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Title: GRAPH PARTITIONING WITH EIGENVECTORS
Creator: Panek, James
Date: 2017, 2017-05
Description: The Cheeger constant of a graph quantities how well a graph can be cut yield- ing two (typically) large vertex sets by a small edge cut. Lower...
Show moreThe Cheeger constant of a graph quantities how well a graph can be cut yield- ing two (typically) large vertex sets by a small edge cut. Lower and upper bounds have been developed using the eigenvalues and eigenvectors of the normalized Laplacian matrix of the graph. Here a classic sweep algorithm is studied using linear combinations of eigenvectors, specifically the columns of approximate discrete Green's functions. It is then shown, statistically on certain families of random graphs following a stochastic block model, that it is enough to use two eigenvalues and vectors to improve this classic algorithm's upper bound in most cases.
M.S. in Applied Mathematics, May 2017
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Title: OPTION PRICING AND HEDGING UNDER JUMP DIFFUSION MODEL WITH DIFFERENTIAL INTEREST RATES
Creator: Fang, Hui
Date: 2014, 2014-07
Description: Classical option pricing schemes have an ideal assumption that a single interest rate is used as risk free discounting rate. This assumption...
Show moreClassical option pricing schemes have an ideal assumption that a single interest rate is used as risk free discounting rate. This assumption has been already relaxed due to deteriorating credit market. In this paper, we assume that risk free lending and borrowing rates are di erent. Under di erential interest rates setup, a no-arbitrage price band rather than a unique price is calculated. We extend this scheme to a jump di usion model. Under mild conditions, option prices can be calculated explicitly. We illustrate the pricing scheme through a European style call option. Numerical results show that funding spread between lending and borrowing interest rates has signi cant impact on the length of the option's no-arbitrage price band.
M.S. in Applied Mathematics, July 2014
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Title: RENEWABLE ENERGY IN MICROGRID: A STOCHASTIC OPTIMIZATION APPROACH
Creator: Jin, Hongwei
Date: 2014, 2014-12
Description: A Microgrid is a group of interconnected loads and distributed energy resources within clearly defined electrical boundaries that acts as a...
Show moreA Microgrid is a group of interconnected loads and distributed energy resources within clearly defined electrical boundaries that acts as a single controllable entity with respect to the grid and that connects and disconnects from such grid to enable it to operate in both grid-connected or island mode. The optimized energy scheduling problem is significant to both utilities and community consumers. In this thesis, I present an approach by analyzing the historical weather data and renewable energy data, and build a two-stage stochastic program with a long term view towards minimizing costs. The underlying stochastic process that generates uncertainty in demand and supply in power network is the local weather, thus understanding solar radiation as a function of weather is significant to us. First, two simple methods, which are majority rule and flexible time selection, are proposed with the purpose of handling noisy raw data and giving a relatively precise prediction of renewable energy consumption and overall energy demands. Then, I implement a deterministic strategy, a two-stage stochastic program and a repeated stochastic program using AMPL, a mathematical modeling language. Every stochastic program is defined as based on 42 scenarios from the weather conditions. In the final step, I solve the model using CPLEX and compare optimal solutions based on a year-long Monte Carlo simulation. Ignoring installation and maintenance costs, the Microgrid can make some profit by an optimized control based on our modeling approach utilizing the stochastic optimization paradigm. Although there are only slight differences between three models, the repeated two-stage stochastic model gives the best long-term results.
M.S. in Applied Mathematics, December 2014
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Title: DISJUNCTNESS PROPERTIES RESULTING FROM CONCATENATION OF GROUP TESTING MATRICES
Creator: Clardy, Melinda Bulin
Date: 2015, 2015-05
Description: This thesis discusses matrix properties as they relate to the idea of non-adaptive group testing. This is accomplished by first considering...
Show moreThis thesis discusses matrix properties as they relate to the idea of non-adaptive group testing. This is accomplished by first considering the history of group testing and then exploring existing results. The next chapter of this thesis discusses taking a given binary matrix and using this as an inner code with some symbol matrix as an outer code to create a new binary matrix. The process is called a concatenation construction and we will cover a few types including the orthogonal array construction, a 𝜆-separating hash family construction, code concatenation, and DNA Sudoku. We conclude by elaborating on primary results coming from orthogonal array construction and 𝜆-separating hash family constructions. These give results pertaining specifically to Steiner systems and cover-free families.
M.S. in Applied Mathematics, May 2015
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Title: PRICING MOUNTAIN RANGE OPTION WITH QUASI-MONTE CARLO AND PRINCIPAL COMPONENT ANALYSIS
Creator: Xiong, Shuo
Date: 2012-12-03, 2012-12
Description: The thesis considers the problem of pricing different types of mountain range options by quasi-Monte Carlo simulation. Principal component...
Show moreThe thesis considers the problem of pricing different types of mountain range options by quasi-Monte Carlo simulation. Principal component analysis (PCA), also known as the singular value decomposition (SVD), has been applied widely to reduce the quasi-Monte Carlo sampling error in many financial problems. This is especially true for pricing single asset options. The purpose of this thesis is to investigate whether the singular component decomposition offers advantages for pricing mountain range options, which are exotic options based on multiple underlying assets. We consider our simulation in a complete, standard Black-Scholes market which has constant risk-free interest rate and covariance matrix. There are main two types of mountain range options simulated in this thesis: Altiplano and Himalayan. Key words : Mountain Range Option; Quasi-Monte Carlo; Singular Value Decompo- sition; Altiplano Option; Himalayan Option
M.S. in Applied Math, December 2012
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Title: ANALYZING REPRODUCING KERNEL APPROXIMATION METHODS VIA A GREEN FUNCTION APPROACH
Creator: Ye, Qi
Date: 2012-04-22, 2012-05
Description: In this thesis, we use Green functions (kernels) to set up reproducing kernels such that their related reproducing kernel Hilbert spaces ...
Show moreIn this thesis, we use Green functions (kernels) to set up reproducing kernels such that their related reproducing kernel Hilbert spaces (native spaces) are isometrically embedded into or even are isometrically equivalent to generalized Sobolev spaces. These generalized Sobolev spaces are set up with the help of a vector distributional operator P consisting of finitely or countably many elements, and possibly a vector boundary operator B. The above Green functions can be computed by the distributional operator L := P TP with possible boundary conditions given by B. In order to support this claim we ensure that the distributional adjoint operator P of P is well-defined in the distributional sense. The types of distributional operators we consider include not only di erential operators but also more general distributional operators such as pseudo-di erential operators. The generalized Sobolev spaces can cover even classical Sobolev spaces and Beppo-Levi spaces. The well-known examples covered by our theories include thin-plate splines, Mat´ern functions, Gaussian kernels, min kernels and others. As an application for high-dimensional approximations, we can use the Green functions to construct a multivariate minimum-norm interpolant s f;X to interpolate the data values sampled from an unknown generalized Sobolev function f at data sites X Rd. Moreover, we also use Green functions to set up reproducing kernel Banach spaces, which can be equivalent to classical Sobolev spaces. This is a new tool for support vector machines. Finally, we show that stochastic Gaussian fields can be well-defined on the generalized Sobolev spaces. According to these Gaussian-field constructions, we find that kernel-based collocation methods can be used to approximate the numerical solutions of high-dimensional stochastic partial differential equations.
Ph.D. in Applied Mathematics
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Title: IMPROVED MAXIMUM LIKELIHOOD ESTIMATION FOR GENERALIZED BASS MODEL
Creator: Razo, Martha
Date: 2017, 2017-05
Description: Today, businesses operate in an interconnected global economy, in which innovation happens on a moment to moment basis. Statistical predictive...
Show moreToday, businesses operate in an interconnected global economy, in which innovation happens on a moment to moment basis. Statistical predictive modeling in marketing and development is emerging as a crucial component to the success of small companies and large corporations alike. The goal of this paper is to analyze the Bass model as it pertains to sales of the Chevy Volt. The Bass model has been shown to be a useful tool for forecasting the sales of new products as they become available in the marketplace, but what are the model's limitations? The widely studied Bass model produces computational problems when we evaluate the model for a larger set data which extends to modified models constructed from the original Bass model. Kijek in [14] and Srinivasan and Mason in [16] alert us of the shortcomings of the Maximum Likelihood approach of solving the Bass model which extends to solving the Generalized Bass model, but these authors limit us to a vague listing without a close analysis. In this paper we present the issues of estimating the Bass model parameters when using the Maximum Likelihood approach. Furthermore, we introduce an improved generalized model which takes into account the shortcomings of the Bass model and our proposed approach of overcoming these. We will illustrate the limitations of the Bass model when using a large data set from the Chevy Volt car data published by Inside EVs[12]. Careful analysis of the Bass model and its current modifications provides a rich tool that has potential in changing the future of a company when introducing new products to the market.
M.S. in Applied Mathematics, May 2017
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Title: ADAPTIVE COVERING CODES IN THE Q-ARY HYPERCUBE
Creator: Tietzer, Daniel
Date: 2011-11-20, 2011-12
Description: We investigate a generalized version of the Pathological Liar Game of Ellis and Yan. In our version, there are nonnegative integer parameters...
Show moreWe investigate a generalized version of the Pathological Liar Game of Ellis and Yan. In our version, there are nonnegative integer parameters M; n; e; a; q with 0 < a < q. The player (Carole) is equipped with M messages, each with an integral error count. Carole performs the following procedure n times in sequence: she numbers the messages in increasing order of error count, divides them into consecutive contiguous size-q blocks, and selects a size-a subset of each block. The messages she does not select have their error count increased by 1. Carole loses i , after this process is complete, there is at least one message with error count e: We allow n to approach in nity and suppose e = bfnc for a xed f 2 (0; 1). We establish an upper bound on the minimum M for which Carole always loses, as a function of n, by extending a technique of Cooper and Ellis; we develop a simple process that approximates the course of the game for any choice by Carole, and bound the difference between this approximating process and any state of the game achievable by Carole. Along the way we generalize several of the results of Cooper and Ellis in ways that suggest future application to similar problems in adaptive coding theory.
M.S. in Applied Mathematics, December 2011
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Title: SOLUTION TO STOKES FLOW DUE TO MOTION OF AN IMMERSED PARTICLE
Creator: Cao, Yu
Date: 2012-04-25, 2012-05
Description: For finding the solution to Stokes flow due to motion of an immersed particle, we introduce two different methods based on boundary integral...
Show moreFor finding the solution to Stokes flow due to motion of an immersed particle, we introduce two different methods based on boundary integral equations (BIE). For the first method, which is based on the first kind BIE, we prove the compactness of the integral operator of which the kernel is the Stokeslet, which implies that the BIE of the first kind in this method are ill-posed problems. For the second method, which is based on the second kind BIE which are well-posed problems, it is shown that the kernel functions in the BIE are generally not smooth. Finally, using the known numerical schemes, we compare the computational cost of these two method for finding the velocity of a given point in the domain in terms of the number of numerical integrations over triangles. It is show that theses two schemes are of same order while the computational cost of the second method is more expensive than that of the first one.
M.S. in Applied Mathematics, May 2012
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Title: GUARANTEED ADAPTIVE UNIVARIATE FUNCTION APPROXIMATION
Creator: Ding, Yuhan
Date: 2015, 2015-12
Description: Numerical algorithms for univariate function approximation attempt to provide approximate solutions that differ from the original function by...
Show moreNumerical algorithms for univariate function approximation attempt to provide approximate solutions that differ from the original function by no more than a user-specified error tolerance. The computational cost is often determined adaptively by the algorithm based on the function values sampled. While adaptive algorithms are widely used in practice, most lack guarantees, i.e., conditions on input functions that ensure the error tolerance is met. In this dissertation we establish guaranteed adaptive numerical algorithms for univariate function approximation using piecewise linear splines. We introduce a guaranteed globally adaptive algorithm, funappxglobal g, in Chapter 2, along with sufficient conditions for the success of funappxglobal g. Two-sided bounds on the computational cost are given in Theorem 1. These bounds are of the same order as the computational cost for an algorithm that knows the infinity norm of the second derivative of the input function as a priori. Lower bound on the complexity of the problem is also provided in Theorem 3. To illustrate the advantages of funappxglobal g, corresponding numerical experiments are presented in Section 2.7. The cost of a globally adaptive algorithm is determined by the most peaky part of the input function. In contrast, locally adaptive algorithms sample more points where the function is peaky and fewer points elsewhere. In Chapter 3, we establish a locally adaptive algorithm, funappx g, with sufficient conditions for its success. An upper bound on the computational cost is also given in Theorem 4. One GUI example is presented to show how funappx g works. Some interesting function approximation problems in computational graphics are also presented. The key to analyzing these adaptive algorithms is looking at the error for cones of input functions rather than balls of input functions. Non-convex cones provide a setting where adaption may be beneficial.
Ph.D. in Applied Mathematics, December 2015
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Title: A BOUNDARY INTEGRAL METHOD FOR SOLVING THE BRINKMAN'S EQUATION IN 3-DIMENSIONAL FLOW
Creator: Gundaboina Harish, Aditya Kiran
Date: 2014, 2014-12
Description: In this paper, we consider the boundary integral equation (BIE) corresponding to the Brinkmans equation. We compute the velocities of the...
Show moreIn this paper, we consider the boundary integral equation (BIE) corresponding to the Brinkmans equation. We compute the velocities of the interacting particles when the point-wise forces acting on them are known. We assume a ctitious ow point inside the bead and eliminate the double-layer in the BIE. This BIE is now numerically solved by covering the surface of the bead by two overlapping patches and de ning their partition of unity. We then discuss the e ect of the overlapping of the patches on the results. We develop a second-order accurate numerical method for the BIE. ix
M.S. in Applied Mathematics, December 2014
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Title: TOPICS IN STATISTICAL MODELING AND OPTIMAL DESIGN
Creator: Li, Yiou
Date: 2014, 2014-07
Description: In this dissertation we discuss several topics in statistics concerning regression models, experimental design and optimization. When it is...
Show moreIn this dissertation we discuss several topics in statistics concerning regression models, experimental design and optimization. When it is expensive to compute a function value via numerical simulation, obtaining gradient values simultaneously can improve model e ciency. In the rst and second parts, polynomial regression models with gradient information are considered. We propose an orthogonal polynomial basis with respect to an inner product involving gradients of functions, to eliminate the illconditioning of the design matrix caused by Hermite polynomial basis. Through a simpli ed nuclear reactor model, we show that compared with Hermite polynomial basis, the orthogonal polynomial basis results in a better-conditioned design matrix, and a signi cant improvement when basis polynomials are chosen adaptively, using a stepwise tting procedure. In the second part, the design problem for polynomial regression models with gradient information is addressed. A theoretical upper bound is derived on the scaled integrated mean squared error in terms of the discrepancy of the design, and this bound can be used to choose designs that are both e cient and robust under model uncertainty. Numerical experiments show that low discrepancy designs, whose empirical distribution functions match a xed target distribution, outperform random and Latin hypercube designs. Considering a speci ed regression model, we propose a relaxed optimization problem, which is a semide nite programming problem, to nd the optimal design that minimizes scaled integrated mean squared error. Numerical examples demonstrate the eligibility of the method by showing that the optimal designs we achieve coincide with the already known optimal designs for regression model without gradient information. In the third part of the dissertation, the optimal layout of wind farm is considered. To maximize the expected annual pro t gained by the wind farm, we seek for both optimal number of wind turbines and optimal positions of wind turbines based on Jensen's model. Wind speed and direction are considered as random variables with distribution approximated by empirical distribution from real data. We propose using particle swarm optimization to solve for the optimal layout of wind farm. At last, the situation that more wind turbines are added to an existing wind farm is discussed.
Ph.D. in Applied Mathematics, July 2014
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Title: TOPICS IN GRAPH FALL-COLORING
Creator: Mitillos, Christodoulos
Date: 2016, 2016-07
Description: Graph fall-coloring, also known as idomatic partitioning or independent domatic partitioning of graphs, was formally introduced by Dunbar,...
Show moreGraph fall-coloring, also known as idomatic partitioning or independent domatic partitioning of graphs, was formally introduced by Dunbar, Hedetniemi, Hedetniemi, Jacobs, Knisely, Laskar, and Rall in 2000 [1] as a simple extension of graph coloring and graph domination. It asks for a partition of the vertex set of a given graph into independent dominating sets. In this thesis, we will study a number of questions related to this concept. In the rst chapter we will give a brief background to graph theory, and introduce the topic of graph fall-coloring, after looking at the fundamental topics it builds on. In the second chapter, we identify the e ects on fall-colorability of various graphical operators, and look at the fall-colorability of certain families of graphs. In the third chapter we will explore certain constructions which create fall-colorable graphs given certain restrictions, and look at the interaction of fall-colorings and non-fall-colorings. Finally, in the fourth chapter, we lay the foundations to establish a connection between fall-coloring and certain existing open problems in graph theory, providing new possible avenues for exploring their solutions. We then provide two applied problems which can be solved with fall-coloring, and which motivate the notion of fall-nearcoloring. We also provide further questions in fall-coloring for future research. Keywords: Graph Fall-coloring, Idomatic Partition, Independent Dominating Sets, Chromatic number, Graph products.
Ph.D. in Applied Mechanics, July 2016
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Title: LEARNING THE STRUCTURE OF PROBABILITY NETWORKS WITH DATA UNCERTAINTY
Creator: Zhang, Sisi
Date: 2014, 2014-07
Description: This paper studies how data uncertainties impact structure learning. Learning the structure of a probabilistic network from observational data...
Show moreThis paper studies how data uncertainties impact structure learning. Learning the structure of a probabilistic network from observational data has been traditionally studied assuming that there are no uncertainties in the data. This paper focuses on the uncertainties that result in “misclassification errors” in the contingency tables based on which the independence tests are carried out. The impact of misclassification errors is investigated through a sensitivity study which focuses on identifying the boundaries of misclassification errors within which the learned structure from erroneous data is identical to the true structure. Mathematical derivations for obtaining this boundary are presented. The analytical results are showed by a case study in epidemiology.
M.S. in Applied Mathematics, July 2014
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Title: RARE EVENT SIMULATION FOR FINANCIAL MODELING WITH INTERACTING PATH SYSTEMS AND QUASI-MONTE CARLO METHODS
Creator: Silva, Tiago M.
Date: 2013, 2013-07
Description: The variance of the estimation of the probability γ that an event occurs using IID Monte Carlo Methods is inversely proportional to the sample...
Show moreThe variance of the estimation of the probability γ that an event occurs using IID Monte Carlo Methods is inversely proportional to the sample size M, Var[ˆγ] = γ(1 − γ)/M. If we fix the relative standard deviation at 10% when estimating the probability of a rare event γ ≪ 1, the necessary sample size may reach values that are computationally not acceptable. This is very common in the Finance Industry when trying to estimate the joint default probabilities of 15 stocks in a portfolio of 100 stocks. De Moral and Patras [1] presented a new variance reduction technique called Interacting Path System (IPaS), which showed to be very efficient in a rare event situation. Our objective is to run some simulations in order to verify its efficiency against other variance reduction techniques found in the literature as well as implement new modifications using quasi-Monte Carlo methods in order to increase its efficiency. We run several simulations of rare and non-rare scenarios using IID Monte Carlo methods, quasi-Monte Carlo Methods and IPaS for the purpose of estimating the efficiency of IPaS. Moreover, we modified the current IPaS algorithm and implemented low discrepancy points in order to obtain a better estimator. We have found that IPaS technique is indeed more efficient than the well-known estimators in rare event scenarios, but quasi Monte Carlo methods is preferable if it is not a rare event. In addition to this, we have shown that the modified IPaS is even better than IPaS technique under rare event cases and it is similar to quasi Monte Carlo methods under non-rare events. However, in problems with higher dimensions, the efficiency of modified IPaS does not differ from IPaS results.
M.S. in Applied Mathematics, July 2013
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Title: DYNAMIC CONIC FINANCE: NO-ARBITRAGE PRICING AND NO-GOOD-DEAL PRICING FOR DIVIDEND-PAYING SECURITIES IN DISCRETE-TIME MARKETS WITH TRANSACTION COSTS
Creator: Rodriguez, Rodrigo
Date: 2012-06-27, 2012-07
Description: This thesis studies no-arbitrage pricing and dynamic conic nance for dividend-paying securities in discrete-time markets with transaction...
Show moreThis thesis studies no-arbitrage pricing and dynamic conic nance for dividend-paying securities in discrete-time markets with transaction costs. The rst part investigates no-arbitrage pricing for dividend-paying securities in discrete-time markets with transaction costs. We introduce the value process and the self- nancing condition in our context. Then, we prove a version of First Fundamental Theorem of Asset Pricing. Speci cally, we prove that the no-arbitrage condition under the e cient friction assumption is equivalent to the existence of a risk-neutral measure. We formulate an appropriate notion of a consistent pricing system in our set-up, and we prove that if there are no transaction costs on the dividends paid by the securities, then the no-arbitrage condition under the e cient friction assumption is equivalent to the existence of a consistent pricing system. We nish the chapter by deriving dual representations for the superhedging ask price and subhedging bid price of a derivative contract. The second part studies dynamic conic nance in the set-up introduced in the rst part. We formulate the no-good-deal condition in terms of a family of dynamic coherent risk measures, and then we prove a version of the Fundamental Theorem of No-Good-Deal Pricing. The Fundamental Theorem of No-Good-Deal Pricing provides a necessary and su cient condition for the no-good-deal condition to hold. Next, we study the no-good-deal ask and bid prices of a derivative contract. We particularize our results to the dynamic Gain-Loss Ratio, and compute the no-good-deal prices of European-style Asian options in a market with transaction costs.
Ph.D. in Applied Mathematics, July 2012
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Title: MAXIMUM INDUCED SUBGRAPHS OF K-TREES WITH COMPONENTS OF ORDER 1 OR 2
Creator: Liu, Yunjiao
Date: 2014, 2014-07
Description: We study induced subgraphs where every component has order 1 or 2. For a graph G, let f(G) be the maximum order of such a subgraph of G....
Show moreWe study induced subgraphs where every component has order 1 or 2. For a graph G, let f(G) be the maximum order of such a subgraph of G. Chappell and Pelsmajer [1] considered a more general parameter for graphs G of bounded treewidth, but were unable to determine f(G) for graphs of treewidth k > 3, even asymptotically. They conjectured that f(G) ≥ ⌈ 2n k+2⌉ for an n-vertex graph of treewidth at most k, but for k > 3, they were only able to show that f(G) ≥ 2n+2 2k+3 . In this thesis, we improve the lower bound to l 8n 5(k+1)m, for n ≥ 2k + 1. In addition, for the case k = 4, we develop methods for an inductive proof, where the cases are verified by computer-checking. If the conjecture is false, then our approach should eventually lead to a counter-example. To facilitate this approach, we come up with the addition structure on 4-trees, where one K4-subgraph is the “root”, and we consider all the different ways that an induced subgraph can intersect with the root separately.
M.S. in Applied Mathematics, July 2014
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Title: DYNAMIC CONIC FINANCE VIA BACKWARD STOCHASTIC DIFFERENCE EQUATIONS AND RECURSIVE CONSTRUCTION OF CONFIDENCE REGIONS
Creator: Chen, Tao
Date: 2016, 2016-07
Description: This thesis consists of two major parts, and it contributes to the fields of mathematical finance and statistics. The contribution to...
Show moreThis thesis consists of two major parts, and it contributes to the fields of mathematical finance and statistics. The contribution to mathematical finance is made via developing new theoretical results in the area of conic finance. Specifically, we have advanced dynamic aspects of conic finance by developing an arbitrage free theoretical framework for modeling bid and ask prices of dividend paying securities using the theory of dynamic acceptability indices. This has been done within the framework of general probability spaces and discrete time. In the process, we have advanced the theory of dynamic sub-scale invariant performance measures. In particular, we proved a representation theorem of such measures in terms of a family of dynamic convex risk measures, and provided a representation of dynamic risk measures in terms of BS Es. The contribution to statistics is of fundamental importance as it initiates the theory underlying recursive computation of confidence regions for finite dimensional parameters in the context of stochastic dynamical systems. In the field of engineering, particularly in the field of control engineering, the area of recursive point estimation came to great prominence in the last forty years. However, there has been no work done with regard to recursive computation of confidence regions. To partially fill this gap, the second part of the thesis is devoted to recursive construction of confidence regions for parameters characterizing the one-step transition kernel of a time-homogeneous Markov chain.
Ph.D. In Applied Mathematics, July 2016
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Title: MACROSCOPIC QUANTITIES FOR STOCHASTIC DIFFERENTIAL EQUATIONS WITH A LEVY NOISE IN TWO DIMENSIONS
Creator: Albert, Hannah
Date: 2016, 2016-05
Description: The mean exit time and transition probability density function are macroscopic quantities used to determine the behavior of stochastic di...
Show moreThe mean exit time and transition probability density function are macroscopic quantities used to determine the behavior of stochastic di↵erential equations (SDEs). The integral-di↵erential equations determining these quantities for SDEs with non- Gaussian ↵-stable L´evy motions involve a nonlocal term consisting of a singular integral, which is a manifestation of the ’flights’ or ’jumps’ due to the non-Gaussian noise. A two-dimensional SDE with radially symmetric ↵-stable L´evy motion is considered, and an efficient second-order accurate numerical scheme is developed for calculating the mean exit time and transition probability density function. The scheme is numerically verified by testing the results of the deterministic integral-di↵erential equations with a known, smooth function u(x) in place of themean exit time, and by calculating an unknown mean exit time u(x).
M.S. in Applied Mathematics, May 2016
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Title: A GUARANTEED, ADAPTIVE, AUTOMATIC ALGORITHM FOR UNIVARIATE FUNCTION MINIMIZATION
Creator: Tong, Xin
Date: 2014, 2014-07
Description: This thesis proposes a guaranteed, adaptive, automatic algorithm for solving univariate function minimization problem on the unit interval....
Show moreThis thesis proposes a guaranteed, adaptive, automatic algorithm for solving univariate function minimization problem on the unit interval. The key to this adaptive algorithm is performing the analysis for cones of input functions that are twice differentiable. This cone is defined in terms of two semi-norms, a stronger one and a weaker one. Three fixed-cost algorithms based on linear splines are used to find the bounds for an input function and its minimum value. The estimated minimum value and possible optimal solution set are given by those bounds. This algorithm is guaranteed to provide either a minimum value within a user-specified tolerance or a possible optimal solution set whose volume is less than another user-specified tolerance.
M.S. in Applied Mathematics, July 2014
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