then used to identify portions of parameter space where neglecting volatility may be acceptable and others where it should be accounted for.
Null Riemannian Geometry and Metrics for Comparing Data
Special time: 1:00 pm
I will present recent and ongoing work on the theoretical and computational aspects of Null Riemannian Geometry (NRG). In general, NRG is a mathematical framework for building metrics on data sets which can ignore unimportant variances. We suppose the data lives on some (relatively) smooth manifold X, and equip X with a non-negative, symmetric 2-tensor field N. This field behaves essentially like a Riemannian metric, except that certain vectors at each point of X may be allowed to have zero length, i.e. sqrt(N(v,v))=0 for some vectors v in the tangent bundle. Examples of potential applications of NRG include face recognition and other classification problems where we would like our metric ignore certain types of differences between data points, and constructing success measures for simulations of dynamical systems which do not penalize variances due to instabilities in the physics. This work was done jointly with my mentor Kevin R. Vixie, and David W. Dreisigmeyer.
Christian Ketelsen
We describe a stable, efficient, parallel algorithm for the solution of symmetric, positive definite, tridiagonal linear systems. Our method is a multilevel recursive technique based on partitioning ideas. The algorithm is well suited for problems on distributed memory machines with a very large number of processors. It can be shown that our method has a communication complexity that is logarithmic in the number of processors.
The need to solve large tridiagonal systems of linear equations arises in many numerical analysis applications. In particular, tridagonal solvers are the workhorse of robust multigrid solvers, such as the parallel BoxMG code, applied to problems on structured grids. In this talk we will describe the implementation of our algorithm in the context of BoxMG.
Xilin Shen
Image restoration is an important preprocessing step for many applications. In this talk, I will present an example-based approach for image denoising.
Restoring an image can be viewed as a minimization problem. In most approaches, image properties such as sharp edges or plausible looking texture are being modelled analytically. However, the richness of real-world images is often oversimplified by analytical models. Therefore we look for an example-based approach, where image properties are learned from a training set of images. The algorithm then applies the knowledge to predict fine details and remove noise of other images. The idea of an example-based approach is appealing because we no longer need to estimate a prior model for images.
In our experiment, we work with one standard database of handwritten digits and one dataset of fingerprint images. Some initial results we obtain for noisy fingerprint images show that the example-based approach efficiently removes the noise, while preserving edges and texture. This approach can be applied to other image restoration tasks too, for example, to enhance images with poor resolution. A restored image obtained by better denoising or resolution enhancement is not only visually more pleasant, but also improves the performance of successive tasks such as recognition and classification.
Jesse Berwald
Examples of networks abound in daily life. I will give an overview of the mathematical tools I have been using to investigate questions that arise with respect to clustering in networks as well as some preliminary results.
Networks come in many sizes. From T-7's server-workstation computer system to the decentralized power grid of the Western United States, all played an integral part in the ability to produce this simple abstract. We hope to develop efficient and robust algorithms to identify geometric structure in large scale networks. This knowledge can be used to apply clustering algorithms to a wide variety a real-world issues. Examples include clusters within scientific collaboration networks, to the identification of social structures in a population in the hopes decreasing the spread of disease.
Maxim Shkarayev
We found a new bisoliton type solutions in optical fiber links with dispersion management, modeled by the Nonlinear Schrodinger equation with dispersion varying periodically along the fiber. We discovered a parametric bifurcation of bisolitons and developed new iterative method with polynomial correction for calculation of these solutions.
Danail Vassilev
In many physical models there are different processes taking place in different parts of the simulation domain. I will present a coupled system of partial differential equations that describes the interaction between surface water and groundwater. In our model we consider an incompressible fluid that can flow across an interface into a porous medium region. A suitable choice of interface conditions leads to well-posedness of the coupled problem. This model can be used to predict how pollution discharged into lakes and rivers makes its way into the water supplies.
One of our goals is to extend the existing numerical schemes to meshes that are suited to approximate the complex geometries arising from the geological models. Due to the highly heterogeneous nature of porous rock formations, a suitable approach to model such geological structures is to use polygonal and polyhedral meshes with curved and degenerate cells. The mimetic finite difference method was designed to handle these meshes, which makes it a very good choice for the porous medium region. In the fluid region the discontinuous Galerkin method is used because of its flexibility and easy implementation.
I will give details on the discretization and the method to solve the associated algebraic system. I will also introduce domain decomposition, a technique that I used to solve the coupled problem on a parallel cluster, and present convergence results from previous numerical experiments.
Sarah Hews
Dengue fever, a vector-borne disease, thrives in tropical and subtropical regions worldwide. We carried out a retrospective analysis of the 2002 dengue epidemic in Colima, Mexico, located on the central Pacific coast. We included 4040 dengue cases diagnosed at the hospitals of the Mexican Institute of Public Health. We analyzed a spatial database containing demographic, epidemiological, and clinical information. We looked for statistical associations between clinical symptoms and gender or age. The distribution of the time from the onset of symptoms to diagnosis was analyzed and associated with other clinical and demographic variables.
We also studied the association between dengue incidence and climatological variables via correlation and multiple regression analyses. Lagged effects from climatological variables to dengue incidence were also explored. Our results were compared with other dengue studies around the world.
Furthermore, we also studied the effects of spatial heterogeneity across the state in the transmissibility of dengue using a minimal model of dengue transmission. Finally, we discuss the implications of our results in terms of dengue control and surveillance.
Alonzo Vera
Signals with a sparse representation are more easily analyzed and processed, thus the search for such a representation has become the focus of intensive research. In this talk we will focus on an efficient implementation of an optimization principle for signal decomposition known as basis pursuit.
An algorithm that implements the basis pursuit principle will search and find the represention whose coefficients have the smallest L1 norm, which in general is a good aproximation of the sparsest one. Such a search leads to large scale optimization algorithms. Primal-dual interior-point methods offer a polynomial-time solution for such a problem, with impact on a wide number of signal processing applications, for example, denoising.
We aim to implement a parametrizable library for fast and efficient implementation of such methods, and test its suitability for practical applications such as signal and image denoising. Valuable insight on implementation issues is provided as well as rough guidlines for application-dependent parameter choosing.
Ethan Coon
A multilevel upscaling (or MLUPS) method for representing fine scale data on coarse scale models is presented to investigate the role of highly heterogeneous media in multiphase geophysical porous flow. This method accurately captures fine scale information about the background medium while enabling computation to be carried out on much coarser-scale domains. In many upscaling methods, the fine scale continuum model is assumed to be valid on coarser levels while rock properties are averaged to form an upscaled material. In contrast, MLUPS forms a true hierarchy of models which accurately represent the coarse scale model on all levels. This method results in basis functions which incorporate more of the fine scale information at each level.
Here, we apply the MLUPS method to the flooding of oil reservoirs to form quantitative predictions of oil production. A standard implicit pressure, explicit saturation, time stepping scheme is combined with the MLUPS finite volume element spatial discretization to model the two-phase flow system. We will present preliminary results of an industry benchmark problem for two-phase porous flow.
Lyubov Chumakova
In this talk I will present an exact analytic result for the forced turbulence with helicity in the limit of infinite Reynolds number---the 2/15 law.
There are very few exact results for the homogeneous isotropic turbulence. One of them is the Kolmogorov's 4/5 law. Since this result is exact, it provides a boundary on numerous theories of turbulence---they should either satisfy the 4/5 law, or explicitly violate the assumptions used in deriving it. In studies of homogeneous isotropic turbulence with no mirror-symmetry, the existing exact result is the 2/15 law, which stems from helicity conservation. However, while the majority of numerically simulated flows have to be forced to reach a high enough Reynolds number, up to now the 2/15 law was proved only for the case of decaying turbulence. Our studies cover that gap and show that the 2/15 law is indeed recovered in the case of forced turbulence.
Yilin Wu
I will present my research on the motion mechanisms of myxobacteria. Our modeling tries to explain a puzzling problem about their cooperative swarming. The study may also help us to understand a large variety of swarming driven by biological interactions.
Myxobacteria are a special type of soil bacteria. When growing on a solid medium with sufficient nutrients, they grow as a swarm and spread outwards. Individual myxobacteria cells have two types of motion mechanism, A(dventurous)-motility and S(ocial)-motility. The two types of motility mutants swarm at different rates. Surprisingly, when the two motility systems work together, the maximum swarming rate is quite larger than the sum of the two separate rates. In order to find out how the cooperative movement can achieve higher swarming efficiency, we build computational models for A- and S-motilites. From preliminary simulation results, we have found out that the role of S-motility may be to align neighboring cells and thus reduce collisions between them.
Milos Ivkovic
In this work we use a new approach to model error events in long-haul optical fiber transmission systems. Existing approaches for obtaining probability density functions (PDFs) rely on numerical simulations or analytical approximations. Numerical simulations make far tails of the PDFs difficult to obtain. Analytical approximations are often inaccurate, as they neglect nonlinear interaction between pulses and noise.
Our approach is to use the instanton method from quantum mechanics to model far tails of the PDFs, and use numerical simulations to refine the middle. We combine them using an orthogonal polynomial expansion constructed specifically for this problem. We will demonstrate the approach with an example for a specific submarine transmission system.
Evelyn Lunasin
In this talk I will discuss analytical and numerical results obtained in the study of two-dimensional Navier-Stokes-$\alpha$ turbulence model. The use of numerical models for the Navier-Stokes equations of fluid dynamics is an established practice in the study of turbulent flows. The calculation of flow from a model, instead of the underlying Navier-Stokes equations, allows for the use of less computing resources for a given flow.
The Navier-Stokes-$\alpha$ model uses a smoothed velocity field to transport a rough velocity field. The expectation is that the smoothed field will be sufficent to recover many of the relevant properties of the flow. Our analysis shows three possible effects of this model on the energy spectrum of two-dimensional turbulence. These three effects stem from the fact that there are three physical timescales in the model equations: one from the smoothed field, the second from the rough field, and a third from a combination of the two. Our numerical simulations reveal that the timescale corresponding to the rough field is responsible for the dominant effects of the model in the small scales. The implication is that the rough field continues to affect the smooth field even though the latter is the postulated candidate for the modeled turbulence.
The main goal of this project is to understand the small scale dynamics of this model. This knowledge will help us determine how to tune the parameter alpha appropriately to simulate the large scale dynamics of fluid flows at a more accessible computational requirement.
Jasleen Kaur
The co-authorship network of scientists represents a prototype of complex evolving networks. It is now becoming possible to study quantitatively the social collective dynamics of innovation and scientific discovery. Quantitative progress in this field is of great interest to foment innovation and plan more efficient funding.
We aim to track and characterize the temporal evolution of the social network (graph) associated with five scientific discoveries namely Prions, Inflation, H5N1 Influenza, Quantum Computing and Carbon nanotubes. This involves the identification of publication sources, the automated extraction of author and publication information, the construction of graphs, and their characterization in terms of diagnostics from complex networks.
The novelty of our project lies in the analyses of the dynamics of co-authorship networks for scientific discoveries in the light of temporal expansion of the field, especially at the onset of a new discovery or breakthrough.
We formulate the hypotheses for sudden increase in number of authors and publications after a breakthrough in scientific field and emergence of giant component based on relation between number of authors, publications, ratio of large component and number of nodes as well as signatures in other structural quantities.
Joyce Lin
We investigated analytically and numerically the impact of nonlinear diffusion and dispersion on various nonlinear, conservative PDEs. These included the Korteweg-de Vries, Burgers, and regularized long-wave equations. Such evolution equations with nonlinear terms balance convective forces with diffusion or dispersion, and give rise to stable exotic waves with coherent structure.
I will give a brief introduction on how to solve for special solutions of these equations and provide video clips (with popcorn!) to demonstrate the rich dynamic behavior. We will see traveling waves and diffusion fronts with compact support. We will also present stable solitary waves of nonintegrable equations that, much like solitons, collide nearly elastically.
Santanu Chatterjee
We developed a general hyperdynamics method based on time-space transformation to simulate long time scale dynamics of water. Conventional molecular dynamics simulations are typically limited to a time scale of less than a microsecond. Therefore many interesting slow processes in chemistry, physics, biology and material science can not be simulated directly. Recently, new methods have been proposed including kinetic Monte Carlo, path sampling algorithms to study slow processes. But these methods require prior knowledge of the system that is often not available. In hyperdynamics, a bias potential, if properly applied, allows us to capture longer time scale and study rare transitional dynamics without assuming any prior knowledge of the system. In our method, stable and transition regions are distinguished by the two-body entropy of the system. Initial studies show us the applicability of our algorithm to study the vapor-liquid phase transition in water.
Amy Bauer
In this talk I will describe a new cell-based model of tumor-induced angiogenesis. Tumor-induced angiogenesis is the formation of new blood vessels from existing vasculature in response to chemical signals from a tumor. This process marks the pivotal transition from avascular to vascular tumor growth, a progressive stage of cancer beyond which cancer becomes extremely difficult to treat and survival rates decrease.
Our model is structured in terms of the dynamics occurring at the extracellular and intercellular levels. At the extracellular level, the model describes diffusion, uptake, and half-life decay of tumor- secreted pro-angiogenic factor (VEGF). At the cellular level, the model uses a discrete lattice Monte Carlo algorithm based on system- energy reduction to describe cell migration, growth, division, cellular adhesion, and the evolving structure of the tissue. This model provides a quantitative framework to test hypotheses on the biochemical and biomechanical mechanisms that cause tumor-induced angiogenesis.
Results from numerical simulations will be presented that demonstrate the model's ability to capture realistic vascular structures and more complex events such as branching (vessel bifurcation) and anastomosis (vessels fusing). Our studies show that different VEGF gradient profiles dramatically affect vessel morphology. We also found that proliferation further from the tip of the vessel yielded faster average rates of vessel extension. Results also suggest that inhomogeneities in the tissue may be important mechanisms leading to vessel branching.
Gowri Srinivasan
This is a study of the phenomenon of light localization in single mode optical fiber arrays as a result of deterministic and random linear and nonlinear effects. The competing effects of non-linearity and randomness in the propagation of light in an optical fiber array have been studied experimentally in recent years. Here we use both numerical and analytical methods to study a hexagonal optical fiber array similar to that used in the experiments. As an approximation, the relevant randomness is only on the linear part of the model. The effect of varying the input power on localization is also studied.
Matthew Buoni
In this talk we present a new, fast algorithm for computing gravitational forces in protoplanet simulations. The discovery of about 170 extrasolar planets over the past decade has raised many fundamental questions about the basic processes that determine planet-system evolution. Numerical simulations have revealed a rich variety of subtle, sometimes competing physical processes on protoplanets' dynamical evolution. In the protoplanet simulations studied here, we model the planet as a discrete point mass embedded in a disk of rotating mass, modeled as a gas. The compressible gas flow equations are solved on the disk with an external force term due to gravity from the planet, while the planet is evolved due to gravitational forces from the disk. Disk self-gravity forces are usually either neglected or crudely approximated (as having only radial dependence, for example), in spite of their physical importance. This is because computing them exactly is very computationally expensive, slowing down simulations by more than 1000 times on highly resolved grids. Here we present a parallel, fast, force-summation algorithm specifically tailored to this calculation. We also discuss modeling the three-dimensional effects of finite disk height within the context of a two-dimensional simulation.
Alexander Danilov
I will talk about our effort to interface an advancing front mesh generator with an open source CAD system. The main purpose of this effort was to get a new simple way to describe complex regions and mesh them using an automatic mesh generator. We used an advancing front mesher developed at Moscow State University. For the CAD system, we used Open CASCADE as a freely available modeler. The two were interfaced using a software called the Common Geometry Module (CGM). CGM can work with different geometric modelers and provides all geometric information in a general way. The use of CGM enables us to switch between different modelers without changing our mesh generator.
I will present some examples of meshing process of a model from Open CASCADE.