Theses and Dissertations - Department of Mathematics

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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.
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.
A bounded and periodic interest rate model
(University of Alabama Libraries, 2013) Cai, Chen; Wang, Pu; University of Alabama Tuscaloosa
In financial market, interest rate is crucially important. Its changes and moves have a great impact on consumer's products, inflation rate, bond and stock market, and almost all the aspects in the financial world. An ideal stochastic model describing the volatility of the short-term interest rate would possess the following nice properties. First it has to have the periodic behavior; this is different from stock price model in which it has an increasing or decreasing trends. Second, it should maintain in a positive range and be bounded. Third, its differential equation should be simple and have an analytical solution so that its density function as well as any moments can be readily derived. In this dissertation, we propose and investigate such a stochastic differential equation. Its solution involves sine/cosine wave functions of Brownian motion that has all these properties. Their statistical properties such as mean, variance and covariance structure of this interest rate at any time are derived; their relation with martingale is established; both analytical and numerical solutions are obtained. From this interest rate model, the term structures and the yield curves will also be demonstrated for various settings.
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.
Capping the variance of cash flow of hedging strategy
(University of Alabama Libraries, 2011) Ginting, Maydison; Wu, Zhijian; University of Alabama Tuscaloosa
This dissertation consolidates previous research on an optimal strategy to reduce the running risk in hedging a long-term supply commitment with short-dated futures contracts. By introducing a cap function, this dissertation defines scenarios of running risk over the hedging horizon. We introduce a linear cap function and wish to find a hedging strategy G with the smallest constant F such that the variance of the cumulative cash flow is less than or equal the multiplication of a cap function and the constant F. The objective is to seek the best function G(s) to cap the variance of cash flow under a given non-negative cap function. We also implement the result in MATLAB by creating a Graphical User Interface application that enables the user to see the various results of the variance of cash flow of the best hedging scenario.
Cofinite graphs and their profinite completions
(University of Alabama Libraries, 2013) Das, Bikash Chandra; Corson, Jon M.; University of Alabama Tuscaloosa
We generalize the work of B. Hartley, to the category of topological graphs. The completion of a topological group in Hartley's work can be thought of as a particular example of the completion of a general uniform space. We consider topologies that are induced by uniformities, in order to topologize our graph structures and talk about their completions. In this dissertation generalize the idea of cofinite groups, due to B. Hartley. First we define cofinite spaces in general. Then, as a special case, we study cofinite graphs and their uniform completions. We are able to show that these completions are also cofinite graphs and being compact Hausdorff and totally disconnected they are rather regarded as profinite graphs. The idea of constructing a cofinite graph starts with defining a uniform topological graph in an appropriate fashion. We endow abstract graphs with uniformities corresponding to separating filter bases of equivalence relations with finitely many equivalence classes over the graph. By taking finitely many equivalence classes, we want to ensure the production of profinite structures over our topological graphs on taking the projective limit of the corresponding quotient graphs. It is established that for any cofinite graph there exists a unique cofinite completion. Generalizing Hartley's idea of cofinite groups and obtaining the structure for cofinite graphs we start establishing a parallel theory of cofinite graphs which in many ways can also be thought of as generalizations of the well-known works on pronite graphs by Pavel Zalesskii and Luis Ribes. Suitably defining the concept of cofinite connectedness of a cofinite graph we find that many of the properties of connectedness of topological spaces have analogs for cofinite connectedness. As an immediate consequence we obtain the following generalized characterization of the connected Cayley graphs of conite groups: G be a cofinite group, X be a cofinite space, then Cayley graph (G, X) is also cofinite graph and it is cofinitely connected if and only if X generates G (topologically). Our immediate next concern is developing group actions on cofinite graphs. Defining the action of an abstract group over a cofinite graph in the most natural way we are able to characterize a unique way of uniformizing an abstract group with a cofinite structure, obtained from the cofinite structure of the graph in the underlying action, so that the afore said action becomes uniformly continuous. We show that the aforesaid actions can actually be extended to the structures' of corresponding cofinite completions, preserving the underlying character of the original group action.
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.
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.
A constructive nullstellensatz for univariate polynomials
(University of Alabama Libraries, 2014) Netyanun, Anupan; Trent, Tavan T.; University of Alabama Tuscaloosa
In this dissertation, we will take an effective approach to prove the Hilbert's Nullstellensatz in a special case where we have univariate polynomials $f_{i}(z)'s$ for $i\in\{1,2,...,m\}$. This approach will explicitly construct polynomials $p_{i}(z)'s$ for $i\in\{1,2,...,m\}.$ Moreover, we will get the best result on the bounds for the degrees of polynomials $p_{i}(z)'s.$ We then use a similar technique to solve the problems in a matrix case. Previous work motivated by algebraic techniques are from {[}2{]} W.D.Brownawell, {[}5{]} J.Kollar. They made a big improvement on the bounded degree of $p_{i}(z)'s$ in solutions. We are also motivated by works done in analysis from L. Carleson (1962), T. Wolff (1979). These are used to get the best result on the bounds on the degrees of $p_{i}'s$ in the solutions obtained in this dissertation. For the matrix case, we are motivated by {[}11{]} T.T. Trent, X. Zhang. This will enable us to derive the results in the matrix case.
A corona theorem for certain subalgebras of H∞(D)
(University of Alabama Libraries, 2009) Ryle, Julie; Trent, Tavan T.; University of Alabama Tuscaloosa
[NOTE: Text or symbols not renderable in plain text are indicated by [...]. See PDF document for full abstract.] The corona theorem for the space of bounded analytic functions on the unit disk, [...], which was proven by Carleson in 1962, states that D is dense in the maximal ideal space of [...]. This theorem can be reduced to the following result: [...]. Furthermore, if we have the additional condition that [...]. In this dissertation, we prove that the corona theorem holds for certain subalgebras of [...], and we provide estimates for the sizes of the given solutions. Among the algebras we consider are those which contain bounded analytic functions whose kth derivatives vanish at 0 for all k in K, a subset of the natural numbers, which we call [...]. We give several properties the set K must have in order for [...] to be an algebra. We then prove the corona theorem in both the vector and matrix cases for these algebras. In fact, in the vector case, we prove the corona theorem using two different techniques. Each gives a unique estimate, and one extends our findings to more general algebras. We also settle a conjecture of Mortini, Sasane, and Wick involving the algebra C+BH∞(D), where B is a Blaschke product. We prove the corona theorem in C+BH∞(D) holds for an infinite number of functions. We end with a few suggestions for future research.
The Corona Theorem for the multiplier algebras on weighted Dirichlet spaces
(University of Alabama Libraries, 2009) Kidane, Berhanu Tekle; Trent, Tavan T.; University of Alabama Tuscaloosa
In this dissertation we give a proof of "The Corona Theorem for Infinitely Many Functions for the Multiplier Algebras on Weighted Dirichlet Spaces", and we obtain explicit estimates on the size of the solution. We denote the open unit disc of the complex plane by D, and for α in (0, 1) we denote by Dα the Weighted Dirichlet Spaces of all holomorphic functions on D, and by
Coverings of profinite graphs
(University of Alabama Libraries, 2013) Acharyya, Amrita; Corson, Jon M.; University of Alabama Tuscaloosa
We define a covering of a profinite graph to be a projective limit of a system of covering maps of finite graphs. With this notion of covering, we develop a covering theory for profinite graphs which is in many ways analogous to the classical theory of coverings of abstract graphs. For example, it makes sense to talk about the universal cover of a profinite graph and we show that it always exists and is unique. We define the profinite fundamental group of a profinite graph and show that a connected cover of a connected profinite graph is the universal cover if and only if its profinite fundamental group is trivial.
Development of interval and non-interval methods for solving multi-objective optimization problems
(University of Alabama Libraries, 2018) Liu, Sijie; Sun, Min; University of Alabama Tuscaloosa
Multi-objective optimization problems are pervasive across many diverse disciplines in engineering, economics, management and design where optimal decisions need to be made in the presence of tradeoffs between two or more conflicting objectives with respect to a set of decisions and subject to a set of constraints (Awad and Khanna 2015). The interval method (as a non-scalarization MOP method) is known as a precise and robust tool for optimization that guarantees to capture all points on the Pareto-front (Kubica and Woźniak 2007) . This research investigated the interval and non-interval methods for multi-objective optimization problems with or without constraints. The underlying framework for our improved interval methods are standard Hansen method and Big Cube and Small Cube method. Listed below are some strategies we have developed: a) Twin interval arithmetic is proposed in an effort to improve the quality of estimates for the range of functions. The main use of twin interval arithmetic (TIA) in the improved interval algorithms is to provide an extra choice for the cut-off value in the discarding process. b) Exclusion zone functions are introduced to further reduce variables' intervals which can be added as a reduction step in the standard Hansen method. c) For constrained global optimization problems with two linear constraints, we outlined a steepest descent procedure to locate a feasible sample point within the framework of Hansen method. d) A hybrid BCSC algorithm supported by the three above-mentioned strategies is described in Chapter 3. A refined sampling step is introduced to the BCSC algorithm and this algorithm was marked as original BCSC for comparison in Chapter 3. Numerical results with respect to each strategy are illustrated to show its effectiveness in Chapter 3. Three non-interval multi-objective linear programming (MOLPs) algorithms are presented in Chapter 4. Numerical examples are used to test their effectiveness. Developing their extensions for solving non-linear optimization systems after linearization would be a part of future work.
Development of modal interval algorithm for solving continuous minimax problems
(University of Alabama Libraries, 2017) Luo, Xin; Sun, Min; University of Alabama Tuscaloosa
While there are a large variety of effective methods developed for solving more traditional minimization problems, much less success has been reported in solving the minimax problem $\displaystyle\min_{u \in U}\displaystyle\max_{v \in V}f(u,v)$ where $U\times V$ is a fixed domain in $\mathbb{R}^n$. Most of the existing work deal with a discrete $V$ or even a finite $V$. Continuous minimax problems can be applied to engineering, finance and other fields. Sainz in 2008 proposed a modal interval algorithm based on their semantic extensions to solve continuous minimax problems. We developed an improved algorithm using modal intervals to solve unconstrained continuous minimax problems. A new interval method is introduced by taking advantage of both the original minimax problem and its dual problem. After theoretical analysis of major issues, the new algorithm is implemented in the framework of uniform partition of the search domain. Various improvement techniques including more bisecting choices, sampling methods and deletion conditions are applied to make the new method more powerful. Preliminary numerical results provide promising evidence of its effectiveness.
Development, analysis and simulation of laboratory scale models of some problems in astrophysical convection
(University of Alabama Libraries, 2017) Aljahdaly, Noufe Hemaid; Hadji, Layachi; University of Alabama Tuscaloosa
This dissertation addresses some open questions in the linear and non-linear theories of thermal convection in regions that are unbounded in the direction of gravity. The first part of the dissertation seeks to model and analyze an instability, the occurrence of which requires the existence of a thin unstably stratified region in an otherwise stably stratified environment that is common in some situations involving astrophysical convection. Convective motion that takes place in stars and planets is characterized by the absence of horizontal boundaries and unstable density stratification of small thickness compared to the full extent of the fluid region. The instability is of buoyancy nature and typically induced by a density stratification that is unstable over a thin part of a vertically unbounded region that is otherwise stably stratified. We model this situation by considering a fluid region having a very large or infinite vertical extent and put forth a mathematical model that can be described as a Rayleigh-B\'{e}nard instability between two stably stratified layers. Thus, we attempt to uncover the instability threshold conditions and corresponding flow patterns when the base state consists of a step-function density profile. This case is investigated in both the Rayleigh-B\'{e}nard (horizontal fluid layer of infinite extent) and Ostroumov (vertical channel) geometries. Our analysis also puts forth the dependence of the threshold instability conditions on the location of the density jump. It predicts new patterns consisting of either symmetric or antisymmetric lens shape instead of the oval shape typically observed in the continuous stratification case. The appearance of the lens shape is attributed to the discontinuity. The linear analysis is extended to the weakly nonlinear regime where we show that the bifurcation from the conduction state is supercritical. Hence, our results are testable experimentally. Furthermore, we derive expressions for the convective thermal flux at the locus of the density jump and put forth a small scale laboratory experimental set-up that can be used to test our theoretical predictions. The experimental set-up describes the situation of a mass of cold fluid that is suddenly made to overly a mass of warm same fluid. The experiment itself serves to quantify the heat transfer between the two masses of fluid upon mixing when they are in direct contact. Do they mix by diffusion alone or by diffusion and convection; and how does the mixing evolve? The second part of the dissertation deals with the three-dimensional Ostroumov problem undergoing rigid body rotation. We carry out both the linear and non-linear theories and derive the threshold instability conditions and corresponding flow patterns for a variety of cells. We consider both closed and open cells either at the top/bottom or in one of the horizontal directions. Our analysis leads to the derivation of non-linear evolution equation for the amplitude of motion, the solution of which yields the stable non-linear solutions as functions of the Taylor and Prandtl numbers. We also examine the question of pattern formation as functions of the main physical parameters.
A difference of composition operators on Bergman space
(University of Alabama Libraries, 2015) Acharyya, Soumyadip; Moen, Kabe; Wu, Zhijian; University of Alabama Tuscaloosa
Let us consider the operator-theoretic difference of two composition operators acting on a weighted Hilbert Bergman space. In 2011, Choe , Hosokawa and Koo [CHK] proved a necessary and sufficient condition under which the difference operator is Hilbert-Schmidt. In this dissertation, we have provided a simpler proof of their result , using a change of variable method. Applying that method, we have also established similar necessary and sufficient integral condition under which a difference operator of a more general form is Hilbert-Schmidt. In 2011, Saukko [S] found a beautiful way of characterizing the boundedness and compactness of the difference of two composition operators , acting in between two weighted Bergman spaces. More precisely, he was able to reduce those problems for a difference operator into corresponding problems for a weighted composition operator, which were already solved in the year 2007. In this work, we have generalized Saukko's results for the same operator where the target function space is more general. Through these generalized results, we have been able to characterize the boundedness and compactness of the previously mentioned difference operator with a more general form.
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.
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.
Fast alternating direction implicit schemes for geometric flow equations and nonlinear poisson equation in biomolecular solvation analysis
(University of Alabama Libraries, 2014) Tian, Wufeng; Zhao, Shan; University of Alabama Tuscaloosa
The present work introduces new alternating direction implicit (ADI) methods to solve potential driven geometric flow partial differential equations (PDEs) for biomolecular surface generation and the nonlinear Poisson equations for electrostatic analysis. For solving potential driven geometric flow PDEs, an extra factor is usually added to stabilize the explicit time integration. However, there are two existing ADI schemes based on a scaled form, which involves nonlinear cross derivative terms that have to be evaluated explicitly. This affects the stability and accuracy of these ADI schemes. To overcome these difficulties, we propose a new ADI algorithm based on the unscaled form so that cross derivatives are not involved. Central finite differences are employed to discretize the nonhomogenous diffusion process of the geometric flow. The proposed ADI algorithm is validated through benchmark examples with analytical solutions, reference solutions, or literature results. Moreover, quantitative indicators of a biomolecular surface, including surface area, surface-enclosed volume, and solvation free energy, are analyzed for various proteins. The proposed ADI method is found to be unconditionally stable and more accurate than the existing ADI schemes in all tests of biomolecular surface generation. The proposed ADI schemes have also been applied in solving the nonlinear Poisson equation for electrostatic solvation analysis. Compared with the existing biconjugate gradient iterative solver, the ADI scheme is more efficient. The CPU time cost is validated through the solvation analysis of an one atom Kirkwood model and a set of 17 small molecules whose experimental measurements are available. Additionally, application of the proposed ADI scheme is considered for electrostatic solvation analysis of a set of 19 proteins. The proposed ADI scheme enables the use of a large time increment in the steady state simulation so that the proposed ADI algorithm is efficient for biomolecular surface generation and solvation analysis.
Fractional Brownian motion and managing risk with short-term futures contracts
(University of Alabama Libraries, 2016) Cui, Wei; Wang, James L.; University of Alabama Tuscaloosa
In this dissertation, we consolidate previous research done by [3], [5], [13], [14] and [22] on an optimal strategy to reduce the running risk in hedging a long-term supply commitment with short-term futures contracts. Under the assumption that the market price of the commodity is modeled by the fractional Brownian motion (fBm), we study the following optimization problem: Under the constraint $$\ \int_{0}^{1}\int_{0}^{1}(1-g(u))(1-g(v))|u-v|^{2H-2}dudv\leq\theta,$$ Which measurable function $g: [0,1]\rightarrow R[0,1]$ will minimize the value of $$\ \sup_{t\in[0,1]}\int_{0}^{t}\int_{0}^{t}(t-g(u))(t-g(v))|u-v|^{2H-2}dudv?$$ where $\theta\in[0,\frac{H^{2H}}{(H+1)^{2H+2}}]$ and $H\in(\frac{1}{2},1)$. Under the fractional market model, we gave the spot risk function by using the hedging strategies provided by Glasserman [5] and found that when the hurst index $H$ is equal to $0.5$ the maximal spot risk is the same as the result given in [5]. The main work in this dissertation is to show that a unique solution to this optimization problem always exists.

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