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 Title
 DYNAMICS OF A THREEDIMENSIONAL FOURBAR LINKAGE SUBJECT TO RANDOM EXTERNAL FORCING
 Creator
 Lytell, Mark R.
 Date
 20111115, 201112
 Description

This thesis explores the dynamics of a threedimensional fourbar mechanical linkage subject to random external forcing. The Lagrangian...
Show moreThis thesis explores the dynamics of a threedimensional fourbar mechanical linkage subject to random external forcing. The Lagrangian formulation of the equations of motion are index3 stochastic di erentialalgebraic 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 index1 formulation, using fourthorder 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 threedimensional 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 index1 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, 201707
 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 highdimensional regression. Chi and Scott (2014) demonstrated by simulation that minimizing the integrated square error or L2 estimation (L2E) is a robust method to fit 2class 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 2class to 3class based on the MM algorithm by Chi and Scott (2014). Then I demonstrated the properties above for 2class categorical models are also applicable to 3class ones.
M.S. in Applied Mathematics, July 2017
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 Title
 COORDINATEEXCHANGE ALGORITHM CONSTRUCTION OF UNIFORM SPACE FILLING DESIGN
 Creator
 Han, Shipeng
 Date
 2014, 201405
 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 spacefilling design that can be used for computer experiments and OK for industrial experiments. The coordinateexchange method we use is onedimensional 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 coordinateexchange algorithm and demonstrate our methodology with numerical examples. Key words: Coordinateexchange, Uniform design, Computer experiment, Spacefilling design, optimal design.
M.S. in Applied Mathematics, May 2014
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 Title
 COMPUTATION AND ANALYSIS OF TUMOR GROWTH
 Creator
 Turian, Emma
 Date
 2016, 201605
 Description

The ability of tumors to metastasize is preceded by morphological instabilities such as chains or fingers that invade the host environment....
Show moreThe ability of tumors to metastasize is preceded by morphological instabilities such as chains or fingers that invade the host environment. Parameters that control tumor morphology may also contribute to its invasive ability. In this thesis, we investigate tumor growth using a twophase Stokes model. We first examine the morphological changes using the surface energy of the tumorhost interface and investigate its nonlinear dynamics using a boundary integral method. In an effort to understand the interface stiffness, we then model the tumorhost interface as an elastic membrane governed by the Helfrich bending energy. Using an energy variation approach, we derive a modified YoungLaplace condition for the stress jump across the interface, and perform a linear stability analysis to evaluate the effects of viscosity, bending rigidity, and apoptosis on tumor morphology. Results show that increased bending rigidity versus mitosis rate contributes to a more stable growth. On the other hand, increased tumor viscosity or apoptosis may lead to an invasive fingering morphology. Comparison with experimental data on glioblastoma spheroids shows good agreement especially for tumors with high adhesion and low proliferation. Next, we evaluate tumor regression during cancer therapy by a combined modality involving chemotherapy and radiotherapy. The goal is to address the complexities of a vascular tumor (e.g. apoptosis and vascularization) during treatment. We introduce an apoptotic time delay and study its impact on tumor regression using numerical and asymptotic techniques. In particular, we implement the linearquadratic model and identify two extreme sets of parameter data, namely the slow, and fast tumor response to therapy. Numerical simulations for the slow response set show good agreements with data representing nonsmall cell lung carcinoma. Using the evolution equation for tumor radius with time delay, we find that tumors with shorter time interval to the onset of apoptosis shrink faster.
Ph.D. in Applied Mathematics, May 2016
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 Title
 DYNAMIC COHERENT ACCEPTABILITY INDICES AND THEIR APPLICATIONS IN FINANCE
 Creator
 Zhang, Zhao
 Date
 20110502, 201105
 Description

This thesis presents a unified framework for studying coherent acceptability indices in a dynamic setup. We study dynamic coherent...
Show moreThis thesis presents a unified framework for studying coherent acceptability indices in a dynamic setup. We study dynamic coherent acceptability indices and dynamic coherent risk measures. In particular, we establish a duality between them. We derive representation theorems for both dynamic coherent acceptability indices and dynamic coherent risk measures in terms of so called dynamically consistent sequence of sets of probability measures. In addition, we present an alternative approach to study dynamic coherent acceptability indices and the representation theorem. Finally, we provide examples and counterexamples of dynamic coherent acceptability indices, and their applications in portfolio management.
Ph.D. in Applied Mathematics, May 2011
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 Title
 PHYSICSPRESERVING FINITE DIFFERENCE SCHEMES FOR THE POISSONNERNSTPLANCK EQUATIONS
 Creator
 Flavell, Allen
 Date
 2014, 201407
 Description

The PoissonNernstPlanck equations are a system of nonlinear di erential equations that describe ow of charged particles in solution. This...
Show moreThe PoissonNernstPlanck equations are a system of nonlinear di erential equations that describe ow of charged particles in solution. This dissertation is about the design of numerical schemes to solve this system which preserves global properties exhibited by the system. There are two major advances presented. The rst is the design of schemes that conserve mass globally when the system is coupled with no ux boundary conditions. Most notably, a scheme using central di erencing and TRBDF2 achieves second order accuracy in both space and time, while also conserving global mass is presented. The second is the design of a more general scheme that preserves the timevarying properties of the free energy of the system. One such a scheme uses central di erencing in space and trapezoidal integration in time to achieve second order accuracy in both space and time, while also preserving the energy dynamics, but at the cost of requiring positivity of the solution. There is also a discussion of solution methods: the classic Newton iteration scheme is compared with a modi ed Gummel iteration scheme for the purpose of solving the transient equations. The intended application of this work is the modeling of ion channels, and many of the simulations presented use parameters consistent with models of ion channels.
Ph.D. in Applied Mathematics, July 2014
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 Title
 THE CHANGE OF KURTOSIS IN IMPORTANCE SAMPLING FOR MONTE CARLO
 Creator
 Zhang, Xiaodong
 Date
 2013, 201312
 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, 201705
 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 quasiMonte Carlo methods for multidimensional integration and computation of means. GAIL computes answers to satisfy userdefined 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 userdefined 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, 201312
 Description

Stochastic differential equations (SDEs) driven by nonGaussian L´evy noises have attracted much attention recently [1, 29]. In [12], the...
Show moreStochastic differential equations (SDEs) driven by nonGaussian L´evy noises have attracted much attention recently [1, 29]. In [12], the authors studied a scalar SDE driven by a nonGaussian 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; nonGaussian L´evy motion; L´evy jump measure; First exit time
M.S. in Applied Mathematics, December 2013
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 Title
 AN ADAPTIVE RESCALING SCHEME FOR COMPUTING HELESHAW PROBLEMS
 Creator
 Zhao, Meng
 Date
 2017, 201707
 Description

In this thesis, we develop efficient adaptive rescaling schemes to investigate interface instabilities associated with moving interface...
Show moreIn this thesis, we develop efficient adaptive rescaling schemes to investigate interface instabilities associated with moving interface problems. The idea of rescaling is to map the current timespace onto a new timespace frame such that the interfaces evolve at a chosen speed in the new frame. We couple the rescaling idea with boundary integral method to demonstrate the efficiency of the rescaling idea, though it can be applied to Cartesiangrid based method in general. As an example, we use the HeleShaw problem to examine the efficiency of the rescaling scheme. First, we apply the rescaling scheme to a slowly expanding interface. In the new frame, the evolution is dramatically accelerated, while the underlying physics remains unchanged. In particular, at long times numerical results reveal that there exist nonlinear, stable, selfsimilarly evolving morphologies. The rescaling idea can also be used to simulate the fast shrinking interface, e.g. the HeleShaw problem with a time dependent gap. In this case, the rescaling scheme slows down the interface evolution in the new frame to remove the severe time step constraint that makes the longtime simulations prohibitive. Finally, we study an analytical solution to the stability of the interface of the HeleShaw problem, assuming a small surface tension under a time dependent flux Q(t). Following [116, 109], we find the motions of daughter singularity ζd and simple singularity ζ0 do not depend on the flux Q(t). We also find a criterion to identify the relation between ζ0 and ζd.
Ph.D. in Applied Mathematics, July 2017
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 Title
 TWO PROBLEMS ON CROSSING NUMBERS
 Creator
 Wang, Lujia
 Date
 20130501, 201305
 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 4edge weighted maps on the annulus. Moreover, we develop methods for solving analogous nedge 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 HELESHAW FLOW
 Creator
 Zhao, Meng
 Date
 2013, 201312
 Description

In this paper, we study a moving interface problem in a HeleShaw cell, where two immiscible reactive fluids meet at the interface and...
Show moreIn this paper, we study a moving interface problem in a HeleShaw cell, where two immiscible reactive fluids meet at the interface and initiate chemical reactions. A new gellike 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 HeleShaw bubble can develop unstable fingering pattern and selfsimilar 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 selfsimilar 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, 201312
 Description

We study hypothesis testing problem for the drift/viscosity coefficient for stochastic fractional heat equation driven by additive spacetime...
Show moreWe study hypothesis testing problem for the drift/viscosity coefficient for stochastic fractional heat equation driven by additive spacetime 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 loglikelihood 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
 A MULTICURVE LIBOR MARKET MODEL WITH UNCERTAINTIES DESCRIBED BY RANDOM FIELDS
 Creator
 Xu, Shengqiang
 Date
 20121219, 201212
 Description

The LIBOR (London Interbank Offered Rate) market model has been widely used as an industry standard model for interest rates modeling and...
Show moreThe LIBOR (London Interbank Offered Rate) market model has been widely used as an industry standard model for interest rates modeling and interest rate derivatives pricing. In this thesis, a multicurve LIBOR market model, with uncertainty described by random fields, is proposed and investigated. This new model is thus called a multicurve random fields LIBOR market model (MRFLMM). First, the LIBOR market model is reviewed and the closedform formulas for pricing caplets and swaptions are provided. It is extended to the case when the uncertainty terms are modeled as random fields and consequently the closedform formulas for pricing caplets and swaptions are derived. This is a new model called the random fields LIBOR market model (RFLMM). Second, local volatility models and stochastic volatility models are combined with the RFLMM to explain the volatility skews or smiles observed in market. Closedform volatility formulas are derived via the lognormal mixture model in local volatility case, while the approximation scheme for the stochastic volatility case is obtained by a stochastic Taylor expansion method. Moreover, the above work is further extended to a multicurve framework, where the curves for generating future forward rates and the curve for discounting cash flows are modeled distinctly but jointly. This multicurve methodology is recently introduced lately by some pioneers to explain the inconsistency of interest rates after the 2008 credit crunch. Both LIBOR market model and RFLMM mentioned above can be categorized as models in singecurve framework. Third, analogous to the singlecurve framework, the multicurve random fields LIBOR market model is derived and caplets and swaptions are priced with closedform formulas that can be reduced to exactly the Black’s formulas. This model is called a multicurve random fields LIBOR market model (MRFLMM). Meanwhile, xii local volatility and stochastic volatility models are also combined with the multicurve LIBOR market model to explain the volatility skews and smiles in the market. Fourth, the calibration of the above models is considered. Taking twocurve setting as an example, four different models, singlecurve LIBOR market model, singlecurve RFLMM, twocurve LIBOR market model and twocurve RFLMM are compared. The calibration is based on the spot market data on one trading day. The four models are calibrated to European cap volatility surface and swaption volatilities, given the specified parameterized form of correlation and instantaneous volatility. The calibration results show that the random fields models capture the volatility smiles better than nonrandom fields models and has less pricing error. Moreover, multicurve models perform better than singlecurve models, especially during/after credit crunch. Finally, the estimation of these four models, including pricing and hedging performance, is considered. The estimation uses time series of forward rates in market. Given a time series of term structure, the parameters of the four models are estimated using unscented Kalman filter (UKF). The results show that the random fields models have better estimation results than nonrandom fields models, with more accurate insample and outsample pricing and better hedging performance. The multicurve models also overperform the singlecurve models. In addition, it is shown theoretically and empirically that the random fields models have advantages that it is unnecessary to determine the number of factors in advance and not needed to recalibrate. The multicurve random fields LIBOR market model has the advantages of both multicurve framework and random fields setting.
PH.D in Applied Mathematics, December 2012
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 Title
 FUNCTION APPROXIMATION WITH KERNEL METHODS
 Creator
 Zhou, Xuan
 Date
 2015, 201512
 Description

This dissertation studies the problem of approximating functions of d variables in a separable Banach space Fd. In particular we are...
Show moreThis dissertation studies the problem of approximating functions of d variables in a separable Banach space Fd. In particular we are interested in convergence and tractability results in the worst case setting and in the average case setting. The symmetric positive definite kernel in both settings is of a product form Kd(x, t) := d =1 1 − α2 + α2 Kγ (x , t ) for all x, t ∈ Rd. The kernel Kd generalizes the anisotropic Gaussian kernel, whose tractability properties have been established in the literature. For a fixed d, we study rates of convergence, which indicate how quickly approximation errors decay. Since rates of convergence can deteriorate quickly as d increases, it is desirable to have dimensionindependent convergence rates, which corresponds to the concept of strong polynomial tractability. We present sufficient conditions on {α }∞ =1 and {γ }∞ =1 under which strong polynomial tractability holds for function approximation problems in Fd. Numerical examples are presented to support the theory and guaranteed automatic algorithms are provided to solve the function approximation problem in a straightforward and efficient way. viii
Ph.D. in Applied Mathematics, December 2015
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 Title
 AN INVESTIGATION OF THE QUASISTANDARD ERROR FOR QUASIMONTE CARLO METHOD
 Creator
 Deng, Siyuan
 Date
 20130501, 201305
 Description

In this thesis we discuss the theory of the QuasiStandard Error(QSE) estimate which plays an important role in the practice of the Quasi...
Show moreIn this thesis we discuss the theory of the QuasiStandard Error(QSE) estimate which plays an important role in the practice of the QuasiMonte 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 QuasiStandard Error, and based on the previous theory we reveal the reason why the QuasiStandardError 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
 MULTILEVEL MONTE CARLO BASED ON THE AUTOMETIC SAMPLE SIZE ALGORITHM
 Creator
 Li, Yao
 Date
 2013, 201312
 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 ddimensional 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 multilevel technique that is based on multigrid ideas. It can be used to reduce the computational complexity for these kind of problems. Moreover, when we apply the multilevel 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 multilevel 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, 201405
 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 equallyspaced projections. With large number of factors, orthogonal Latin hyper cube designs enable researchers to estimate uncorrelated rstorder regression e ects, as well as higherorder 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
 NONGAUSSIAN STOCHASTIC DYNAMICS: MODELING, SIMULATION, QUANTIFICATION AND ASSIMILATION
 Creator
 Gao, Ting
 Date
 2015, 201505
 Description

Motivated by real world applications, three topics  deterministic quantities, uncertainty quantification and data assimilation, are...
Show moreMotivated by real world applications, three topics  deterministic quantities, uncertainty quantification and data assimilation, are considered for nonGaussian stochastic dynamics. More specifically, three problems are formulated to investigate nonGaussian dynamics: (i) exit problem and timedependent probability density; (ii) parameter and function estimation for stochastic differential equations driven by L´evy motion; and (iii) nonlinear data assimilation to infer transition phenomena. First, numerical algorithms are developed to study important metrics: mean exit time, escape probability and timedependent probability density, which can be utilized to quantify dynamical behaviors of stochastic differential equations with non Gaussian stable L´evy motion. Moreover, detailed numerical analysis work is done to ensure the algorithms accurate, fast and stable considering the singular nature of the L´evy jump measure. Second, new approaches on parameter and function estimation in stochastic dynamical systems are devised. Taking advantage of observations on mean exit time, escape probability or probability density, model uncertainty can be quantified by some optimization methods. These methods are beneficial to systems for which mean exit time, escape probability or probability density are feasible to observe. Finally, nonlinear data assimilation on nonGaussian models is studied. For continuousdiscrete filtering, a recursive Bayesian approach is used, and for continuous filtering, Zakai equation is solved to provide the system state estimation. In both cases, timedependent transition probability between metastable states are investigated. xiMotivated by real world applications, three topics  deterministic quantities, uncertainty quantification and data assimilation, are considered for nonGaussian stochastic dynamics. More specifically, three problems are formulated to investigate nonGaussian dynamics: (i) exit problem and timedependent probability density; (ii) parameter and function estimation for stochastic differential equations driven by L´evy motion; and (iii) nonlinear data assimilation to infer transition phenomena. First, numerical algorithms are developed to study important metrics: mean exit time, escape probability and timedependent probability density, which can be utilized to quantify dynamical behaviors of stochastic differential equations with non Gaussian stable L´evy motion. Moreover, detailed numerical analysis work is done to ensure the algorithms accurate, fast and stable considering the singular nature of the L´evy jump measure. Second, new approaches on parameter and function estimation in stochastic dynamical systems are devised. Taking advantage of observations on mean exit time, escape probability or probability density, model uncertainty can be quantified by some optimization methods. These methods are beneficial to systems for which mean exit time, escape probability or probability density are feasible to observe. Finally, nonlinear data assimilation on nonGaussian models is studied. For continuousdiscrete filtering, a recursive Bayesian approach is used, and for continuous filtering, Zakai equation is solved to provide the system state estimation. In both cases, timedependent transition probability between metastable states are investigated.
Ph.D. in Applied Mathematics, May 2015
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 Title
 ANALYSIS OF VARIATION IN MULTISTAGE MANUFACTURING PROCESS BASED ON TREE REGRESSION
 Creator
 Chen, Zhefu
 Date
 2014, 201405
 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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