Random convex hulls and kernel quadrature

Discretization of probability measures is ubiquitous in the field of applied mathematics, from classical numerical integration to data compression and algorithmic acceleration in machine learning. In this thesis, starting from generalized Tchakaloff-type cubature, we investigate random convex hulls and kernel quadrature. In the first two chapters after the introduction, we investigate the probability that a given vector θ is contained in the convex hull of independent copies of a random vector X. After deriving a sharp inequality that describes the relationship between the said probability and Tukey’s halfspace depth, we explore the case θ = E[X] by using moments of X and further the case when X enjoys some additional structure, which are of primary interest from the context of cubature. In the subsequent two chapters, we study kernel quadrature, which is numerical integration where integrands live in a reproducing kernel Hilbert space. By explicitly exploiting the spectral properties of the associated integral operator, we derive convex kernel quadrature with theoretical guarantees described by its eigenvalue decay. We further derive practical variants of the proposed algorithm and discuss their theoretical and computational aspects. Finally, we briefly discuss the applications and future work of the thesis, including Bayesian numerical methods, in the concluding chapter

Calculation of Realistic Charged-Particle Transfer Maps

The study and computation of nonlinear charged-particle transfer maps is fundamental to understanding single-particle beam dynamics in accelerator devices. Transfer maps for individual elements of the beamline can in general depend sensitively on nonlinear fringe-field and high-multipole effects. The inclusion of these effects requires a detailed and realistic model of the interior and fringe magnetic fields, including knowledge of high spatial derivatives. Current methods for computing such maps often rely on idealized models of beamline elements. This Dissertation describes the development and implementation of a collection of techniques for computing realistic (as opposed to idealized) charged-particle transfer maps for general beamline elements, together with corresponding estimates of numerical error. Each of these techniques makes use of 3-dimensional measured or numerical field data on a grid as provided, for example, by various 3-dimensional finite element field codes. The required high derivatives of the corresponding vector potential A, required to compute transfer maps, cannot be reliably computed directly from this data by numerical differentiation due to numerical noise whose effect becomes progressively worse with the order of derivative desired. The effect of this noise, and its amplification by numerical differentiation, can be overcome by fitting on a bounding surface far from the axis and then interpolating inward using the Maxwell equations. The key ingredients are the use of surface data and the smoothing property of the inverse Laplacian operator. We explore the advantages of map computation using realistic field data on surfaces of various geometry. Maps obtained using these techniques can then be used to compute realistically all derived linear and nonlinear properties of both single pass and circular machines. Although the methods of this Dissertation have been applied primarily to magnetic beamline elements, they can also be applied to electric and radio-frequency beamline elements

Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition