Theses and Dissertations - Department of Mathematics

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    Weighted Norm Inequalities for the Maximal Operator on Variable Lebesgue Spaces Over Spaces of Homogeneous Type
    (University of Alabama Libraries, 2020) Cummings, Jeremy; Cruz-Uribe, David; University of Alabama Tuscaloosa
    Given a space of homogeneous type $(X,\mu,d)$, we prove strong-type weighted norm inequalities for the Hardy-Littlewood maximal operator over the variable exponent Lebesgue spaces $L^\pp$. We prove that the variable Muckenhoupt condition $\App$ is necessary and sufficient for the strong type inequality if $\pp$ satisfies log-H\"older continuity conditions and $1 < p_- \leq p_+ < \infty$. Our results generalize to spaces of homogeneous type the analogous results in Euclidean space proved in [14].
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    Topological transformation groups with a fixed end point
    (University of Alabama Libraries, 1965) Gray, William Jesse; University of Alabama Tuscaloosa
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    A compiler for the Bama-Bell floating point interpretive system
    (University of Alabama Libraries, 1962) Gray, William J.; University of Alabama Tuscaloosa
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    A Classifying Family of Spaces for the Cohomology of Profinite Groups
    (University of Alabama Libraries, 2021) Lee, Evan Matthew; Corson, Jon; University of Alabama Tuscaloosa
    In the study of homological algebra, one useful tool for studying the cohomology of a discrete group is that group's classifying space. In some sense, the classifying space captures both the group itself and a description of its cohomology for any action of the group on any coefficient module. While some constructions for a classifying space also apply to topological groups, the relationship of the resulting space to the group's cohomology is unclear. Profinite groups are a special case of topological groups, determined entirely as the limits of inverse systems of finite, discrete groups. The goal of this work is to construct for profinite groups as close an analog as possible to the classifying space of a discrete group. In particular, we are interested in the construction for a finite, discrete group's classifying space achieved by first producing the nerve of the group as a category and then taking its geometric realization to obtain a space with isomorphic cohomology groups. We proceed by extending each step of this process to apply to a profinite group using inverse limits, followed by correcting for a lack of continuity (in the sense of compatibility with inverse limits) in singular cohomology by applying alternative cohomology theories to the resulting sequence of spaces. The end result has a promising isomorphism to the cohomology of the group, with the possibility of a further isomorphism.
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    Coordinate Descent Methods for Sparse Optimal Scoring and its Applications
    (University of Alabama Libraries, 2021) Ford, Katie Wood; Ames, Brendan P; University of Alabama Tuscaloosa
    Linear discriminant analysis (LDA) is a popular tool for performing supervised classification in a high-dimensional setting. It seeks to reduce the dimension by projecting the data to a lower dimensional space using a set of optimal discriminant vectors to separate the classes. One formulation of LDA is optimal scoring which uses a sequence of scores to turn the categorical variables into quantitative variables. In this way, optimal scoring creates a generalized linear regression problem from a classification problem. The sparse optimal scoring formulation of LDA uses an elastic-net penalty on the discriminant vectors to induce sparsity and perform feature selection. We propose coordinate descent algorithms for finding optimal discriminant vectors in the sparse optimal scoring formulation of LDA, along with parallel implementations for large-scale problems. We then present numerical results illustrating the efficacy of these algorithms in classifying real and simulated data. Finally, we use Sparse Optimal Scoring to analyze and classify visual comprehension of Deaf persons based on EEG data.
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    Augmented matched interface and boundary (AMBI) method for solving interface and boundary value problems
    (University of Alabama Libraries, 2021) Feng, Hongsong; Zhao, Shan; University of Alabama Tuscaloosa
    This dissertation is devoted to the development the augmented matched interface and boundary(AMIB) method and its applications for solving interface and boundary value problems. We start with a second order accurate AMIB introduced for solving two-dimensional (2D) elliptic interface problems with piecewise constant coefficients, which illustrates the theory of AMIB illustrated in details. AMIB method is different from its ancestor matched interface and boundary (MIB) method in employing fictitious values to restore the accuracy of central differences for interface and boundary value problems by approximating the corrected terms in corrected central differences with these fictitious values. Through the augmented system and Schur complement, the total computational cost of the AMIB is about $O(N \log N)$ for degree of freedom $N$ on a Cartesian grid in 2D when fast Fourier transform(FFT) based Poisson solver is used. The AMIB method achieves $O(N \log N)$ efficiency for solving interface and boundary value problems, which is a significant advance compared to the MIB method. Following the theory of AMIB in chapter 2, chapter 3 to chapter 6 cover the development of AMIB for a high order efficient algorithm in solving Poisson boundary value problems and a fourth order algorithm for elliptic interface problems as well as efficient algorithm for parabolic interface problems. The AMIB adopts a second order FFT-based fast Poisson solver in solving elliptic interface problems. However, high order FFT-based direct Poisson solver is not available in the literature, which imposes a grand challenge in designing a high order efficient algorithm for elliptic interface problems. The AMIB method investigates efficient algorithm of Poisson boundary value problem (BVP) on rectangular and cubic domains by converting Poisson BVP to an immersed boundary problem, based on which a high order FFT algorithm is proposed. This naturally allows for fulfilling a fourth order fast algorithm for solving elliptic interface problems. Besides the FFT algorithm, a multigrid method is also considered to achieve high efficiency in solving parabolic interface problems. Extensive numerical results are included in each chapter of the concerned problem, and are used to show the robustness and efficiency of AMIB method.
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    Regularization methods on solving Poisson’s equation and Poisson Boltzmann equation with singular charge sources and diffuse interfaces
    (University of Alabama Libraries, 2021) Wang, Siwen; Zhao, Shan; University of Alabama Tuscaloosa
    Numerical treatment of singular charges is a grand challenge in solving Poisson-Boltzmann (PB) equation for analyzing electrostatic interactions between the solute biomolecules and the surrounding solvent with ions. For diffuse interface models in which solute and solvent are separated by a smooth boundary, no effective algorithm for singular charges has been developed, because the fundamental solution with a space dependent dielectric function is intractable. In this research work, regularization formulations are introduced to capture the singularity analytically, which are the first of their kind for diffuse interface Poisson's equation and PB models. The success lies in a dual decomposition -- besides decomposing the potential into Coulomb and reaction field components, the dielectric function is also split into a constant base plus space changing part. Using the constant dielectric base, the Coulomb potential is represented analytically via Green's functions. After removing the singularity, the reaction field potential satisfies a regularized PB equation with a smooth source. Some diffuse interface models including a Gaussian convolution surface (GCS) are also introduced. The GCS efficiently generates a diffuse interface for three-dimensional realistic biomolecules. The performance of the proposed regularization is examined by considering both analytical and GCS diffuse interfaces, and compared with the trilinear method. Moreover, the proposed GCS-regularization algorithm is validated by calculating electrostatic free energies for a set of proteins and by estimating salt affinities for seven protein complexes. The results are consistent with experimental data and estimates of sharp interface PB models.
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    Adaptive pseudo-time methods for the Poisson-Boltzmann equation with eulerian solvent excluded surface
    (University of Alabama Libraries, 2021) Jones, Benjamin Daniel; Zhao, Shan; University of Alabama Tuscaloosa
    This work further improves the pseudo-transient approach for the Poisson Boltzmann equation (PBE) in the electrostatic analysis of solvated biomolecules. The numerical solution of the nonlinear PBE is known to involve many difficulties, such as exponential nonlinear term, strong singularity by the source terms, and complex dielectric interface. Recently, a pseudo-time ghost-fluid method (GFM) has been developed in [S. Ahmed Ullah and S. Zhao, Applied Mathematics and Computation, 380, 125267, (2020)], by analytically handling both nonlinearity and singular sources. The GFM interface treatment not only captures the discontinuity in the regularized potential and its flux across the molecular surface, but also guarantees the stability and efficiency of the time integration. However, the molecular surface definition based on the MSMS package is known to induce instability in some cases, and a nontrivial Lagrangian-to-Eulerian conversion is indispensable for the GFM finite difference discretization. In this paper, an Eulerian Solvent Excluded Surface (ESES) is implemented to replace the MSMS for defining the dielectric interface. The electrostatic analysis shows that the ESES free energy is more accurate than that of the MSMS, while being free of instability issues. Moreover, this work explores, for the first time in the PBE literature, adaptive time integration techniques for the pseudo-transient simulations. A major finding is that the time increment $\Delta t$ should become smaller as the time increases, in order to maintain the temporal accuracy. This is opposite to the common practice for the steady state convergence, and is believed to be due to the PBE nonlinearity and its time splitting treatment. Effective adaptive schemes have been constructed so that the pseudo-time GFM methods become more efficient than the constant $\Delta t$ ones.
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    On the Theory of Structures in Sets
    (University of Alabama Libraries, 1970) Hall, Japheth Jr.; University of Alabama Tuscaloosa
    In many branches of mathematics the property of being a subspace receives considerable attention. The subspaces of a given space might be regarded as values of a structure in a set, that is, a function P on the subsets of a set V such that P(X) ⊆ V for all X ⊆ V. Thus, structures in sets form a basis for an abstract treatment of the property of being a subspace.
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    Conjugate operator on variable harmonic Bergman space
    (University of Alabama Libraries, 2020) Wang, Xuan; Ferguson, Timothy; University of Alabama Tuscaloosa
    Complex analytic functions have astonishing and amazing properties. Their real parts and imaginary parts are deeply connected by the Cauchy-Riemann equations. It is natural to ask if we obtain some information about the real part, what can we conclude about the imaginary part, which is called the harmonic conjugate of the real part? Treating the relationship as an operation, the question becomes how well behaved is the harmonic conjugate operator? In this paper, by modifying some classical methods in constant exponent Hardy and Bergman spaces and developing new ways for the modern variable exponent spaces, we will study the harmonic conjugate operator on variable exponent Bergman spaces and prove that the operator is bounded when the exponent has positive minimum and finite maximum and satisfies the log-Holder condition.
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    A case-study of using tensors in multi-way electroencephalograme (EEG) data analysis
    (University of Alabama Libraries, 2020) Milligan-Williams, Essence; Sidje, Roger B.; University of Alabama Tuscaloosa
    Tensors are multi-dimensional arrays that can represent large datasets. Acquiring large data sets has its pros and cons; a pro being the bigger the data set, the more information could potentially be generated. A con would be the amount of labor needed to process this information. To combat this con, researchers in a variety of fields rely on tensor decomposition. Tensor decomposition's goal is to compress the data without losing any signficant information. Tensor decompositon is also known for its ability to extract underlying features that could not have been seen at face value. One such field is electroencephalography, which is the study of electrograms (EEG). An electrogram is a brain imaging tool that measures brain electrical activity. Having the abilitiy to be continously recorded for long periods of time, this could be hours, days, even weeks, EEG tends to have massive multi-dimensional datasets. In order to process the data, tensor decompositon methods such as Parallel Factor Analysis (PARAFAC) and Tucker decomposition can be executed on these large datasets.
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    Bounds on the number of generators of a module
    (University of Alabama Libraries, 2019) Morris, Peyton; Evans, Martin; University of Alabama Tuscaloosa
    The aim of this work is to present the Forster-Swan bound on the number of generators of a module. It is not our intention to present a novel finding or research discovery. Instead, we will develop the commutative algebra necessary to understand its proof, as well as the significance of the result. As a consequence, we will not present merely a series of prerequisites to the proof of the Forster-Swan bound, but rather everything which is strictly necessary to obtain a genuine understanding of the nature of the theorem. With this goal in mind, we will develop not only many of the fundamental results of commutative algebra, but in addition present results which are in a more profound way related to or even go beyond the Forster-Swan theorem itself. In section one, we describe the basic properties of commutative rings and modules over them. In particular, we will give the standard results on prime and maximal ideals, finitely generated modules, exact sequences, tensor products and flatness. In section two we define the notion of the spectrum of a ring, show that this is a geometric object associated with the ring, and present a few examples of spectra of rings. In section three, the theory of localization, the driving force behind the Forster-Swan theorem, is developed and several local-global principles of commutative algebra are demonstrated. In section four, the theory of Noetherian rings and modules, the dimension of rings, the concept of an algebraic variety, and Hilbert's basis and zero-locus theorems are presented. Section five contains the full proof of the Forster-Swan theorem. We conclude in section 6 with a discussion of topics surrounding Serre's problem on projective modules and the Eisenbud-Evans conjectures which give an improvement on the Forster-Swan bound.
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    A combinatorial proof of the invariance of tangle floer homology
    (University of Alabama Libraries, 2019) Homan, Timothy Adam; Roberts, Lawrence; University of Alabama Tuscaloosa
    The aim of this work is to take the combinatorial construction put forward by Petkova and Vértesi for tangle Floer homology and show that many of the arguments that apply to grid diagrams for knots can be applied to grid diagrams for tangles. In particular, we showed that the stabilization and commutation arguments used in combinatorial knot Floer homology can be applied mutatis mutandis to combinatorial tangle Floer homology, giving us an equivalence of chain complexes (either exactly in the case of commutations or up to the size of the grid in stabilizations). We then added a new move, the stretch move, and showed that the same arguments which work for commutations work for this move as well. We then extended these arguments to the context of A-infinity structures. We developed for our stabilization arguments a new type of algebraic notation and used this notation to demonstrate and simplify useful algebraic results. These results were then applied to produce type D and type DA equivalences between grid complexes and their stabilized counterparts. For commutation moves we proceeded more directly, constructing the needed type D homomorphisms and homotopies as needed and then showing that these give us a type D equivalence between tangle grid diagrams and their commuted counterparts. We also showed that these arguments can also be applied to our new stretch move. Finally, we showed that these grid moves are sufficient to accomplish the planar tangle moves required to establish equivalence of the tangles themselves with the exception of one move.
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    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.
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    Matrix groups and Lie algebras: an algebraic treatment
    (University of Alabama Libraries, 2019) Ervin, Tucker Jerome; Evans, Martin J.; University of Alabama Tuscaloosa
    There exist two correspondences between groups and Lie algebras. One occurs between matrix Lie groups and Lie algebras. The other concerns itself with complete groups of unitriangular matrices and Lie algebras. Titled the Lie and Mal'cev correspondences respectively, the purpose of this paper is to explore the two. We begin with an introduction to the basic properties of Lie algebras and other preliminary material followed by a construction of free Lie rings and algebras as well as by other interesting material discovered along the way. We then dive into the Lie correspondence, with which we contrast the Mal'cev correspondence in the section after.
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    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.
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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.
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    Efficient algorithms for solving three dimensional parabolic interface problem with variable coefficients
    (University of Alabama Libraries, 2018) Wei, Zhihan; Zhao, Shan; University of Alabama Tuscaloosa
    The dissertation consists of two parts, in the first part, a new matched alternating direction implicit (ADI) method is proposed for solving three-dimensional (3D) parabolic interface problems with discontinuous jumps, piecewise constant diffusion coefficients and complex interfaces. This scheme inherits the merits of its ancestor of two-dimensional problems, while possesses several novel features, such as a non-orthogonal local coordinate system for decoupling the jump conditions, two-side estimation of tangential derivatives at an interface point, and a new Douglas-Rachford ADI formulation that minimizes the number of perturbation terms, to attack more challenging 3D problems. In time discretization, this new ADI method is found to be first order and stable in numerical experiments. In space discretization, the matched ADI method achieves a second order of accuracy based on simple Cartesian grids for various irregularly-shaped surfaces and spatial-temporal dependent jumps. Computationally, the matched ADI method is as efficient as the fastest implicit scheme based on the geometrical multigrid for solving 3D parabolic equations, in the sense that its complexity in each time step scales linearly with respect to the spatial degree of freedom $N$, i.e., $O(N)$. Furthermore, unlike iterative methods, the ADI method is an exact or non-iterative algebraic solver which guarantees to stop after a certain number of computations for a fixed $N$. Therefore, the proposed matched ADI method provides an efficient tool for solving 3D parabolic interface problems. In the second part, instead of constant diffusion coefficients, improved schemes for variable diffusion coefficient are also performed in the work. A comparison of proposed ADI method with different other time splitting methods, including locally one-dimensional implicit Euler(LOD-IE), locally one-dimensional Crank-Nicolson(LOD-CN) and Trapezoidal Splitting(TS) method will be implemented, coupled with different variation of matched interface and boundary (MIB) method in spatial discretization. These large scale computational studies facilitate the further development of matched ADI algorithms for 3D parabolic interface problems.
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    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.
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    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.