A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces

In this paper we present a high-order kernel method for numerically solving diffusion and reaction-diffusion partial differential equations (PDEs) on smooth, closed surfaces embedded in $\mathbb{R}^d$. For two-dimensional surfaces embedded in $\mathbb{R}^3$, these types of problems have received growing interest in biology, chemistry, and computer graphics to model such things as diffusion of chemicals on biological cells or membranes, pattern formations in biology, nonlinear chemical oscillators in excitable media, and texture mappings. Our kernel method is based on radial basis functions (RBFs) and uses a semi-discrete approach (or the method-of-lines) in which the surface derivative operators that appear in the PDEs are approximated using collocation. The method only requires nodes at "scattered" locations on the surface and the corresponding normal vectors to the surface. Additionally, it does not rely on any surface-based metrics and avoids any intrinsic coordinate systems, and thus does not suffer from any coordinate distortions or singularities. We provide error estimates for the kernel-based approximate surface derivative operators and numerically study the accuracy and stability of the method. Applications to different non-linear systems of PDEs that arise in biology and chemistry are also presented

A Fast and Accurate Algorithm for Spherical Harmonic Analysis on HEALPix Grids with Applications to the Cosmic Microwave Background Radiation

The Hierarchical Equal Area isoLatitude Pixelation (HEALPix) scheme is used extensively in astrophysics for data collection and analysis on the sphere. The scheme was originally designed for studying the Cosmic Microwave Background (CMB) radiation, which represents the first light to travel during the early stages of the universe's development and gives the strongest evidence for the Big Bang theory to date. Refined analysis of the CMB angular power spectrum can lead to revolutionary developments in understanding the nature of dark matter and dark energy. In this paper, we present a new method for performing spherical harmonic analysis for HEALPix data, which is a central component to computing and analyzing the angular power spectrum of the massive CMB data sets. The method uses a novel combination of a non-uniform fast Fourier transform, the double Fourier sphere method, and Slevinsky's fast spherical harmonic transform (Slevinsky, 2019). For a HEALPix grid with $N$ pixels (points), the computational complexity of the method is $\mathcal{O}(N\log^2 N)$, with an initial set-up cost of $\mathcal{O}(N^{3/2}\log N)$. This compares favorably with $\mathcal{O}(N^{3/2})$ runtime complexity of the current methods available in the HEALPix software when multiple maps need to be analyzed at the same time. Using numerical experiments, we demonstrate that the new method also appears to provide better accuracy over the entire angular power spectrum of synthetic data when compared to the current methods, with a convergence rate at least two times higher

Computing with functions in spherical and polar geometries I. The sphere

A collection of algorithms is described for numerically computing with smooth functions defined on the unit sphere. Functions are approximated to essentially machine precision by using a structure-preserving iterative variant of Gaussian elimination together with the double Fourier sphere method. We show that this procedure allows for stable differentiation, reduces the oversampling of functions near the poles, and converges for certain analytic functions. Operations such as function evaluation, differentiation, and integration are particularly efficient and can be computed by essentially one-dimensional algorithms. A highlight is an optimal complexity direct solver for Poisson's equation on the sphere using a spectral method. Without parallelization, we solve Poisson's equation with $100$ million degrees of freedom in one minute on a standard laptop. Numerical results are presented throughout. In a companion paper (part II) we extend the ideas presented here to computing with functions on the disk.Comment: 23 page

A High-Order Radial Basis Function (RBF) Leray Projection Method for the Solution of the Incompressible Unsteady Stokes Equations

A new projection method based on radial basis functions (RBFs) is presented for discretizing the incompressible unsteady Stokes equations in irregular geometries. The novelty of the method comes from the application of a new technique for computing the Leray-Helmholtz projection of a vector field using generalized interpolation with divergence-free and curl-free RBFs. Unlike traditional projection methods, this new method enables matching both tangential and normal components of divergence-free vector fields on the domain boundary. This allows incompressibility of the velocity field to be enforced without any time-splitting or pressure boundary conditions. Spatial derivatives are approximated using collocation with global RBFs so that the method only requires samples of the field at (possibly scattered) nodes over the domain. Numerical results are presented demonstrating high-order convergence in both space (between 5th and 6th order) and time (up to 4th order) for some model problems in two dimensional irregular geometries.Comment: 34 pages, 8 figure

A Phylogenetic Analysis of the African Plant Genus Palisota (family Commelinaceae) based on Chloroplast DNA Sequences

The plant genus Palisota (family Commelinaceae, or spiderwort family) consists of approximately 20 species and is distributed throughout the forests of tropical Africa. The genus exhibits several unusual morphological characteristics, and as a result has been difficult to classify based on morphology. Molecular phylogenetic studies have placed it near the base of Commelinaceae, but the exact placement of Palisota within the family is not clear. As the African continent has become more arid in recent geological times, the forests have receded, reducing the habitat for Palisota species and potentially impacting speciation and extinction rates within the genus. The goal of this study is to sequence the chloroplast-encoded gene rbcL in several additional species of Palisota and its relatives in order to: 1) determine the phylogenetic relationship of the genus with respect to other members of Commelinaceae; 2) evaluate phylogenentic relationships among species of Palisota; and 3) infer relative speciation/extinction rates within the genus. Additionally, we are exploring the use of other molecular regions for phylogenetic analysis with the genus

Six Pillars of Social Policy: The State of Pensions and Health Care in Canada

William B.P. Robson, a co-author with David Slater of a series of papers on pension issues, has written an ambitious survey of the state of Canadian economic policy in the areas of pensions and health care. He argues that it is appropriate to tackle both issues in the same paper because they are both major spending programs strongly related to the life cycle of Canadians, and face challenges arising from the aging of the population. Robson notes that the pension debate uses the metaphor of three pillars to describe a comprehensive pension system: a safety net to guard against destitution in old age; a mandatory employment-related system to provide basic replacement income; and a voluntary system supported by provisions that reduce the double-taxation of saving. The main elements of public policy related to pensions in Canada cover these pillars. He recognizes that all three of the pillars cannot be directly applied to health care, but he argues that the three-pillar metaphor is still a fruitful perspective because it facilitates constructive responses to the pressures confronting Canada’s health system and illuminates interactions between the pension and health systems. Hence his title “six pillars of social policy”. Based on his examination of Canada’s pension and health-care systems, Robson makes a number of recommendations. First, he advocates more prefunding in both the pension and health areas to cover the future cost of the aging baby-boom cohort. Second, he recommends a gradual increase in the normal age of eligibility for pension benefits. Third, he recommends the creation of a second pillar, a mandatory contribution scheme in the health area as a way to avoid the development of a means-tested system that would exacerbate the disincentives to work and save. Fourth, he puts forward the idea of a new type of saving vehicle that provides tax-relief on distributions rather than on contributions so that Canadians can avoid the high marginal effective tax rates associated with means-tested programs.Health, Health Care, Health-care, Healthcare, Canada, Pensions, CPP, Retirement, Mandatory Contribution, Aging, Ageing

Review of research in primary reading at Boston University: 1954-1959.

Thesis (Ed.M.)--Boston Universit