Search results

Title: Hilbert Polynomial of the Kimura 3-Parameter Model, AS2012 Special Volume, part 1: This issue includes a second series of papers from talks, posters and collaborations resulting from and inspired by the Algebraic Statistics in the Alleghenies Conference at Penn State, which took place in July 2012.
Creator: Kubjas, Kaie, inspired by the Algebraic Statistics in the Alleghenies Conference at Penn State, which took place in July
Description: In [2] Buczyn ́ska and Wi ́sniewski showed that the Hilbert polynomial of the algebraic variety associated to the Jukes-Cantor binary model on...
Show moreIn [2] Buczyn ́ska and Wi ́sniewski showed that the Hilbert polynomial of the algebraic variety associated to the Jukes-Cantor binary model on a trivalent tree depends only on the number of leaves of the tree and not on its shape. We ask if this can be generalized to other group-based models. The Jukes-Cantor binary model has Z2 as the underlying group. We consider the Kimura 3-parameter model with Z2 × Z2 as the underlying group. We show that the generalization of the statement about the Hilbert polynomials to the Kimura 3-parameter model is not possible as the Hilbert polynomial depends on the shape of a trivalent tree.
Show less
Collection: Journal of Algebraic Statistics

Title: Mathematics of Civil Infrastructure Network Optimization
Creator: Rumpf, Adam Andrew
Date: 2020
Description: We consider a selection of problems from civil infrastructure network design that are of great importance in modern urban planning but have,...
Show moreWe consider a selection of problems from civil infrastructure network design that are of great importance in modern urban planning but have, until relatively recently, gone largely ignored in mathematical literature. Each of these problems is approached from the perspective of network optimization-based modeling, with a major focus placed on the development of efficient solution algorithms.We begin with a study of the phenomenon of interdependent civil infrastructure networks, wherein the functionality of one network (such as a telecommunications system) requires the input of resources from another network (such as the electrical power grid). We first consider a linear relaxation of an established binary interdependence minimum-cost network flows model, including its unique modeling applications and its use as part of a randomized rounding approximation algorithm for the mixed integer model. We also develop a generalized network simplex algorithm for the efficient solution of this generalized minimum-cost network flows problem. We then move on to consider a trilevel network interdiction game for use in planning the fortification of interdependent networks subject to targeted attacks. A variety of solution algorithms are developed for both the binary and the linear interdependence models, and the linear interdependence model is used to develop an approximation algorithm for the more computationally expensive binary model.We then develop a public transit network design model which incorporates a social access objective in addition to traditional operator cost and user cost objectives. The model is meant for use in planning minor modifications to a public transit network capable of improving equity of access to important services while guaranteeing that service levels remain within a specified tolerance of their initial values. A hybrid tabu search/simulated annealing algorithm is developed to solve this model, which is then applied to a test case based on the Chicago public transit network with the objective of improving equity of primary health care access across the city.
Show less

Title: AN ACCELERATING COUETTE FLOW IN NEK5000: APPLICATIONS IN OCEANOGRAPHY AND MAGNETOHYDRODYNAMICS
Creator: Miksis, Zachary M.
Date: 2017, 2017-05
Description: Nek5000 is a highly scalable spectral element code used in a broad array of problems in computational fluid dynamics. In this thesis, we focus...
Show moreNek5000 is a highly scalable spectral element code used in a broad array of problems in computational fluid dynamics. In this thesis, we focus on applying the code to a model problem of an accelerating Couette flow, or a hydrodynamic flow between two plates, of which the top plate is accelerating and the bottom plate is stationary, and verifying the numerical methods as applied to this problem. We obtain an analytical solution to the hydrodynamic flow problem, and use this to analyze the effects of changing time step length, the size of the computational mesh, and the computational polynomial order on the accuracy and stability of Nek5000. Additionally, we discuss the addition of an applied magnetic field to the hydrodynamic Couette flow, and provide a formulation for an exact solution to this magnetohydrodynamic problem that can be used to further verify Nek5000 in a similar fashion to the hydrodynamic problem.
M.S. In Applied Mathematics, May 2017
Show less

Title: RUNTIME FOR PERFORMING EXACT TESTS ON THE PI STATISTICAL MODEL FOR RANDOM GRAPHS
Creator: Dillon, Martin
Date: 2016, 2016-05
Description: In statistics, we ask whether some statistical model ts observed data. We use a Markov chain proposed by Gross, Petrovi c, and Stasi to...
Show moreIn statistics, we ask whether some statistical model ts observed data. We use a Markov chain proposed by Gross, Petrovi c, and Stasi to perform exact testing for the p1 random graph model. By comparing it to the simple switch Markov chain, we prove that it mixes rapidly on many classes of degree sequences, and we discuss why it is sometimes better suited than the simple switch chain, and try to easily introduce the concepts from the general theory along the way.
M.S. in Applied Mathematics, May 2016
Show less

Title: A Gröbner Basis Database
Creator: Mojsilović, Jelena
Date: 2022

Title: Stephen Fienberg's influence on algebraic statistics, Special Volume in honor of memory of S.E.Fienberg
Creator: Petrović, Sonja, Slavkovic, Aleksandra, Yoshida, Ruriko
Date: 2019, 2019-04-12
Description: Stephen (Steve) E. Fienberg (1942-2016) was an eminent statistician, whose impact on research, education and the practice of statistics, and...
Show moreStephen (Steve) E. Fienberg (1942-2016) was an eminent statistician, whose impact on research, education and the practice of statistics, and many other fields is astonishing in its breadth. He was a visionary when it came to linking many different areas to address real scientific issues. He professed the importance of statistics in many disciplines, but recognized that true interdisciplinary work requires joining of the expertise across different areas, and it is in this spirit that he helped steer algebraic statistics toward becoming a thriving subject. Many of his favorite topics in the area are covered in this special issue. We are grateful to all authors for contributing to this volume to honor him and his influence on the field. During the preparation of this issue, we learned about the tragic killing of his widow, Joyce Fienberg, during the Tree of Life Synagogue massacre in Pittsburgh, PA on October 27, 2018. This issue is dedicated to their memory.
Show less
Collection: Journal of Algebraic Statistics

Title: Algorithms for Discrete Data in Statistics and Operations Research
Creator: Schwartz, William K.
Date: 2021-11-19, 2021-12
Publisher: ProQuest, https://www.proquest.com/docview/2622985712
Description: Sponsorship: The Air Force Office of Scientific Research's grant FA9550-14-1-0141 supported Prof. Petrović's and my initial work on this project.

Title: Applications of Optimal Contract Theory in Brokerage
Creator: Alonso Alvarez, Guillermo
Date: 2023
Description: In this thesis we study optimal brokerage problems in different scenarios. The thesisis structured in two parts:...
Show moreIn this thesis we study optimal brokerage problems in different scenarios. The thesisis structured in two parts: In the first part of this thesis, corresponding to Chapter 2 and 3, we construct optimal brokerage contracts for multiple (heterogeneous) clients trading a single asset whose price follows the Almgren-Chriss model. The distinctive features of this work are as follows: (i) the reservation values of the clients are determined endogenously, and (ii) the broker is allowed to not offer a contract to some of the potential clients, thus choosing her portfolio of clients strategically. We find a computationally tractable characterization of the optimal portfolios of clients (up to a digital optimization problem, which can be solved efficiently if the number of potential clients is small) and conduct numerical experiments which illustrate how these portfolios, as well as the equilibrium profits of all market participants, depend on the price impact coefficients. In the second part of this thesis, corresponding to Chapter 4, we establish existence of a solution to the optimal contract problem in models where the state process is given by a multidimensional diffusion with linearly controlled drift. Then, under certain concavity assumptions, we show that the optimal contracts in the relaxed formulation also solve the associated strong optimal contract problem. The main advantages of this approach, relative to the existing methods, are due to the fact that it allows (i) to obtain the existence of an optimal contract (as a limit point of epsilon-optimal ones), and (ii) to include various additional constraints on the associated control problems (e.g., state constraints, difference in filtrations of the agent and of the principal, etc.). Finally, we apply our results to the problem of brokerage fees when the agent has access to a larger filtration.
Show less

Title: Latent Price Model for Market Microstructure: Estimation and Simulation
Creator: Yin, Yuan
Date: 2023
Description: This thesis focuses on exploring and solving several problems based on partiallyobserved diffusion models. The thesis has two parts....
Show moreThis thesis focuses on exploring and solving several problems based on partiallyobserved diffusion models. The thesis has two parts. In the first part we present a tractable sufficient condition for the consistency of maximum likelihood estimators (MLEs) in partially observed diffusion models, stated in terms of stationary distributions of the associated test processes, under the assumption that the set of unknown parameter values is finite. We illustrate the tractability of this sufficient condition by verifying it in the context of a latent price model of market microstructure. Finally, we describe an algorithm for computing MLEs in partially observed diffusion models and test it on historical data to estimate the parameters of the latent price model. In the second part we provide a thorough analysis of the particle filtering algorithm for estimating the conditional distribution in partially observed diffusion models. Specifically, we focus on estimating the distribution of unobserved processes using observed data. The algorithm involves several steps and assumptions, which are described in detail. We also examine the convergence of the algorithm and identify the sufficient conditions under which it converges. Finally, we derive an explicit upper bound of the convergence rate of the algorithm, which depends on the set of parameters and the choice of time frequency. This bound provides a measure of the algorithm’s performance and can be used to optimize its parameters to achieve faster convergence.
Show less

Title: Phase field modeling and computation of vesicle growth or shrinkage
Creator: Tang, Xiaoxia
Date: 2023
Description: Lipid bilayers are the basic structural component of all biological cell membranes. It is a semipermeable barrier to most solutes, including...
Show moreLipid bilayers are the basic structural component of all biological cell membranes. It is a semipermeable barrier to most solutes, including ions, glucoses, proteins and other molecules. Vesicles formed by a bilayer lipid membrane are often used as a model system for studying fundamental physics underlying complicated biological systems such as cells and microcapsules. Mathematical modeling of membrane deformation has become an important topic in biological and industrial system for a long time. In this thesis, we develop a phase field model for vesicle growth or shrinkage based on osmotic pressure that arises due to a chemical potential gradient. This thesis consists of three main parts.In the first part, we establish a phase field model for vesicle growth or shrinkage without flow. It consists of an Allen-Cahn equation, which describes the evolution of the phase field parameter (the shape of the vesicle), and a Cahn-Hilliard-type equation, which simulates the evolution of the ionic fluid. The model is mass conserved and surface area constrained during the membrane deformation. Conditions for vesicle growth or shrinkage are analyzed via the common tangent construction. We develop the numerical computing in two-dimensional space using a nonlinear multigrid method which is a combination of nonlinear Gauss-Seidel relaxation operator and V-cycles multigrid solver, and perform convergence tests that suggest an $\mathcal{O}(t+h^2)$ accuracy. Numerical results demonstrate the growth and shrinkage effects graphically and numerically, which agree with the conditions analyzed via the common tangent construction.In the second part, we present a model for vesicle growth or shrinkage with flow. The dynamical equations considered are an Allen-Cahn equation, which describes the phase field evolution, a Cahn-Hilliard-type equation, which simulates the fluid concentration, and a Stokes-type equation, which models the flow. The numerical scheme in two-dimensional space includes a nonlinear multigrid method comprised of a standard FAS method for the Allen-Cahn and Cahn-Hilliard part, and the Vanka smoothing strategy for the Stokes part. Convergence tests imply an $\mathcal{O}(t+h^2)$ accuracy. Numerical results are demonstrated under zero velocity boundary condition and with boundary-driven shear flows, respectively.In the last part, we give an unconditionally energy stable and uniquely solvable finite difference scheme for the model established in the first part. The finite difference scheme is based on a convex splitting of the discrete energy and is semi-implicit. One key difficulty associated with the energy stability is due to the fact that some nonlinear energy functional terms in the expansion is neither convex nor concave. To overcome this subtle difficulty, we add auxiliary terms to make the combined term convex, which in turn yields a convex–concave decomposition of the physical energy. As a result, both the unique solvability and energy stability of the proposed numerical scheme are assured. In addition, we show the scheme is stable in the defined discrete norm.
Show less

Title: Linear Systems Analysis of Molecular Dynamics
Creator: Nicholson, Stanley Anselm
Date: 2023
Description: Most proteins reduce the complexity of atomic motion to stable and coherent structures. Molecular dynamics (MD) has provided swaths of...
Show moreMost proteins reduce the complexity of atomic motion to stable and coherent structures. Molecular dynamics (MD) has provided swaths of trajectory data of proteins. We analyze these trajectories using classical stochastic signal analysis, well established and utilized by engineers. Linear systems analysis operates to uncover linearities given an input and output signal. The coherence function says an input and output are linearly related if and only if the coherence equals one. Analyzing protein motion in the frequency domain allows us to extract a frequency function relating the modes of motion as determined by atomic power spectra. Motivated by biochemistry, we analyze classical interactions like hydrogen bonds and salt bridges and find they act like a linear system, or effective spring. We test our analysis on two protein systems: crambin and the Mu Opioid Receptor (MOR). We extend our results to all pairwise interaction and determine coherent communities of atoms within the MOR. We present various community detection algorithms and demonstrate their validity using common metrics in MD. Identifying rigid and tightly correlated regions of the protein offers great potential in coarse graining protein structure and understanding protein motion.
Show less

Title: Machine Learning On Graphs
Creator: He, Jia
Date: 2022
Description: Deep learning has revolutionized many machine learning tasks in recent years.Successful applications range from computer vision, natural...
Show moreDeep learning has revolutionized many machine learning tasks in recent years.Successful applications range from computer vision, natural language processing to speech recognition, etc. The success is partially due to the availability of large amounts of data and fast growing computing resources (i.e., GPU and TPU), and partially due to the recent advances in deep learning technology. Neural networks, in particular, have been successfully used to process regular data such as images and videos. However, for many applications with graph-structured data, due to the irregular structure of graphs, many powerful operations in deep learning can not be readily applied. In recent years, there is a growing interest in extending deep learning to graphs. We first propose graph convolutional networks (GCNs) for the task of classification or regression on time-varying graph signals, where the signal at each vertex is given as a time series. An important element of the GCN design is filter design. We consider filtering signals in either the vertex (spatial) domain, or the frequency (spectral) domain. Two basic architectures are proposed. In the spatial GCN architecture, the GCN uses a graph shift operator as the basic building block to incorporate the underlying graph structure into the convolution layer. The spatial filter directly utilizes the graph connectivity information. It defines the filter to be a polynomial in the graph shift operator to obtain the convolved features that aggregate neighborhood information of each node. In the spectral GCN architecture, a frequency filter is used instead. A graph Fourier transform operator or a graph wavelet transform operator first transforms the raw graph signal to the spectral domain, then the spectral GCN uses the coe"cients from the graph Fourier transform or graph wavelet transform to compute the convolved features. The spectral filter is defined using the graph’s spectral parameters. There are additional challenges to process time-varying graph signals as the signal value at each vertex changes over time. The GCNs are designed to recognize di↵erent spatiotemporal patterns from high-dimensional data defined on a graph. The proposed models have been tested on simulation data and real data for graph signal classification and regression. For the classification problem, we consider the power line outage identification problem using simulation data. The experiment results show that the proposed models can successfully classify abnormal signal patterns and identify the outage location. For the regression problem, we use the New York city bike-sharing demand dataset to predict the station-level hourly demand. The prediction accuracy is superior to other models. We next study graph neural network (GNN) models, which have been widely used for learning graph-structured data. Due to the permutation-invariant requirement of graph learning tasks, a basic element in graph neural networks is the invariant and equivariant linear layers. Previous work by Maron et al. (2019) provided a maximal collection of invariant and equivariant linear layers and a simple deep neural network model, called k-IGN, for graph data defined on k-tuples of nodes. It is shown that the expressive power of k-IGN is equivalent to k-Weisfeiler-Lehman (WL) algorithm in graph isomorphism tests. However, the dimension of the invariant layer and equivariant layer is the k-th and 2k-th bell numbers, respectively. Such high complexity makes it computationally infeasible for k-IGNs with k > 3. We show that a much smaller dimension for the linear layers is su"cient to achieve the same expressive power. We provide two sets of orthogonal bases for the linear layers, each with only 3(2k & 1) & k basis elements. Based on these linear layers, we develop neural network models GNN-a and GNN-b, and show that for the graph data defined on k-tuples of data, GNN-a and GNN-b achieve the expressive power of the k-WL algorithm and the (k + 1)-WL algorithm in graph isomorphism tests, respectively. In molecular prediction tasks on benchmark datasets, we demonstrate that low-order neural network models consisting of the proposed linear layers achieve better performance than other neural network models. In particular, order-2 GNN-b and order-3 GNN-a both have 3-WL expressive power, but use a much smaller basis and hence much less computation time than known neural network models. Finally, we study generative neural network models for graphs. Generative models are often used in semi-supervised learning or unsupervised learning. We address two types of generative tasks. In the first task, we try to generate a component of a large graph, such as predicting if a link exists between a pair of selected nodes, or predicting the label of a selected node/edge. The encoder embeds the input graph to a latent vector space via vertex embedding, and the decoder uses the vertex embedding to compute the probability of a link or node label. In the second task, we try to generate an entire graph. The encoder embeds each input graph to a point in the latent space. This is called graph embedding. The generative model then generates a graph from a sampled point in the latent space. Di↵erent from the previous work, we use the proposed equivariant and invariant layers in the inference model for all tasks. The inference model is used to learn vertex/graph embeddings and the generative model is used to learn the generative distributions. Experiments on benchmark datasets have been performed for a range of tasks, including link prediction, node classification, and molecule generation. Experiment results show that the high expressive power of the inference model directly improves latent space embedding, and hence the generated samples.
Show less

Title: KERNEL FREE BOUNDARY INTEGRAL METHOD AND ITS APPLICATIONS
Creator: Cao, Yue
Date: 2022
Description: We developed a kernel-free boundary integral method (KFBIM) for solving variable coefficients partial differential equations (PDEs) in a...
Show moreWe developed a kernel-free boundary integral method (KFBIM) for solving variable coefficients partial differential equations (PDEs) in a doubly-connected domain. We focus our study on boundary value problems (BVP) and interface problems. A unique feature of the KFBIM is that the method does not require an analytical form of the Green’s function for designing quadratures, but rather computes boundary or volume integrals by solving an equivalent interface problem on Cartesian mesh. We decompose the problem defined in a doubly-connected domain into two separate interface problems. Then we evaluate integrals using a Krylov subspace iterative method in a finite difference framework. The method has second-order accuracy in space, and its complexity is linearly proportional to the number of mesh points. Numerical examples demonstrate that the method is robust for variable coefficients PDEs, even for cases when diffusion coefficients ratio is large and when two interfaces are close. We also develop two methods to compute moving interface problems whose coefficients in governing equations are spatial functions. Variable coefficients could be a non-homogeneous viscosity in Hele-Shaw problem or an uptake rate in tumor growth problems. We apply the KFBIM to compute velocity of the interface which allows more flexible boundary condition in a restricted domain instead of free space domain. A semi-implicit and an implicit methods were developed to evolve the interface. Both methods have few restrictions on the time step regardless of numerical stiffness. Theyalso could be extended to multi-phase problem, e.g., annulus domain. The methods have second-order accuracy in both space and time. Machine learning techniques have achieved magnificent success in the past decade. We couple the KFBIM with supervised learning algorithms to improve efficiency. In the KFBIM, we apply a finite difference scheme to find dipole density of the boundary integral iteratively, which is quite costly. We train a linear model to replace the finite difference solver in GMRES iterations. The cost, measured in CPU time, is significantly reduced. We also developed an efficient data generator for training and derived an empirical rule for data set size. In the future work, the model could be expanded to moving interface problems. The linear model will be replaced by neural network models, e.g., physics-informed neural networks (PINNs).
Show less

Title: Thermal Effects in Fluid Dynamics
Creator: Sulzbach, Jan-Eric
Date: 2021
Description: In this thesis we propose a mathematical framework modeling non-isothermal fluids.The framework is based on a coupling between non-equilibrium...
Show moreIn this thesis we propose a mathematical framework modeling non-isothermal fluids.The framework is based on a coupling between non-equilibrium thermodynamics and an energetic variational approach for the mechanical parts of the system. From this general model we derive and analyze three separate systems.The first application is the Brinkman-Fourier model. This is related to the ideal gas system, where the pressure and internal energy depend linearly on the product of density and temperature. This is a subsystem to the general Navier-Stokes-Fourier system. We prove the existence of local-in-time weak solutions via compensated compactness arguments.The next model we study is a non-isothermal diffusion system involving chemical reactions. For a system close to chemical equilibrium we show the well-posedness of classical solution using a fixed-point argument involving theory of homogeneous Besov spaces.The third application of the general theory is for another general diffusion system with a Cahn-Hilliard energy. In this framework, we study in detail how the temperature can affect the system on different scales, leading to different models. For the analysis, we focus on one case and show the well-posedness of classical solutions. The proof relies on methods from the theory of Besov spaces and paradifferential calculus.
Show less

Title: Numerical Analysis and Deep Learning Solver of the Non-local Fokker-Planck Equations
Creator: Jiang, Senbao
Date: 2022
Description: This thesis is divided into three mutually connected parts. ...
Show moreThis thesis is divided into three mutually connected parts. In the first part, we introduce and analyze arbitrarily high-order quadrature rules for evaluating the two-dimensional singular integrals of the forms \begin{align*} I_{i,j} = \int_{\mathbb{R}^2}\phi(x)\frac{x_ix_j}{|x|^{2+\alpha}} \d x, \quad 0< \alpha < 2 \end{align*} where $i,j\in\{1,2\}$ and $\phi\in C_c^N$ for $N\geq 2$. This type of singular integrals and its quadrature rule appear in the numerical discretization of fractional Laplacian in non-local Fokker-Planck Equations in 2D. The quadrature rules are trapezoidal rules equipped with correction weights for points around singularity. We prove the order of convergence is $2p+4-\alpha$, where $p\in\mathbb{N}_{0}$ is associated with total number of correction weights. We present numerical experiments to validate the order of convergence of the proposed modified quadrature rules. In the second part, we propose and analyze a general arbitrarily high-order modified trapezoidal rule for a class of weakly singular integrals of the forms $I = \int_{\R^n}\phi(x)s(x)\d x$ in $n$ dimensions, where $\phi$ and $s$ is the regular and singular part respectively. The admissible class requires $s$ satisfies three hypotheses and is large enough to contain singular kernel of the form $P(x)/|x|^r,\ r > 0$ where $P(x)$ is any monomial with degree strictly less than $r$. The modified trapezoidal rule is the singularity-punctured trapezoidal rule plus correction terms involving the correction weights for grid points around singularity. Correction weights are determined by enforcing the quadrature rule to exactly evaluate some monomials and solving corresponding linear systems. A long-standing difficulty of these types of methods is establishing the non-singularity of the linear system, despite strong numerical evidence. By using an algebraic-combinatorial argument, we show the non-singularity always holds and prove the general order of convergence of the modified quadrature rule. We present numerical experiments to validate the order of convergence. In the final part, we propose \emph{trapz-PiNN}, a physics-informed neural network incorporated with a modified trapezoidal rule and solve the space-fractional Fokker-Planck equations in 2D and 3D. We verify the modified trapezoidal rule has the second-order accuracy for evaluating the fractional laplacian. We demonstrate trapz-PiNNs have high expressive power through predicting solutions with low $\mathcal{L}^2$ relative error on a variety of numerical examples. We also use local metrics such as point-wise absolute and relative errors to analyze where could be further improved. We present an effective method for improving performance of trapz-PiNN on local metrics, provided that physical observations of high-fidelity simulation of the true solution are available. Besides the usual advantages of the deep learning solvers such as adaptivity and mesh-independence, the trapz-PiNN is able to solve PDEs with fractional laplacian with arbitrary $\alpha\in (0,2)$ and specializes on rectangular domains. It also has potential to be generalized into higher dimensions.
Show less

Title: Modeling, Analysis and Computation of Tumor Growth
Creator: Lu, Min-Jhe
Date: 2022
Description: In this thesis we investigate the modeling, analysis and computation of tumor growth.The sharp interface model we considered is to understand...
Show moreIn this thesis we investigate the modeling, analysis and computation of tumor growth.The sharp interface model we considered is to understand how the two key factors of (1) the mechanical interaction between the tumor cells and their surroundings, and (2) the biochemical reactions in the microenvironment of tumor cells can influence the dynamics of tumor growth. From this general model we give its energy formulation and solve it numerically using the boundary integral methods and the small-scale decomposition under three different scenarios.The first application is the two-phase Stokes model, in which tumor cells and the extracellular matrix are both assumed to behave like viscous fluids. We compared the effect of membrane elasticity on the tumor interface and the curvature-weakening one and found the latter would promote the development of branching patterns.The second application is the two-phase nutrient model under complex far-field geometries, which represents the heterogeneous vascular distribution. Our nonlinear simulations reveal that vascular heterogeneity plays an important role in the development of morphological instabilities that range from fingering and chain-like morphologies to compact,plate-like shapes in two-dimensions.The third application is for the effect of angiogenesis, chemotaxis and the control of necrosis. Our nonlinear simulations reveal the stabilizing effects of angiogenesis and the destabilizing ones of chemotaxisand necrosis in the development of tumor morphological instabilities if the necrotic core is fixed. We also perform the bifurcation analysis for this model.In the end, as a future work, we propose new models through Energetic Variational Approach (EnVarA) to shed light on the modeling issues.
Show less

Title: Algorithms for Discrete Data in Statistics and Operations Research
Creator: Schwartz, William K.
Date: 2021
Description: This thesis develops mathematical background for the design of algorithms for discrete-data problems, two in statistics and one in operations...
Show moreThis thesis develops mathematical background for the design of algorithms for discrete-data problems, two in statistics and one in operations research. Chapter 1 gives some background on what chapters 2 to 4 have in common. It also defines some basic terminology that the other chapters use.Chapter 2 offers a general approach to modeling longitudinal network data, including exponential random graph models (ERGMs), that vary according to certain discrete-time Markov chains (The abstract of chapter 2 borrows heavily from the abstract of Schwartz et al., 2021). It connects conditional and Markovian exponential families, permutation- uniform Markov chains, various (temporal) ERGMs, and statistical considerations such as dyadic independence and exchangeability. Markovian exponential families are explored in depth to prove that they and only they have exponential family finite sample distributions with the same parameter as that of the transition probabilities. Many new statistical and algebraic properties of permutation-uniform Markov chains are derived. We introduce exponential random ?-multigraph models, motivated by our result on replacing ? observations of a permutation-uniform Markov chain of graphs with a single observation of a corresponding multigraph. Our approach simplifies analysis of some network and autoregressive models from the literature. Removing models’ temporal dependence but not interpretability permitted us to offer closed-form expressions for maximum likelihood estimators that previously did not have closed-form expression available. Chapter 3 designs novel, exact, conditional tests of statistical goodness-of-fit for mixed membership stochastic block models (MMSBMs) of networks, both directed and undirected. The tests employ a ?²-like statistic from which we define p-values for the general null hypothesis that the observed network’s distribution is in the MMSBM as well as for the simple null hypothesis that the distribution is in the MMSBM with specified parameters. For both tests the alternative hypothesis is that the distribution is unconstrained, and they both assume we have observed the block assignments. As exact tests that avoid asymptotic arguments, they are suitable for both small and large networks. Further we provide and analyze a Monte Carlo algorithm to compute the p-value for the simple null hypothesis. In addition to our rigorous results, simulations demonstrate the validity of the test and the convergence of the algorithm. As a conditional test, it requires the algorithm sample the fiber of a sufficient statistic. In contrast to the Markov chain Monte Carlo samplers common in the literature, our algorithm is an exact simulation, so it is faster, more accurate, and easier to implement. Computing the p-value for the general null hypothesis remains an open problem because it depends on an intractable optimization problem. We discuss the two schools of thought evident in the literature on how to deal with such problems, and we recommend a future research program to bridge the gap those two schools. Chapter 4 investigates an auctioneer’s revenue maximization problem in combinatorial auctions. In combinatorial auctions bidders express demand for discrete packages of multiple units of multiple, indivisible goods. The auctioneer’s NP-complete winner determination problem (WDP) is to fit these packages together within the available supply to maximize the bids’ sum. To shorten the path practitioners traverse from from legalese auction rules to computer code, we offer a new wdp formalism to reflect how government auctioneers sell billions of dollars of radio-spectrum licenses in combinatorial auctions today. It models common tie-breaking rules by maximizing a sum of bid vectors lexicographically. After a novel pre-solving technique based on package bids’ marginal values, we develop an algorithm for the WDP. In developing the algorithm’s branch-and-bound part adapted to lexicographic maximization, we discover a partial explanation of why classical WDP has been successful in using the linear programming relaxation: it equals the Lagrangian dual. We adapt the relaxation to lexicographic maximization. The algorithm’s dynamic-programming part retrieves already computed partial solutions from a novel data structure suited specifically to our WDP formalism. Finally we show that the data structure can “warm start” a popular algorithm for solving for opportunity-cost prices.
Show less

Title: Choice-Distinguishing Colorings of Cartesian Products of Graphs
Creator: Tomlins, Christian James
Date: 2022
Description: A coloring $f: V(G)\rightarrow \mathbb N$ of a graph $G$ is said to be \emph{distinguishing} if no non-identity automorphism preserves every...
Show moreA coloring $f: V(G)\rightarrow \mathbb N$ of a graph $G$ is said to be \emph{distinguishing} if no non-identity automorphism preserves every vertex color. The distinguishing number, $D(G)$, of a graph $G$ is the smallest positive integer $k$ such that there exists a distinguishing coloring $f: V(G)\rightarrow [k]$ and was introduced by Albertson and Collins in their paper ``Symmetry Breaking in Graphs.'' By restricting what kinds of colorings are considered, many variations of distinguishing numbers have been studied. In this paper, we study proper list-colorings of graphs which are also distinguishing and investigate the choice-distinguishing number $\text{ch}_D(G)$ of a graph $G$. Primarily, we focus on the choice-distinguishing number of Cartesian products of graphs. We determine the exact value of $\text{ch}_D(G)$ for lattice graphs and prism graphs and provide an upper bound on the choice-distinguishing number of the Cartesian products of two relatively prime graphs, assuming a sufficient condition is satisfied. We use this result to bound the choice distinguishing number of toroidal grids and the Cartesian product of a tree with a clique. We conclude with a discussion on how, depending on the graphs $G$ and $H$, we may weaken the sufficient condition needed to bound $\text{ch}_D(G\square H)$.
Show less

Title: Stochastic dynamical systems with non-Gaussian and singular noises
Creator: Zhang, Qi
Date: 2022
Description: In order to describe stochastic fluctuations or random potentials arising from science and engineering, non-Gaussian or singular noises are...
Show moreIn order to describe stochastic fluctuations or random potentials arising from science and engineering, non-Gaussian or singular noises are introduced in stochastic dynamical systems. In this thesis we investigate stochastic differential equations with non-Gaussian Lévy noise, and the singular two-dimensional Anderson model equation with spatial white noise potential. This thesis consists of the following three main parts. In the first part, we establish a linear response theory for stochastic differential equations driven by an α-stable Lévy noise (1<α<2). We first prove the ergodic property of the stochastic differential equation and the regularity of the corresponding stationary Fokker-Planck equation. Then we establish the linear response theory. This result is a general fluctuation-dissipation relation between the response of the system to the external perturbations and the Lévy type fluctuations at a steady state.In the second part, we study the global well-posedness of the singular nonlinear parabolic Anderson model equation on a two-dimensional torus. This equation can be viewed as the nonlinear heat equation with a random potential. The method is based on paracontrolled distribution and renormalization. After splitting the original nonlinear parabolic Anderson model equation into two simpler equations, we prove the global existence by some a priori estimates and smooth approximations. Furthermore, we prove the uniqueness of the solution by classical energy estimates. This work improves the local well-posedness results in earlier works.In the third part, we investigate the variation problem associated with the elliptic Anderson model equation in a two-dimensional torus in the paracontrolled distribution framework. The energy functional in this variation problem is arising from the Anderson localization. We obtain the existence of minimizers by a direct method in the calculus of variations, and show that the Euler-Lagrange equation of the energy functional is an elliptic singular stochastic partial differential equation with the Anderson Hamiltonian. We further establish the L2 estimates and Schauder estimates for the minimizer as weak solution of the elliptic singular stochastic partial differential equation. Therefore, we uncover the natural connection between the variation problem and the singular stochastic partial differential equation in the paracontrolled distribution framework.Finally, we summarize our results and outline some research topics for future investigation.
Show less

Title: Dynamic Risk and Dynamic Performance Measures Generated by Distortion Functions and Diversification Benefits Optimization
Creator: Liu, Hao
Date: 2023
Description: This thesis consists of two major parts, and it contributes to the fields of risk management and optimization.One contribution to risk...
Show moreThis thesis consists of two major parts, and it contributes to the fields of risk management and optimization.One contribution to risk management is made via developing dynamic risk measures and dynamic acceptability indices that can be characterized by distortion functions. In particular, we proved a representation theorem illustrating that the class of dynamic coherent risk measures generated by distortion functions coincides with a specific type of dynamic risk measures, the dynamic WV@R. We also investigate thoroughly various types of time consistencies for dynamic risk measures and dynamic acceptability indices in terms of distortion functions. Another contribution to risk management is proving strong consistency and asymptotic normality of two estimators of dynamic WV@R. In contrast to the exist- ing literature, our results do not rely on the assumptions of distribution of random variables. Instead, we investigate the asymptotic normality of estimators in terms of the generating distortion functions. Last but not least, we give counterexample to show that a sufficient condition of asymptotic normality is not necessary. The contribution to optimization is twofold. On the one hand, we formulate the (scalar) diversification optimization problem as a vector optimization problem (VOP), and show that a set-valued Bellman principle is satisfied by this VOP. On the other hand, we derive explicit policy gradient formula and implement the deep neural network to solve diversification optimization problem numerically. This deep learning technique allows to overcome computation difficulty caused by the non-convexity of VOP.
Show less

repository.iit

Search the repository

Pages

Pages

Enabled Filters

Refine Results

Type

Date

Academic status of creator

Department

Subject

Creator

Part of

Rights