Poisson convergence in the restricted kk-partioning problem

The randomized $k$-number partitioning problem is the task to distribute $N$ i.i.d. random variables into $k$ groups in such a way that the sums of the variables in each group are as similar as possible. The restricted $k$-partitioning problem refers to the case where the number of elements in each group is fixed to $N/k$. In the case $k=2$ it has been shown that the properly rescaled differences of the two sums in the close to optimal partitions converge to a Poisson point process, as if they were independent random variables. We generalize this result to the case $k>2$ in the restricted problem and show that the vector of differences between the $k$ sums converges to a $k-1$-dimensional Poisson point process.Comment: 31pp, AMSTe

A domain decomposition matrix-free method for global linear stability

This work is dedicated to the presentation of a matrix-free method for global linear stability analysis in geometries composed of multi-connected rectangular subdomains. An Arnoldi technique using snapshots in subdomains of the entire geometry combined with a multidomain linearized Direct Numerical Finite difference simulations based on an influence matrix for partitioning are adopted. The method is illustrated by three benchmark problems: the lid-driven cavity, the square cylinder and the open cavity flow. The efficiency of the method to extract large-scale structures in a multidomain framework is emphasized. The possibility to use subset of the full domain to recover the perturbation associated with the entire flow field is also highlighted. Such a method appears thus a promising tool to deal with large computational domains and three-dimensionality within a parallel architecture

A domain decomposition matrix-free method for global linear stability

This work is dedicated to the presentation of a matrix-free method for global linear stability analysis in geometries composed of multi-connected rectangular subdomains. An Arnoldi technique using snapshots in subdomains of the entire geometry combined with a multidomain linearized Direct Numerical Finite difference simulations based on an influence matrix for partitioning are adopted. The method is illustrated by three benchmark problems: the lid-driven cavity, the square cylinder and the open cavity flow. The efficiency of the method to extract large-scale structures in a multidomain framework is emphasized. The possibility to use subset of the full domain to recover the perturbation associated with the entire flow field is also highlighted. Such a method appears thus a promising tool to deal with large computational domains and three-dimensionality within a parallel architecture

Local energy statistics in disordered systems: a proof of the local REM conjecture

Recently, Bauke and Mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should in most circumstances be the same as in the random energy model. Here we give necessary conditions for this hypothesis to be true, which we show to hold in wide classes of examples: short range spin glasses and mean field spin glasses of the SK type. We also show that, under certain conditions, the conjecture holds even if energy levels that grow moderately with the volume of the system are considered

Optimization for Urban Mobility Systems

In the recent decades, new modes of transportation have been developed due to urbanization, highly dense population, and technological advancement. As a result, design and operation of urban transportation have become increasingly important to better utilize the resources and efficiently meet demand. This dissertation was motivated by two problems on optimizing design and control of urban transportation. In the first one, we consider a problem of dynamically matching heterogeneous market parcitipants so as to maximize the total number of matching, which was motivated by practices of ride-sharing platforms. In the other problem, we study efficient design of elevator zoning system in high-rises with uncertainty in customer batching.In Chapter 1, we consider a multiperiod stochastic optimization of a market that matches heterogeneous and impatient agents. The model was mainly motivated from carpooling products run by ride-sharing platforms such as Uber and Lyft, and kidney exchange market, where market participants are heterogeneous in terms of how likely they can be matched with others. In the case of a ride-sharing platform, one of the key operational decisions for carpooling is to efficiently match riders and clear the market in a timely manner. In doing so, the platform needs to take into account the heterogeneity of riders in terms of their trip types(e.g origin-destination pair) and different matching compatibility. For example, some customers may request rides within San Francisco, while others may request rides from San Francisco to outside the city. Since picking up and dropping off a customer within the city can be done within relatively short amount of time, those who want to travel within the city can be matched with any other riders for carpooling. However, the destinations of those who want to travel to outside the city may be very different, and in order to maintain customers' additional transit time due to carpooling, it is likely that they can be only matched with those who want to travel within the city. In the case of kidney exchange where market participants arrive in the form of patient-donor pair, pairs with donor who can donate her kidney to most of patients (for example, blood type O) and patient who can get kidney from most of donors (for example, blood type AB) can be easily matched to other pairs. The opposite case would be hard-to-match pair that is incompatible for matching with most of other pairs. Our model is an abstraction of these two motivating examples, and considers two types of agents: easy-to-match agents that can be matched with either type of agents, and hard-to-match agents that can be only matched with easy-to-match ones. We first formulate a dynamic program to solve for optimal matching decisions over infinite time horizon in a discrete time setting, and characterize structure of optimal stationary policies. Inspired by practices in kidney exchange where the market is cleared for every fixed time interval, we connect the discrete time model to a continuous time setting by investigating the effect of the length of matching intervals on the matching performance. Results from numerical experiments indicate certain patterns in the relationship between the length of matching intervals and the maximum number of matching achieved, and provides valuable insights for future direction of research. In Chapter 2, we consider a zoning problem for elevator dispatching systems in high-rises. In practice, zoning is frequently used to improve efficiency of elevator systems. The idea of zoning is to prevent different elevators from stopping at common floors, which may result in long service times of elevators and thus long waiting times of customers. Our goal is to provide a mathematical framework that can help a system planner decide optimal zoning design with some performance guarantee. To this end, we focus on uppeak traffic situation during morning rush hour, which is in general the heaviest traffic during the day. The performance in the uppeak traffic situation can be considered as the system's capacity, because if the system can handle uppeak traffic well, it can also serve other types of traffic with good performance. Thus, the performance measure in the uppeak traffic situation can be used as a metric to choose the optimal zoning configuration. One of the components that complicate the problem is customer batching, on which the system may not have a control. In view of this, we formulate an adversarial optimization problem that can measure the system performance of different zoning decisions. By considering the heaviest traffic situation of the day and using the adversarial framework, we provide a model that can be used for capacity planning of elevator systems. We formulate mixed-integer linear program(MILP)s to find the optimal zoning configuration. To solve the MILPs, we show that we can use simple greedy algorithms and solve smaller linear programs. We also provide a few illustrative examples as well as numerical experiments to verify the theoretical results and obtain insights for further analysis

Bayesian Item Response Modeling in R with brms and Stan

Item Response Theory (IRT) is widely applied in the human sciences to model persons' responses on a set of items measuring one or more latent constructs. While several R packages have been developed that implement IRT models, they tend to be restricted to respective prespecified classes of models. Further, most implementations are frequentist while the availability of Bayesian methods remains comparably limited. We demonstrate how to use the R package brms together with the probabilistic programming language Stan to specify and fit a wide range of Bayesian IRT models using flexible and intuitive multilevel formula syntax. Further, item and person parameters can be related in both a linear or non-linear manner. Various distributions for categorical, ordinal, and continuous responses are supported. Users may even define their own custom response distribution for use in the presented framework. Common IRT model classes that can be specified natively in the presented framework include 1PL and 2PL logistic models optionally also containing guessing parameters, graded response and partial credit ordinal models, as well as drift diffusion models of response times coupled with binary decisions. Posterior distributions of item and person parameters can be conveniently extracted and post-processed. Model fit can be evaluated and compared using Bayes factors and efficient cross-validation procedures.Comment: 54 pages, 16 figures, 3 table

Doctor of Philosophy

dissertationShape analysis is a well-established tool for processing surfaces. It is often a first step in performing tasks such as segmentation, symmetry detection, and finding correspondences between shapes. Shape analysis is traditionally employed on well-sampled surfaces where the geometry and topology is precisely known. When the form of the surface is that of a point cloud containing nonuniform sampling, noise, and incomplete measurements, traditional shape analysis methods perform poorly. Although one may first perform reconstruction on such a point cloud prior to performing shape analysis, if the geometry and topology is far from the true surface, then this can have an adverse impact on the subsequent analysis. Furthermore, for triangulated surfaces containing noise, thin sheets, and poorly shaped triangles, existing shape analysis methods can be highly unstable. This thesis explores methods of shape analysis applied directly to such defect-laden shapes. We first study the problem of surface reconstruction, in order to obtain a better understanding of the types of point clouds for which reconstruction methods contain difficulties. To this end, we have devised a benchmark for surface reconstruction, establishing a standard for measuring error in reconstruction. We then develop a new method for consistently orienting normals of such challenging point clouds by using a collection of harmonic functions, intrinsically defined on the point cloud. Next, we develop a new shape analysis tool which is tolerant to imperfections, by constructing distances directly on the point cloud defined as the likelihood of two points belonging to a mutually common medial ball, and apply this for segmentation and reconstruction. We extend this distance measure to define a diffusion process on the point cloud, tolerant to missing data, which is used for the purposes of matching incomplete shapes undergoing a nonrigid deformation. Lastly, we have developed an intrinsic method for multiresolution remeshing of a poor-quality triangulated surface via spectral bisection