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- Title
- COMPUTATION AND ANALYSIS OF TUMOR GROWTH
- Creator
- Turian, Emma
- Date
- 2016, 2016-05
- 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 two-phase Stokes model. We first examine the morphological changes using the surface energy of the tumor-host interface and investigate its nonlinear dynamics using a boundary integral method. In an effort to understand the interface stiffness, we then model the tumor-host interface as an elastic membrane governed by the Helfrich bending energy. Using an energy variation approach, we derive a modified Young-Laplace 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 linear-quadratic 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 non-small 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
- 2011-05-02, 2011-05
- Description
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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
- PHYSICS-PRESERVING FINITE DIFFERENCE SCHEMES FOR THE POISSON-NERNST-PLANCK EQUATIONS
- Creator
- Flavell, Allen
- Date
- 2014, 2014-07
- Description
-
The Poisson-Nernst-Planck equations are a system of nonlinear di erential equations that describe ow of charged particles in solution. This...
Show moreThe Poisson-Nernst-Planck 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 TR-BDF2 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 time-varying 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
- AN ADAPTIVE RESCALING SCHEME FOR COMPUTING HELE-SHAW PROBLEMS
- Creator
- Zhao, Meng
- Date
- 2017, 2017-07
- 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 time-space onto a new time-space 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 Cartesian-grid based method in general. As an example, we use the Hele-Shaw 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, self-similarly evolving morphologies. The rescaling idea can also be used to simulate the fast shrinking interface, e.g. the Hele-Shaw 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 long-time simulations prohibitive. Finally, we study an analytical solution to the stability of the interface of the Hele-Shaw 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
- A MULTI-CURVE LIBOR MARKET MODEL WITH UNCERTAINTIES DESCRIBED BY RANDOM FIELDS
- Creator
- Xu, Shengqiang
- Date
- 2012-12-19, 2012-12
- 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 multi-curve LIBOR market model, with uncertainty described by random fields, is proposed and investigated. This new model is thus called a multi-curve random fields LIBOR market model (MRFLMM). First, the LIBOR market model is reviewed and the closed-form 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 closed-form 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 multi-curve framework, where the curves for generating future forward rates and the curve for discounting cash flows are modeled distinctly but jointly. This multi-curve 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 singe-curve framework. Third, analogous to the single-curve framework, the multi-curve 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 multi-curve random fields LIBOR market model (MRFLMM). Meanwhile, xii local volatility and stochastic volatility models are also combined with the multi-curve LIBOR market model to explain the volatility skews and smiles in the market. Fourth, the calibration of the above models is considered. Taking two-curve setting as an example, four different models, single-curve LIBOR market model, single-curve RFLMM, two-curve LIBOR market model and two-curve 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 non-random fields models and has less pricing error. Moreover, multi-curve models perform better than single-curve 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 non-random fields models, with more accurate in-sample and out-sample pricing and better hedging performance. The multi-curve models also over-perform the single-curve 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 re-calibrate. The multi-curve random fields LIBOR market model has the advantages of both multi-curve 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, 2015-12
- 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 dimension-independent 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
- NON-GAUSSIAN STOCHASTIC DYNAMICS: MODELING, SIMULATION, QUANTIFICATION AND ASSIMILATION
- Creator
- Gao, Ting
- Date
- 2015, 2015-05
- 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 non-Gaussian stochastic dynamics. More specifically, three problems are formulated to investigate non-Gaussian dynamics: (i) exit problem and time-dependent 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 time-dependent 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 non-Gaussian models is studied. For continuous-discrete filtering, a recursive Bayesian approach is used, and for continuous filtering, Zakai equation is solved to provide the system state estimation. In both cases, time-dependent transition probability between metastable states are investigated. xiMotivated by real world applications, three topics - deterministic quantities, uncertainty quantification and data assimilation, are considered for non-Gaussian stochastic dynamics. More specifically, three problems are formulated to investigate non-Gaussian dynamics: (i) exit problem and time-dependent 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 time-dependent 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 non-Gaussian models is studied. For continuous-discrete filtering, a recursive Bayesian approach is used, and for continuous filtering, Zakai equation is solved to provide the system state estimation. In both cases, time-dependent transition probability between metastable states are investigated.
Ph.D. in Applied Mathematics, May 2015
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- Title
- CONSTRUCTIONS IN NON-ADAPTIVE GROUP TESTING STEINER SYSTEMS AND LATIN SQUARES
- Creator
- Balint, Gergely `greg' T.
- Date
- 2014, 2014-05
- Description
-
This thesis explores and introduces new constructions for non-adaptive group testing which are particulary important for the parameter range...
Show moreThis thesis explores and introduces new constructions for non-adaptive group testing which are particulary important for the parameter range we encounter in real life problems. After a summary of existing results, the rst part of this thesis introduces our own constructions, the Latin Square Construction and the Column Augmented Concatenation. Both of these constructions take existing good group testing matrices to create test matrices of larger dimensions. These new matrices are easy to nd for the practical small parameter range we are most interested in. We also address and prove asymptotic results of our Latin Square Construction. In case of the Column Augmented Concatenation the asymptotic results depend greatly on the codes used for the construction. The second part of our work is to address possible ways of augmentation of the Latin Square Construction. Here we explore the di erence in augmentation based on the properties of the starting matrix. In the appendices we give tables of best matrices coming from our constructions with xed, small column weights. We also give a list of the known best 2-disjunct matrices for small row numbers.
PH.D in Applied Mathematics, May 2014
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- Title
- Quantitative Tools for Stochastic Dynamical Systems: Invariant Structures and Escape Probabilities
- Creator
- Kan, Xingye
- Date
- 2012-07-16, 2012-07
- Description
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Three types of quantitative structures, stochastic inertial manifolds, random invariant foliations, and escape probabilities, are investigated...
Show moreThree types of quantitative structures, stochastic inertial manifolds, random invariant foliations, and escape probabilities, are investigated to study stochastic dynamical systems. Invariant structures for stochastic dynamical systems are reviewed and detailed techniques for their simulation, approximation and construction are presented with several illustrative examples. First, a numerical approach for the simulation of inertial manifolds of stochastic evolutionary equations with multiplicative noise is presented and illustrated. After splitting the stochastic evolutionary equations into a backward and a forward part, a numerical scheme is devised for solving this backward-forward stochastic system, and an ensemble of graphs representing the inertial manifold is consequently obtained. This numerical approach is tested in two illustrative examples: one is for a stochastic differential equation and the other is for a stochastic partial differential equation. Second, invariant foliations for dynamical systems with small white noisy perturbation are approximated via asymptotic analysis. In other words, random invariant foliations are represented as a perturbation of the corresponding deterministic invariant foliations, with deviation errors estimated. The escape probability is a deterministic concept making methods of partial differential equations theory attainable to stochastic dynamics. Finally, the escape probability p(x) for dynamical systems driven by non-Gaussian L´evy motions, especially symmetric α-stable L´evy motions, is considered and characterized. More precisely, it is represented as the solution of the Balayage-Dirichlet problem of a certain partial differential-integral equation. This issue has been investigated previously for dynamical systems driven by Wiener process. Differences between escape probabilities for dynamical systems driven by Gaussian and non-Gaussian noises are highlighted.
Ph.D. in Applied Mathematics, July 2012
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- Title
- TOPICS IN COUNTERPARTY RISK AND DYNAMIC CONIC FINANCE
- Creator
- Iyigunler, Ismail
- Date
- 2012-11-02, 2012-12
- Description
-
This thesis consists of three essays about modeling counterparty risk and pricing derivative securities. In the rst essay, we analyze the...
Show moreThis thesis consists of three essays about modeling counterparty risk and pricing derivative securities. In the rst essay, we analyze the counterparty risk embedded in CDS contracts, in presence of a bilateral margin agreement. We focus on the pricing of collateralized counterparty risk, and we derive the bilateral Credit Valuation Adjustment (CVA), unilateral Credit Valuation Adjustment (UCVA), and Debt Valuation Adjustment (DVA). We propose a model for the collateral by incorporating all related factors such as the thresholds, haircuts and margin period of risk. We derive the dynamics of the bilateral CVA in a general form with related jump martingales. Counterparty risky and the counterparty risk-free spread dynamics are derived and the dynamics of the Spread Value Adjustment (SVA) is found as a consequence. We nally employ a Markovian copula model for default intensities and illustrate our ndings with numerical results. In the second essay we address the issue of computation of the bilateral CVA under rating triggers in presence of ratings-linked margin agreements. We consider collateralized OTC contracts, that are subject to rating triggers, between two parties { an investor and a counterparty. Moreover, we model the margin process as a function of the credit ratings of the counterparty and the investor. We employ a Markovian approach for modeling of the rating transitions and of the default probabilities of the counterparties. In this framework, we derive the representation for bilateral CVA. We also introduce a new component in the decomposition of the counterparty risky price: namely the rating valuation adjustment (RVA) that accounts for the rating triggers. We consider several dynamic collateralization schemes where the margin thresholds are linked to the credit ratings of the counterparties. We account for the rehypothecation risk in the presence of independent amounts. Our results are ix illustrated in terms of a CDS contract and an IRS contract. In the third essay, we study the problem of pricing in incomplete markets with risk measures and acceptability indices. We propose a model for nding the dynamic ask and bid prices of derivative securities using Dynamic Coherent Acceptability Indices (DCAI) in the presence of transaction costs. In this framework, we de ne and prove a representation theorem for dynamic bid ask prices. We show that our prices can be computed using dynamic Gain-Loss Ratio (dGLR), which is a DCAI. To illustrate our results, we provide several numerical examples, by pricing barrier options with dGLR.
PH.D in Applied Mathematics, December 2012
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- Title
- CONTRIBUTIONS TO ALGORITHMIC MATROID PROBLEMS
- Creator
- Huang, Jinyu
- Date
- 2015, 2015-07
- Description
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In this thesis, we obtain several algorithms for problems related to matroids, a structure that generalizes the concept of linear independence...
Show moreIn this thesis, we obtain several algorithms for problems related to matroids, a structure that generalizes the concept of linear independence in a vector space and an acyclic subgraph structure in a graph. Matroids have been widely applied in combinatorial optimization, graph theory, coding theory and so forth. Specifically, our results include: In this thesis, we present a constant-competitive online algorithm of the matroid secretary problem for the partition matroids without information of the partition and for the paving matroids. We also introduce the multi-objective matroid secretary problem that extends the matroid secretary problem, in which the weight function is a k-vector w = [w1, · · · , wk]. We show a constant competitive algorithm of the multiobjective matroid secretary problem for the uniform matroids and for the paving matroids. Since bases of a matroid generalize many important combinatorial structures, many counting problems can be expressed as a problem that counts the number of bases of a matroid. An efficient approximate counting algorithm can be designed provided that a rapidly-mixing Markov chain that samples bases of a matroid can be constructed. Let Φ(G) be the conductance of the base-exchange graph G. Matroid- Expansion Conjecture (1989, Mihai and Vazirani) states that Φ(G) ≥ 1 for any base-exchange graph G of a matroid, which implies an FPRAS (fully-polynomial randomized approximation scheme) for counting the number of bases of a matroid. We use λ2, the second smallest eigenvalue of L, the discrete Laplacian matrix of G, to prove the Matroid-Expansion Conjecture for any paving matroid, for any balanced matroid, and for the direct sum of a paving matroid with a balanced matroid. A maximum linear matroid parity set is called a basic matroid parity set, if its size is the rank of the matroid. We show that determining the existence of a common base (basic matroid parity set) for linear matroid intersection (linear matroid parity) is in NC2, provided that there are polynomial number of common bases (basic matroid parity sets). For graphic matroids, we show that finding a common base for matroid intersection is in NC2, if the number of common bases is polynomial bounded. We also give a new RNC2 algorithm that finds a common base for graphic matroid intersection. Finally, we prove that if there is a black-box NC algorithm for PIT (Polynomial Identity Testing), then there is an NC algorithm to determine the existence of a common base (basic matroid parity set) for linear matroid intersection (linear matroid parity).
Ph.D. in Applied Mathematics, July 2015
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- Title
- DYNAMICS OF VESICLES IN VISCOUS FLUID
- Creator
- Liu, Kai
- Date
- 2014, 2014-12
- Description
-
Modeling vesicle dynamics involves a complicated moving boundary problem where uids, thermal uctuations, and vesicle morphology are intimately...
Show moreModeling vesicle dynamics involves a complicated moving boundary problem where uids, thermal uctuations, and vesicle morphology are intimately coupled. In this thesis, we study the dynamics of a two-dimensional membrane in linear viscous ows. In the asymptotic analysis section, we derive deterministic and stochastic equations describing the motion of a slightly perturbed membrane interface. Using a 2nd order Runge-Kutta method, we solve these equations numerically, and explain the formation and development of wrinkling patterns. We then develop a boundary integral method and an immersed boundary method for simulating the nonlinear wrinkling dynamics of a homogenous vesicle in viscous ows. The nonlinear results agree with the asymptotic theory for a nearly circular vesicle, and also agree with experimental results for an elongated vesicle. Using a stochastic immersed boundary method, we investigate the e ects of thermal uctuations in vesicle dynamics. Comparing with the deterministic results, thermal uctuation can lead to the development of odd modes and asymmetric wrinkles. Finally, we investigate the nonlinear wrinkling dynamics of a multi-component vesicle. The model includes a 4th order Cahn-Hilliard type equation describing the phase transitions on the vesicle surface. We nd that for an elongated vesicle with large excess arc length, the inhomogeneous bending introduces nontrivial asymmetric wrinkling and buckling dynamics.
Ph.D. in Applied Mathematics, December 2014
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- Title
- AN ENERGY-PRESERVING SCHEME FOR THE POISSON-NERNST-PLANCK EQUATIONS
- Creator
- Kabre, Julienne
- Date
- 2017, 2017-07
- Description
-
Transport of ionic particles is ubiquitous in all biology. The Poisson-Nernst- Planck (PNP) equations have recently been used to describe the...
Show moreTransport of ionic particles is ubiquitous in all biology. The Poisson-Nernst- Planck (PNP) equations have recently been used to describe the dynamics of ion transport through biological ion channels (besides being widely employed in semiconductor industry). This dissertation is about the design of a numerical scheme to solve the PNP equations that preserves exactly (up to roundoff error) a discretized form of the energy dynamics of the system. The proposed finite difference scheme is of second-order accurate in both space and time. Comparisons are made between this energy dynamics preserving scheme and a standard finite difference scheme, showing a difference in satisfying the energy law. Numerical results are presented for validating the orders of convergence in both time and space of the new scheme for the PNP system. The energy preserving scheme presented here is one dimensional in space. A highlight of an extension to the multi-dimensional case is shown.
Ph.D. in Applied Mathematics, July 2017
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- Title
- ADAPTIVE QUASI-MONTE CARLO CUBATURE
- Creator
- Jimenez Rugama, Lluis Antoni
- Date
- 2016, 2016-12
- Description
-
In some definite integral problems the analytical solution is either unknown or hard to compute. As an alternative, one can approximate the...
Show moreIn some definite integral problems the analytical solution is either unknown or hard to compute. As an alternative, one can approximate the solution with numerical methods that estimate the value of the integral. However, for high dimensional integrals many techniques suffer from the curse of dimensionality. This can be solved if we use quasi-Monte Carlo methods which do not suffer from this phenomenon. Section 2.2 describes digital sequences and rank-1 lattice node sequences, two of the most common points used in quasi-Monte Carlo. If one uses quasi-Monte Carlo, there is still another problem to address: how many points are needed to estimate the integral within a particular absolute error tolerance. In this dissertation, we propose two automatic cubatures based on digital sequences and rank-1 lattice node sequences that estimate high dimensional problems. These new algorithms are constructed in Chapter 3 and the user-specified absolute error tolerance is guaranteed to be satisfied for a specific set of integrands. In Chapter 4 we define a new estimator that satisfies a generalized tolerance function and includes a relative error tolerance option. An important property of quasi-Monte Carlo methods is that they are effective when the function has low effective dimension. In [1], Sobol’ defined the global sensitivity indices, which measure what part of the variance is explained by each dimension. We can use these indices to measure the effective dimensionality of a function. In Chapter 5 we extend our digital sequences cubature to estimate first order and total effect Sobol’ indices.
Ph.D. in Applied Mathematics, December 2016
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- Title
- GUARANTEED ADAPTIVE MONTE CARLO METHODS FOR ESTIMATING MEANS OF RANDOM VARIABLES
- Creator
- Jiang, Lan
- Date
- 2016, 2016-05
- Description
-
Monte Carlo is a versatile computational method that may be used to approximate the means, μ, of random variables, Y , whose distributions are...
Show moreMonte Carlo is a versatile computational method that may be used to approximate the means, μ, of random variables, Y , whose distributions are not known explicitly. This thesis investigates how to reliably construct fixed width confidence intervals for μ with some prescribed absolute error tolerance, "a, relative error tolerance, "r or some generalized error criterion. To facilitate this, it is assumed that the kurtosis, , of the random variable, Y , does not exceed a user specified bound max. The key idea is to confidently estimate the variance of Y by applying Cantelli’s Inequality. A Berry-Esseen Inequality makes it possible to determine the sample size required to construct such a confidence interval. When relative error is involved, this requires an iterative process. This idea for computing μ = E(Y ) can be used to develop a numerical integration method by writing the integral as μ = E(f(x)) = RRd f(x)⇢(x)dx, where x is a d dimensional random vector with probability density function ⇢. A similar idea is used to develop an algorithm for computing p = E(Y) where Y is a Bernoulli random variable. All of the algorithms have been implemented in the Guaranteed Automatic Integration Library (GAIL).
Ph.D. in Applied Mathematics, May 2016
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- Title
- The Relationship Between Default and Volatility and Its Impact on Counterparty Credit Risk
- Creator
- Yang, Jiarui
- Date
- 2012-07-16, 2012-07
- Description
-
This thesis presents a uni ed framework for studying the impact of the correlation between interest rate volatility and counterparty default...
Show moreThis thesis presents a uni ed framework for studying the impact of the correlation between interest rate volatility and counterparty default probability on the credit risk of collateralized interest-rate derivative contracts. A defaultable term structure model is proposed in which the default risk is correlated with interest rate volatility. In particular, an existence and uniqueness theorem of this model is proved. The pricing formula of credit derivatives under the proposed model is derived and the stochastic interest rate model and credit model are calibrated together . Finally, given all the parameters calibrated by the unscented Kalman lter, a sensitivity analysis of the impact of the correlation between interest rate volatility and a counterparty's default probability on the credit risk of collateralized interest-rate derivative contracts is presented.
Ph.D. in Applied Mathematics, July 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
- 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
- TOPICS IN STATISTICAL MODELING AND OPTIMAL DESIGN
- Creator
- Li, Yiou
- Date
- 2014, 2014-07
- Description
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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
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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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