How Fast Can You Escape a Compact Polytope?

The Continuous Polytope Escape Problem (CPEP) asks whether every trajectory of a linear differential equation initialised within a convex polytope eventually escapes the polytope. We provide a polynomial-time algorithm to decide CPEP for compact polytopes. We also establish a quantitative uniform upper bound on the time required for every trajectory to escape the given polytope. In addition, we establish iteration bounds for termination of discrete linear loops via reduction to the continuous case

Inverse Materials Design Employing Self-folding and Extended Ensembles

The development of new technology is made possible by the discovery of novel materials. However, this discovery process is often tedious and largely consists of trial and error. In this thesis, I present methods to aid in the design of two distinct model systems. In the first case study, I model the 43,380 nets belonging to the five platonic solids to elucidate a universal folding mechanism. I then correlate geometric and topological features of the nets with folding propensity for simple shapes (i.e., tetrahedron, cube, and octahedron), in order to predict the folding propensity of nets belonging to more complex shapes (i.e., dodecahedron and icosahedron). In the second case study, I develop Monte Carlo techniques to sample the alchemical ensemble of hard polyhedra. In general, the anisotropy dimensions (e.g, faceting, branching, patchiness, etc.) of material building blocks are fixed attributes in experimental systems. In the alchemical ensemble, anisotropy dimensions are treated as thermodynamic variables and the free energy of the system in this ensemble is minimized to find the equilibrium particle shape for a given colloidal crystal at a given packing fraction. The method can sample millions of unique shapes within a single simulation, allowing for efficient particle design for crystal structures. Finally, I employ the method to explore how glasses formed from hard polyhedra, which are geometrically frustrated systems, can utilize extra dimensions to escape the glassy state in the extended ensemble.PHDChemical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/146005/1/pdodd_1.pd

An Introduction To The Web-Based Formalism

This paper summarizes our rather lengthy paper, "Algebra of the Infrared: String Field Theoretic Structures in Massive ${\cal N}=(2,2)$ Field Theory In Two Dimensions," and is meant to be an informal, yet detailed, introduction and summary of that larger work.Comment: 50 pages, 40 figure

Computational and Analytical Modelling of Droplet-Macroion Interactions

Charged droplets involving macromolecules undergo distinct disintegration mechanisms and shape deformations as a consequence of droplet-macroion interactions. Three general classes of droplet-macroion interactions that have been identified in the Consta group are: contiguous extrusion of a linear macroion from a droplet, pearl-necklace droplet conformations, and star -shaped droplets. This dissertation probes in a systematic manner the onset and various outcomes of macroion-droplet interactions, using atomistic molecular dynamics and realistic examples of solvent and macromolecules. When the charge-squared-to-volume ratio of a droplet is below but near a threshold value, certain flexible macromolecules, such as poly(ethylene glycol), extrude from a droplet, induced by the charging of the macromolecules. An analytical model is constructed based on the simulation data to suggest that the droplet surface electric field may play a role in the extrusion of the macroion. The effect of different solvents is studied to show that the final charge state of the macroion is determined by complicated macromolecule-ion-solvent interactions. Beyond this threshold, the charge-induced instability evolves to certain droplet deformations that lead to new stable states. These include pear-shaped lobes of solvent at the termini of a linear macroion, such as unstructured proteins, and conical protrusions of dielectric solvent surrounding a macroion regardless of its shape. In the former, such droplet conformation may emerge due to the interplay of a number of factors, subject to the constraint that each sub-droplet should be below a certain charge-squared-to-volume ratio. In the latter, the overall star geometry is determined by the amount of the macroion charge. As the next level of system complexity, different factors that affect the stability of weak transient protein complexes in droplets are examined. A multiscale approach is devised to model a protein in an evaporating droplet where its acidity constantly changes. A methodology is then developed to compute the dissociation rate and the error in the dissociation constant measured in mass spectrometry experiments. A possible charging mechanism of the macroion due to the star structure of solvent is also proposed

A computational geometric approach for an ensemble-based topological entropy calculation in two and three dimensions

From the stirring of dye in viscous fluids to the availability of essential nutrients spreading over the surface of a pond, nature is rife with examples of mixing in two-dimensional fluids. The long-time exponential growth rate of a thin filament of dye stretched by the fluid is a well-known proxy for the quality of mixing in two dimensions. This growth rate in turn gives a lower bound on the flow's topological entropy, a measure quantifying the complexity of chaotic dynamics. In the real-world study of mixing, topological entropy may be hard to compute; the velocity field may not be known or may be expensive to recover or approximate, thus limiting our knowledge of the governing system and underlying mechanics driving the mixing. Central to this study are two questions: \emph{How can stretching rates in two-dimensional planar flows best be computed using only trajectory data?}, and \emph{Can a method for computing stretching rates in higher dimensions from only trajectory data be developed?}. In this spirit, we introduce the Ensemble-based Topological Entropy Calculation (E-tec), a method to derive a lower-bound on topological entropy that requires only finite number of system trajectories, like those obtained from ocean drifters, and no detailed knowledge of the velocity field. E-tec is demonstrated to be computationally more efficient than other competing methods in two dimensions that accommodate trajectory data. This is accomplished by considering the evolution of a ``rubber band" wrapped around the data points and evolving with their trajectories. E-tec records the growth of this band as the collective motion of trajectories strike, deform, and stretch it. This exponential growth rate acts as a lower bound on the topological entropy. In this manuscript, I demonstrate convergence of E-tec's approximation with respect to both the number of trajectories (ensemble size) and the duration of trajectories in time. Driving the efficiency of E-tec in two dimensions is the use of computational geometry tools. Not only this, by computing stretching rates in this new computational geometry framework, I extend E-tec to three dimensions using two methods. First, I consider a two-dimensional rubber sheet stretched around a collection of points in a three-dimensional flow. Similar to the band-stretching component of two-dimensional E-tec, a three-dimensional triangulation is used to record the growth of the sheet as it is stretched and deformed by points evolving in time. Second, I calculate the growth rates of one-dimensional rubber strings as they are stretched by the edges of this dynamic, moving triangulation

Understanding Modern Techniques in Optimization: Frank-Wolfe, Nesterov's Momentum, and Polyak's Momentum

In the first part of this dissertation research, we develop a modular framework that can serve as a recipe for constructing and analyzing iterative algorithms for convex optimization. Specifically, our work casts optimization as iteratively playing a two-player zero-sum game. Many existing optimization algorithms including Frank-Wolfe and Nesterov's acceleration methods can be recovered from the game by pitting two online learners with appropriate strategies against each other. Furthermore, the sum of the weighted average regrets of the players in the game implies the convergence rate. As a result, our approach provides simple alternative proofs to these algorithms. Moreover, we demonstrate that our approach of optimization as iteratively playing a game leads to three new fast Frank-Wolfe-like algorithms for some constraint sets, which further shows that our framework is indeed generic, modular, and easy-to-use. In the second part, we develop a modular analysis of provable acceleration via Polyak's momentum for certain problems, which include solving the classical strongly quadratic convex problems, training a wide ReLU network under the neural tangent kernel regime, and training a deep linear network with an orthogonal initialization. We develop a meta theorem and show that when applying Polyak's momentum for these problems, the induced dynamics exhibit a form where we can directly apply our meta theorem. In the last part of the dissertation, we show another advantage of the use of Polyak's momentum -- it facilitates fast saddle point escape in smooth non-convex optimization. This result, together with those of the second part, sheds new light on Polyak's momentum in modern non-convex optimization and deep learning.Comment: PhD dissertation at Georgia Tech. arXiv admin note: text overlap with arXiv:2010.0161