Department of Mathematics

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Efficient approximation of the stationary solution to the chemical master equation
(University of Alabama Libraries, 2019) Reid, Brandon M.; Sidje, Roger B.; University of Alabama Tuscaloosa
When studying chemical reactions on the cellular level, it is often helpful to model the system using the continuous-time Markov chain (CTMC) that results from the chemical master equation (CME). It is frequently instructive to compute the probability distribution of this CTMC at statistical equilibrium, thereby gaining insight into the stationary, or long-term, behavior of the system. Computing such a distribution directly is problematic when the state space of the system is large. To alleviate this difficulty, it has become popular to constrain the computational burden by using a finite state projection (FSP), which aims only to capture the most likely states of the system, rather than every possible state. We propose efficient methods to further narrow these states to those that remain highly probable in the long run, after the transient behavior of the system has dissipated. Our strategy is to quickly estimate the local maxima of the stationary distribution using the reaction rate formulation, which is of considerably smaller size than the full-blown chemical master equation, and from there develop adaptive schemes to profile the distribution around the maxima. The primary focus is on constructing an efficient FSP; however, we also examine how some of our initial estimates perform on their own and discuss how they might be applied to tensor-based methods. We include numerical tests that show the efficiency of our approaches.
Heuristics for large-scale semidefiniite programming for the K disjoint clique problem
(University of Alabama Libraries, 2018) Barnes, Alexander Putnam; Ames, Brendan; University of Alabama Tuscaloosa
Large-scale semidefinite programming has many applications, including optimal control, computer vision, and machine learning. However, current algorithms for solving semidefinite programs (SDPs) can be time consuming and memory intensive. We look at new heuristics for solutions of the K disjoint clique problem. We model the K disjoint clique optimization problem as a SDP based on non-convex low rank factorization, and solve using Alternating Direction Method of Multipliers, augmented Lagrangian, and alternating direction. We will present numerical results illustrating the efficacy of our approach for clustering of real and simulated data and pose future questions of interest.
Implementation of some parallel algorithms arising in sparse matrix and other applications
(University of Alabama Libraries, 2017) Lass, David Francis; Sidje, Roger B.; University of Alabama Tuscaloosa
Generally the processing time of a program can be improved by splitting the program into several portions and executing each portion on its own computing core. This process allows us to take advantage of as much of the computer system hardware as we can. However, this also can cause some complexities to arise such as dependency issues. Sparse matrices are, in the most general form, matrices with a large number of zero entries relative to their number of nonzero entries. These matrices are commonly occurring in applications like in a Partial Differential Equation (PDE). Naturally, systems of linear equations involving these sparse matrices are common and there are several methods for solving these systems. As most of these matrices are relatively large, it can be inefficient to solve them directly. Instead many ways to solve these systems involve the use of iterative methods. Combining parallel processing with these iterative methods, we can quickly and efficiently solve large sparse systems. The following thesis contains the use of two types of parallel processing, provides applications of both, shows the value of using iterative methods with sparse matrices, and examines solving partial differential equations using several iterative methods with parallel processing.
Parallel stochastic simulation of biochemical reaction systems
(University of Alabama Libraries, 2019) Cook, Keisha; Sidje, Roger B.; University of Alabama Tuscaloosa
Chemical reactions of various scales occur in nature and in our bodies. As technology has improved, researchers have gained access to in-depth knowledge about the relationships between the moving parts of a chemical reaction system. This has led to a multitude of studies by researchers who strive to understand the background and behavior of these systems both experimentally and mathematically. Computational biology allows us the opportunity to study chemical processes from a model-based approach, in which algorithms are used to simulate and interpret biological systems to validate our models with data when available. A number of biological processes such as interactions between molecules, cells, organs, and tissues in the body can be modeled mathematically, making it useful in medicine, biology, chemistry, biophysics, statistics, genomics, and more. Mathematically, biochemical processes can be modeled deterministically and stochastically. The Reaction Rate Equations (RREs), in the form of a system of ODEs, are used to model deterministically. The Chemical Master Equation (CME), in the form of a Markov Chain, is used to solve stochastically. When the CME becomes computationally expensive, methods such as the Stochastic Simulation Algorithm (SSA), the Tau-Leap Method (Tau-Leap), the First Reaction Method (FRM), and the Delay Stochastic Simulation Algorithm (DSSA) are used to simulate the change in population of the species in a system over time. For accuracy when examining the resulting data, models are simulated many times in order to produce probability distributions of the involved species. An increase in the size and complexity of a system, leads to an increase in the computational time needed to simulate a model. Parallel processing is used to speed up the computational time of simulating biochemical processes via the aforementioned methods. The numerical results can be illustrated for various models found in science.
Pseudo-transient ghost fluid methods for the Poisson-Boltzmann equation with a two-component regularization
(University of Alabama Libraries, 2019) Ahmed Ullah, Sheik; Zhao, Shan; University of Alabama Tuscaloosa
The Poisson Boltzmann equation (PBE) is a well-established implicit solvent continuum model for the electrostatic analysis of solvated biomolecules. The numerical solution of the nonlinear PBE is still a challenge due to its strong singularity by the source terms, dielectrically distinct regions, and exponential nonlinear terms. In this dissertation, a new alternating direction implicit method (ADI) is proposed for solving the nonlinear PBE using a two-component regularization. This scheme inherits all the advantages of the two-component regularization and the pseudo-time solution of the PBE while possesses a novel approach to combine them. A modified Ghost Fluid Method (GFM) has been introduced to incorporate the nonzero jump condition into the ADI framework to construct a new GFM-ADI method. It produced better results in terms of spatial accuracy and stability compared to the existing ADI methods for PBE and it is simpler to implement by circumventing the work necessary to apply the rigorous 3D interface treatments with the regularization. Moreover, the stability of the GFM-ADI method has been significantly improved in comparing with the non-regularized ADI method, so that stable and efficient protein simulations can be carried out with a pretty large time step size. Two locally one-dimensional (LOD) methods have also been developed for the time-dependent regularized PBE, which are unconditionally stable. Finally, for numerical validation, we have evaluated the solvation free energy for a collection of 24 proteins with various sizes and the salt effect on the protein-protein binding energy of protein complexes.
Sparse regression of textual analysis
(University of Alabama Libraries, 2018) Carter, Phylisicia N.; Ames, Brendan; University of Alabama Tuscaloosa
We consider sparse regression techniques as tools for classification of sentiment within Twitter posts. Analysis of Twitter usage suffers from several unique challenges. For example, the 140-character limit severely limits the amount of information contained in each post; this causes most tweets to contain an extremely small subset of the dictionary, presenting challenges for learning schemes based on dictionary usage. To remedy this undersampling issue, we propose usage of penalized regression. Here, we employ logistic regularization to avoid any degeneracy caused by the sparse usage of the dictionary in each tweet, while simultaneously learning which terms are most associated with each sentiment. Accelerated sparse discriminant analysis is also used to combat the issues of degeneracy and overfitting of the training data while providing dimension reduction. As illustrative examples, we employ sparse logistic regression to classify tweets based on the users’ perception of a connection between vaccination and autism, and we examine the Twitter users' sentiment of the use of autonomous cars.
A super-Gaussian Poisson-Boltzmann model for electrostatic solvation energy calculation: smooth dielectric distributions for protein cavities and in both water and vacuum states
(University of Alabama Libraries, 2018) Hazra, Tania; Zhao, Shan; University of Alabama Tuscaloosa
Calculations of electrostatic potential and solvation energy of macromolecules are essential for understanding the mechanism of many biological processes. In the classical implicit solvent Poisson-Boltzmann (PB) model, the macromolecule and water are modeled as two-dielectric media with a sharp border. However, the dielectric property of interior cavities and ion-channels is difficult to model in a two-dielectric setting. In fact, whether there are water molecules or cavity-fluid inside a protein cavity remains to be an experimental challenge. Physically, this uncertainty affects the subsequent solvation free energy calculation. In order to compensate this uncertainty, a novel super-Gaussian dielectric PB model is introduced in this work, which devices an inhomogeneous dielectric distribution to represent the compactness of atoms and characterize empty cavities via a gap dielectric value. Moreover, the minimal molecular surface level set function is adopted so that the dielectric profile remains to be smooth when the protein is transfer from water phase to vacuum. A nice feature of this new model is that as the order of super-Gaussian function approaches the infinity, the dielectric distribution reduces a piecewise constant of the two-dielectric model. Mathematically, a simple effective dielectric constant analysis is introduced in this work to benchmark the dielectric model and select optimal parameter values. Computationally, a pseudo-time alternative direction implicit (ADI) algorithm is utilized for solving the super-Gaussian PB equation, which is found to be unconditionally stable in a smooth dielectric setting. Solvation free energy calculation of a Kirkwood sphere and various proteins is carried out to validate the super-Gaussian model and ADI algorithm. One macromolecule with both cavity-fluids and empty cavities is employed to demonstrate how the cavity uncertainty in protein structure can be bypassed through dielectric modeling in the biomolecular electrostatic analysis.
Weakly nonlinear convection induced by the sequestration of co$_2$ in a perfectly impervious geological formation
(University of Alabama Libraries, 2017) Vo, Liet Anh; Hadji, Layachi; University of Alabama Tuscaloosa
This thesis examines the problem of convection that occurs during the geological sequestration of carbon dioxide. The mathematical model accounts for both diffusion and convection in a geological formation that has anistropic permeability and anisotropic carbon diffusion. The permeability is modeled as a decaying exponential to describe its slow decrease with depth. We also account for a first order reaction between CO$_2$ and porous medium. We investigate the onset of convection and its weakly nonlinear evolution. We consider a base state that mimics a Rayleigh-Taylor configuration with a carbon-rich heavy layer overlying a carbon-free lighter layer and determine the thickness at which the configuration becomes unstable. The analysis is performed using the classical normal modes and the weakly nonlinear analysis is performed using long-wavelength asymptotics. We derive the threshold instability conditions and associated flow patterns. Our analysis leads to the derivation of the convective flux conditions at the interface and the resulting fingering patterns. Finally, we put forth the conditions expressed in terms of formation and fluid parameters of parameters for the onset of convective shutdown.

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Browsing Department of Mathematics by Author "Ames, Brendan"

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