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Title: DYNAMICS OF A THREE-DIMENSIONAL FOUR-BAR LINKAGE SUBJECT TO RANDOM EXTERNAL FORCING
Creator: Lytell, Mark R.
Date: 2011-11-15, 2011-12
Description: This thesis explores the dynamics of a three-dimensional four-bar mechanical linkage subject to random external forcing. The Lagrangian...
Show moreThis thesis explores the dynamics of a three-dimensional four-bar mechanical linkage subject to random external forcing. The Lagrangian formulation of the equations of motion are index-3 stochastic di erential-algebraic equations (SDAE) that describe the time evolution of the sample paths of the generalized coordinates, velocities, and Lagrange multipliers as stochastic processes. We solve the SDAEs using two di erent approaches: inverse dynamics, Case Study 1, via independent, successive solution of the nonlinear equations for each kinematic variable, where the time evolution of one generalized coordinate is prescribed; and direct dynamics, Case Study 2, via direct solution of the SDAEs in the index-1 formulation, using fourth-order stochastic backward di erentiation formula (BDF) with modi ed Newton iteration and position and velocity stabilization (Ascher and Petzold [2]), where the (deterministic) input driving torque is prescribed. For the particular application of a three-dimensional swing gate security system, we conduct numerical experiments for both approaches. In Case Study 1, we simulate the random external forcing as a Gaussian wind speed process that applies stochastic wind drag onto the gate. The kinematic variables are deterministic, while the required input driving torque is a stochastic process. In Case Study 2, we apply the external forcing as a resistive torque with additive Gaussian noise modeling the wind drag; the kinematic variables are stochastic processes. For both cases, we apply four mean wind speeds: 0 mph (deterministic only), 10 mph, 20 mph, and 30 mph, from which we compute the deterministic solution and three stochastic sample paths for each stochastic process. The overall conclusions are that direct solution is possible for inverse dynamics, that the solution of index-1 SDAEs in multibody dynamics is tractable since the mass matrix is symmetric and positive de nite, and that the deterministic solution is the expectation of the sample paths.
M.S. in Applied Mathematics, December 2011
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Title: L2E ESTIMATOR FOR THE CATEGORICAL MODEL WITH ELASTIC NET PENALTY
Creator: Wang, Yuan
Date: 2017, 2017-07
Description: The logistic regression model is an important generalized linear model for the categorical data. The maximum likelihood estimation is mostly...
Show moreThe logistic regression model is an important generalized linear model for the categorical data. The maximum likelihood estimation is mostly used in estimating the parameters of the logistic regression model. However, the maximum likelihood estimation is very sensitive to outliers which will cause the inaccuracies of the fitted parameters and model selection in high-dimensional regression. Chi and Scott (2014) demonstrated by simulation that minimizing the integrated square error or L2 estimation (L2E) is a robust method to fit 2-class categorical models. They also showed that the L2E estimation method can select the right model even in the presence of many outliers in high dimensional scenarios. In my thesis, I extended the L2E estimation method from 2-class to 3-class based on the MM algorithm by Chi and Scott (2014). Then I demonstrated the properties above for 2-class categorical models are also applicable to 3-class ones.
M.S. in Applied Mathematics, July 2017
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Title: COORDINATE-EXCHANGE ALGORITHM CONSTRUCTION OF UNIFORM SPACE FILLING DESIGN
Creator: Han, Shipeng
Date: 2014, 2014-05
Description: Many scientific phenomena are now investigated by complex computer models. A computer experiment is a sequence of runs with various inputs....
Show moreMany scientific phenomena are now investigated by complex computer models. A computer experiment is a sequence of runs with various inputs. The uniform experimental design seeks its design points to be uniformly scattered on the experimental domain and is one kind of a space-filling design that can be used for computer experiments and OK for industrial experiments. The coordinate-exchange method we use is one-dimensional constrained optimization, searching for the optimal coordinate to make the points filling the space uniformly. This method saves large amounts of calculation compared to the multivariate optimization problems. In this thesis we provide the coordinate-exchange algorithm and demonstrate our methodology with numerical examples. Key words: Coordinate-exchange, Uniform design, Computer experiment, Space-filling design, optimal design.
M.S. in Applied Mathematics, May 2014
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Title: THE CHANGE OF KURTOSIS IN IMPORTANCE SAMPLING FOR MONTE CARLO
Creator: Zhang, Xiaodong
Date: 2013, 2013-12
Description: The Mont e Carlo (II IC) Method is commonly used to approximat e mult ivariat e integrals, which can be interpreted as means of random variab...
Show moreThe Mont e Carlo (II IC) Method is commonly used to approximat e mult ivariat e integrals, which can be interpreted as means of random variab les. The IIIC method uses th e sample mean to estimate the tr ue mean. In this thesis, we focus on minimizing th e sample size in MC simulat ion needed to sat isfy the specified error tolerance. Based on t he algorithm proposed by [5], we explain that t he cost of reliable IIIe est imat ion depends not only on variance but also on kurtosis. T herefore, when we try to improve th e efficiency of MC simulation by reducing variance, such as with Importance Sampling (IS), we need also look into the change of kurtosis. We analyze the change of cost in terms of the change of kur tosis and the change of variance. For a special case of IS we explore how to find th e optimal density in order to reduce variance.
M.S. in Applied Mathematics, December 2013
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Title: SIMULATING THE HESTON MODEL VIA THE QE METHOD WITH A SPECIFIED ERROR TOLERANCE
Creator: Zhao, Xiaoyang
Date: 2017, 2017-05
Description: The Quadratic Exponential (QE) model is a market standard simulation method for the Heston stochastic volatility model. We identify certain...
Show moreThe Quadratic Exponential (QE) model is a market standard simulation method for the Heston stochastic volatility model. We identify certain numerical problems with the standard discretization and modify the original method to correct these problems. We implement our modified QE scheme for the Heston model in the Guaranteed Automatic Integration Library (GAIL)|a suite of algorithms that includes Monte Carlo and quasi-Monte Carlo methods for multidimensional integration and computation of means. GAIL computes answers to satisfy user-defined error tolerances. We also implement variance reduction techniques for our modified QE scheme in GAIL. The numerical results show that our modified scheme is fast and accurate, and satisfies the user-defined error tolerances.
M.S. in Applied Mathematics, May 2017
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Title: MEAN EXIT TIME FOR RADIALLY SYMMETRICAL DYNAMICAL SYSTEMS DRIVEN
Creator: Luan, Yuanchao
Date: 2013, 2013-12
Description: Stochastic differential equations (SDEs) driven by non-Gaussian L´evy noises have attracted much attention recently [1, 29]. In [12], the...
Show moreStochastic differential equations (SDEs) driven by non-Gaussian L´evy noises have attracted much attention recently [1, 29]. In [12], the authors studied a scalar SDE driven by a non-Gaussian L´evy motion, and numerically investigate mean exit time and escape probability for arbitrary noise intensity in one dimensional case. In the present thesis, we utilize a different strategy to explore a numerical method for the problem in two dimensional cases. To be specific, we assume the solution u(x) is radially symmetric with respect to the origin, and then represent the equation using radial coordinate, reducing the problem into one dimensional case. Then main difficulty is that, in the integral term, appears a Gauss Hypergeometric function and the unknown function u(r), which makes the error estimates complicated. We exploit some properties of Gauss Hypergeometric function, and finally make out a way for estimating the error [19]. Up to now we are only able to deal with this problem with 0 < α ≤ 1, since our numerical scheme does not converge when 1 < α < 2. Then we compare our numerical solutions with the analytical ones which are given in [3], and they coincide very well. KeyWords: Stochastic dynamical systems; non-Gaussian L´evy motion; L´evy jump measure; First exit time
M.S. in Applied Mathematics, December 2013
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Title: TWO PROBLEMS ON CROSSING NUMBERS
Creator: Wang, Lujia
Date: 2013-05-01, 2013-05
Description: The crossing number of a graph G, cr(G) is the minimum number of intersections among edges over all possible drawings on a plane. The pairwise...
Show moreThe crossing number of a graph G, cr(G) is the minimum number of intersections among edges over all possible drawings on a plane. The pairwise crossing number pcr(G) is the the minimum number of pairs of edges that cross at least once over drawings. In the rst part of this survey, we deal with the conjecture that pcr(G) = cr(G), and prove that this is true for 4-edge weighted maps on the annulus. Moreover, we develop methods for solving analogous n-edge problems including the classi cation of permutations on a circle. In the second part, we de ne the generalized crossing number cri(G) as the crossing number of a graph on the orientable surface of genus i. The crossing sequence is de ned as (cri(G))g(G) i=0 , where g(G) is the genus of the graph. This part aims at the conjecture that for each sequence of four numbers decreasing to 0, there is some graph with such numbers as its crossing sequence. We come up with a particular family of graphs which have concave crossing sequences of length 4, but partially prove it.
M.S. in Applied Mathematics, May 2013
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Title: NUMERICAL SIMULATIONS OF CURVATURE WEAKENING MODEL OF REACTIVE HELE-SHAW FLOW
Creator: Zhao, Meng
Date: 2013, 2013-12
Description: In this paper, we study a moving interface problem in a Hele-Shaw cell, where two immiscible reactive fluids meet at the interface and...
Show moreIn this paper, we study a moving interface problem in a Hele-Shaw cell, where two immiscible reactive fluids meet at the interface and initiate chemical reactions. A new gel-like phase is produced at the interface and may modify the elastic bending property there. We model the interface as an elastic membrane with a local curva- ture dependent bending rigidity. In the first part of this paper, we review the linear stability analysis on a curvature weakening model, and derive critical flux conditions such that a Hele-Shaw bubble can develop unstable fingering pattern and self-similar morphology. In the second part of this report, we develop a boundary integral nu- merical algorithm to perform nonlinear simulations. Preliminary numerical results show that in the nonlinear regime, there also exist stable self-similar solutions.
M.S. in Applied Mathematics, December 2013
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Title: HYPOTHESIS TESTING FOR STOCHASTIC PDES DRIVEN BY ADDITIVE NOISE
Creator: Xu, Liaosha
Date: 2013, 2013-12
Description: We study hypothesis testing problem for the drift/viscosity coefficient for stochastic fractional heat equation driven by additive space-time...
Show moreWe study hypothesis testing problem for the drift/viscosity coefficient for stochastic fractional heat equation driven by additive space-time white noise colored in space. Since it is the first attempt to deal with hypothesis testing in SPDEs, we assume that the first N Fourier modes of the solution are observed continuously over time interval [0, T], similar methodology could be developed later for discrete sampling. The highlight of this article lies in the notion of “asymptotically the most powerful test” we introduce, which is a brand new idea for hypothesis testing not only in stochastic PDEs but in general stochastic processes. This conception provides a definite criterion how we compare the convergence rates of errors of two tests and how we maximize this convergence rate in a given rejection class when T or N is near infinity. And also we will give some equally important results for controlling the errors with finite T and N. We will build up asymptotic rejection class and find explicit forms of “the most powerful test” in two asymptotic regimes: large time asymptotics T →∞, and increasing number of Fourier modes N → ∞. The proposed statistics are derived based on Maximum Likelihood Ratio. We first consider a simple hypothesis testing, for which we exploit the key technic, by which we continue considering for more general issues. Over the course of proving the main results, we obtain a series of technical results on the asymptotic behaviors of the probabilities related to likelihood ratio, which are also, in some sense, of high value for study in probability theory. In particular, we find the cumulant generating function of the log-likelihood ratio, we obtain some sharp large deviation type results for both T → ∞ and N → ∞, and develop some useful strategies in probability convergence for studying asymptotic properties of the power of the likelihood ratio type tests.
M.S. in Applied Mathematics, December 2013
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Title: AN INVESTIGATION OF THE QUASI-STANDARD ERROR FOR QUASI-MONTE CARLO METHOD
Creator: Deng, Siyuan
Date: 2013-05-01, 2013-05
Description: In this thesis we discuss the theory of the Quasi-Standard Error(QSE) estimate which plays an important role in the practice of the Quasi...
Show moreIn this thesis we discuss the theory of the Quasi-Standard Error(QSE) estimate which plays an important role in the practice of the Quasi-Monte Carlo method. In the first part the deduction using Walsh series reveals an expression for the Quasi- Standard Error for digital nets. The second part of this thesis a special class of functions has been designed to fool the Quasi-Standard Error, and based on the previous theory we reveal the reason why the Quasi-Standard-Error can be fooled. The third part, apply the theory we developed to some actual application in financial mathematics, to see if the QSE works well in practice. There are mixed results.
M.S. in Applied Mathematics, May 2013
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Title: MULTI-LEVEL MONTE CARLO BASED ON THE AUTOMETIC SAMPLE SIZE ALGORITHM
Creator: Li, Yao
Date: 2013, 2013-12
Description: This research's purpose is to optimize an existing method to simulate stochas- tic integrals using Monte Carlo when the cost of function...
Show moreThis research's purpose is to optimize an existing method to simulate stochas- tic integrals using Monte Carlo when the cost of function evaluation is dimension dependent. In the area of mathematical nance, we often need to price a path- dependent nancial derivative. This will result in the computation of E[g(B( ))], where g stands for a payoff function, and B is the Brownian Motion. A simple way to approximate this expectation is to take the average of the functional over a large num- ber of sample paths. Each path is approximated by a d-dimensional random vector. A larger d will provide a more accurate result. However, due to the limitation in time cost and computer memory, some large dimensions are not easy to be implemented. Therefore, we introduce the multi-level technique that is based on multi-grid ideas. It can be used to reduce the computational complexity for these kind of problems. Moreover, when we apply the multi-level technique, the proper sample size for each subspace integration needs to be computed in order to satisfy our guaranteed conser- vative xed width con dence intervals. Thus, the automatic sample size algorithm (two stage con dence interval algorithm) is used in conjunction with the multi-level method.
M.S. in Applied Mathematics, December 2013
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Title: A REVIEW OF ORTHOGONAL LATIN HYPERCUBE DESIGNS FOR COMPUTER EXPERIMENTS
Creator: Jiang, Yin
Date: 2014, 2014-05
Description: Computer models can describe complicated physical phenomena. Due to the highly nonlinear and complex nature, they require specially designed...
Show moreComputer models can describe complicated physical phenomena. Due to the highly nonlinear and complex nature, they require specially designed experimental inputs. One direction of computer experiment design is orthogonal Latin hypercube design which is widely used. This thesis reviews the most resent methods of orthogo- nal Latin hypercube designs, their constructing algorithms and important theoretical properties. These designs are easy to construct and preserve the orthogonal and equally-spaced projections. With large number of factors, orthogonal Latin hyper- cube designs enable researchers to estimate uncorrelated rst-order regression e ects, as well as higher-order e ects. For large number of runs, we reviewed general design method to construct large orthogonal Latin hypercubes from small orthogonal Latin hypercubes. A similar construction method for nearly orthogonal Latin hypercubes was discussed as well.
M.S. in Applied Mathematics, May 2014
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Title: ANALYSIS OF VARIATION IN MULTISTAGE MANUFACTURING PROCESS BASED ON TREE REGRESSION
Creator: Chen, Zhefu
Date: 2014, 2014-05
Description: In a multistage manufacturing process, variation propagates in the process when we produce products from stage to stage. Since there are...
Show moreIn a multistage manufacturing process, variation propagates in the process when we produce products from stage to stage. Since there are limits of reducing variation through process in traditional industrial management and it is hard to monitor the process when the engineering domain knowledge is insufficient. We investigated statistical methods based on piecewise tree regression model including: CART, Bayesian CART, Bayesian treed linear model and Bayesian treed Gaussian process to multistage manufacturing process. The difference performances between models were discussed in a wafer manufacturing process case as a result comparison. Key Words: Multistage process, variation propagation, process monitoring, piecewise tree regression
M.S. in Applied Mathematics, May 2014
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Title: SPECTRALLY ACCURATE BOUNDARY INTEGRAL METHOD FOR FREE SURFACES IN STOKES FLOW
Creator: Kuang, Yin
Date: 2011-04-18, 2011-05
Description: spectrally accurate boundary integral method is developed for solving the velocity of the film flow with suspended particles down an inclined...
Show morespectrally accurate boundary integral method is developed for solving the velocity of the film flow with suspended particles down an inclined plane in Stokes flow. The problem is a two-dimensional gravity-driven film flow with a rigid particle flowing down an inclined plane. We present the governing equations and the numerical methods for solving it with the help of periodic Green’s function. To obtain the system of discretized equations, we discuss the smoothness of each integrand appearing in the boundary integral formulation and use the composite trapezoidal rule for smooth periodic integrands to achieve the spectral accuracy. For the weakly singular integral, we approximate it by a special spectrally accurate quadrature. The Krylov subspace iterative method GMRES is employed to solve the resulting linear system. This method can be also applied to compute the velocity of interface in some other cases of film flow.
M.S. in Applied Mathematics, May 2011
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Title: MULTILEVEL ALGORITHMS FOR PHASE RETRIEVAL
Creator: Tanoue, Cullen
Date: 2015, 2015-05
Description: Phase retrieval is an important optimization problem that arises in di raction imaging, where the original structure of an object needs to be...
Show morePhase retrieval is an important optimization problem that arises in di raction imaging, where the original structure of an object needs to be reconstructed from its measured di raction data that does not have information concerning the phase of the object. Multilevel algorithms can be used to compute solutions to the standard phase retrieval optimization problem by constructing a hierarchy of problems using a series of restriction and prolongation operations. The coarser problems have a quarter of the variables as the ner problems, and hence, there are much less linear algebra requirements for solving the coarser problems. Further, the prolongation of the solutions computed for the coarser problems yield good starting points for the ner problems. We can also use an approach that alternates between solving the coarse and ne problem. Parameters for these methods include the number of levels, prolongation and restriction operations, and the number of iterations to perform at each level. We study the solutions to the standard phase retrieval optimization problem that result from exploring these parameters and compare them to the results obtained from using single-level methods.
M.S. in Applied Mathematics, December 2014
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Title: NONLINEAR SIMULATIONS OF MULTI-VESICLE DYNAMICS
Creator: Hamiilton, Caleb
Date: 2015, 2015-07
Description: Vesicles in biology are closed forms of membranes. The dimensions of vesicles can vary in terms of surface area and enclosed volume. Examples...
Show moreVesicles in biology are closed forms of membranes. The dimensions of vesicles can vary in terms of surface area and enclosed volume. Examples range from small organelles to large cell bodies which all play a variety of resource transportation roles in biological systems. Research from the fields of chemistry and physics helps mathematical modeling by providing the mechanisms behind certain observed morphologies. Mathematical models and methods for simulating vesicle dynamics have produced accurate numerical solutions to verify experimental data and can be used to design new experiments that lead to more discoveries. The most researched case has been a single vesicle under shear flow. However, recent numerical and experimental results consider extensional flows on a single vesicle and hydrodynamic interactions among multiple vesicles. This thesis extends work on hydrodynamic interactions between vesicles in viscous fluid. We investigate numerically cases with multiple vesicles relaxing in asymmetrical configurations, time-dependent flow with more oscillation, and stochastic dynamics. Subjecting vesicles to these various cases reveals sensitivity to initial conditions such as distance and relative orientation. The effects from adding more vesicles are: increased time before equilibrium for the relaxation tests, and distributive wrinkling dynamics for the extensional flow tests. In stochastic cases, there are similar wrinkling distributions. However, initial conditions like distance and orientation have less important effects when competing with influence from thermal fluctuations. Additionally, in the presence of other vesicles under extensional flow, a vesicle may change the number and amplitude of wrinkles it would have experienced alone.
M.S. in Applied Mathematics, July 2015
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Title: MEAN-VARIANCE HEDGING WITH TIME CHANGED LEVY PROCESS
Creator: Liu, J Ingran
Date: 2012-11-17, 2012-12
Description: The goal of this thesis is to consider asset pricing model which driven by an exponential time changed process: Brownian motion with time...
Show moreThe goal of this thesis is to consider asset pricing model which driven by an exponential time changed process: Brownian motion with time changing process{ Poisson process. We rst present the characteristic function of the time change exponential Brown motion and its ltration. Second we exhibit the explicit European call pricing formula then discuss the mean-variance hedging method in this thesis.
M.S. in Applied Mathematics, December 2012
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Title: COMPUTATIONAL COST OF SIMULATING MEAN EXIT TIME USING STOCHASTIC DIFFERENTIAL EQUATIONS
Creator: Liu, Fanjing
Date: 2016, 2016-05
Description: Stochastic di erential equations play an important role in modern science, including engineering, physics, computer science and nance. It has...
Show moreStochastic di erential equations play an important role in modern science, including engineering, physics, computer science and nance. It has been shown that numerically solving stochastic di erential equation is a productive methodto deal with such problems. In this work, we try to analyze the procedure of numerically computing the mean exit time of some stochastic processes from a given boundary using Monte Carlo simulations. The two methods, including the Euler-Maruyama Method and Milstein's higher order method, will be explained and used extensively when we simulate paths of the random process. The simulated processes generated through the methods will then be used to identify the exit times. Later we use the average of the exit times as a numerical solution of Mean Exit Time. We compare the e ciency of the above two methods by evaluating their computational complexity and CPU cost of reaching the same level of accuracy.
M.S. in Applied Mathematics, May 2016
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Title: A STUDY OF HIGH FREQUENCY TRADING IN LIMIT ORDER BOOKS
Creator: Jiang, Yuan
Date: 2013, 2013-12
Description: In the thesis we study the high frequency trading and its applications in limit order books. We discuss the basic concepts and review the...
Show moreIn the thesis we study the high frequency trading and its applications in limit order books. We discuss the basic concepts and review the models in the limit order books. The review section focuses on the queues in the limit order books, optimal trading strategies, short-term volatilities and multi-agent problems in the scenario of limit order markets. Discussions on the shortage of some prevalent models of limit order books are addressed thereafter. For the main results of the thesis, market data are calibrated to facilitate the comparison between a theoretical model and the empirical behaviors in terms of order flows, price changes and diffusion limit of prices.
M.S. in Applied Mathematics, December 2013
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Title: MOTION OF BUBBLY FLUID IN A TANK
Creator: Langman, Michael
Date: 2014, 2014-07
Description: Computational uid dynamics is the numerical study of the motion of uids. In this thesis, an introduction to uid mechanics is presented and the...
Show moreComputational uid dynamics is the numerical study of the motion of uids. In this thesis, an introduction to uid mechanics is presented and the governing equations of uid mechanics are derived. The open-source computational uid dynamics library OpenFOAM is then used to simulate uid dynamics and to model the formation and movement of bubbles in a tank.
M.S. in Applied Mathematics, July 2014
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