Closed paths of light trapped in a closed Fermat curve

Geometric constructions have previously been shown that can be interpreted as rays of light trapped either in polygons or in conics, by successive reflections. The same question, trapping light in closed Fermat curves, is addressed here. Numerical methods are used to study the behaviour of the reflection points of a triangle when the degree of the curve varies, including a generalization to non integer powers.     

A note on conjugate gradient convergence – Part III

Summary. In this paper we again consider the rate of convergence of the conjugate gradient method. We start with a general analysis of the conjugate gradient method for uniformly bounded solutions vectors and matrices whose eigenvalues are uniformly bounded and positive. We show that in such cases a fixed finite number of iterations of the method gives some fixed amount of improvement as the the size of the matrix tends to infinity. Then we specialize to the finite element (or finite difference) scheme for the problem . We show that for some classes of function we see this same effect. For other functions we show that the gain made by performing a fixed number of iterations of the method tends to zero as the size of the matrix tends to infinity. Received July 9, 1998 / Published online March 16, 2000     

A note on conjugate gradient convergence

Summary. The one-dimensional discrete Poisson equation on a uniform grid with points produces a linear system of equations with a symmetric, positive-definite coefficient matrix. Hence, the conjugate gradient method can be used, and standard analysis gives an upper bound of ) on the number of iterations required for convergence. This paper introduces a systematically defined set of solutions dependent on a parameter , and for several values of , presents exact analytic expressions for the number of steps ) needed to achieve accuracy . The asymptotic behavior of these expressions has the form )} as and )} as . In particular, two choices of corresponding to nonsmooth solutions give , i.e., iteration counts independent of ; this is in contrast to the standard bounds. The standard asymptotic convergence behavior, , is seen for a relatively smooth solution. Numerical examples illustrate and supplement the analysis. Received August 30, 1995 / Revised version received January 23, 1996     

Multivariate Functional Kernel Machine Regression and Sparse Functional Feature Selection

Motivated by mobile devices that record data at a high frequency, we propose a new methodological framework for analyzing a semi-parametric regression model that allow us to study a nonlinear relationship between a scalar response and multiple functional predictors in the presence of scalar covariates. Utilizing functional principal component analysis (FPCA) and the least-squares kernel machine method (LSKM), we are able to substantially extend the framework of semi-parametric regression models of scalar responses on scalar predictors by allowing multiple functional predictors to enter the nonlinear model. Regularization is established for feature selection in the setting of reproducing kernel Hilbert spaces. Our method performs simultaneously model fitting and variable selection on functional features. For the implementation, we propose an effective algorithm to solve related optimization problems in that iterations take place between both linear mixed-effects models and a variable selection method (e.g., sparse group lasso). We show algorithmic convergence results and theoretical guarantees for the proposed methodology. We illustrate its performance through simulation experiments and an analysis of accelerometer data.     

Some Simple U-Statistic Tests for Uniformity on Certain Homogeneous Spaces

We develop some examples of degenerate U-statistic tests for uniformity in certain compact topological groups and homogeneous spaces, namely, the orthogonal group and the Grassmann manifold. These tests have the attractive properties that they involve simple matrix calculations and their asymptotic null distributions are easy to describe, in that the eigenspace decompositions that characterize these distributions are quite elementary. The tests are shown to be locally most powerful against suitable families of alternatives.     

A note on conjugate gradient convergence – Part II

Summary. Continuing our previous analysis, we derive the exact number of conjugate gradient iterations needed (to achieve a given tolerance) for the one-dimensional discrete Poisson equation on a uniform grid, and a particularly smooth solution vector. Received July 29, 1998 / Published online March 16, 2000     

The effect of porosity on shock interaction with a rigid, porous barrier

This work investigates the pressure amplification experienced behind a rigid, porous barrier that is exposed to a planar shock. Numerical simulations are performed in two dimensions using the full Navier–Stokes equations for a M = 1.3 incoming shock wave. An array of cylinders is positioned at some distance from a solid wall and the shock wave is allowed to propagate past the barrier and reflect off the wall. Pressure at the wall is recorded and the flowfield is examined using numerical schlieren images. This work is intended to provide insight into the interaction of a shock wave with a cloth barrier shielding a solid boundary, and therefore the Reynolds number is small (i.e., Re = 500 to 2000). Additionally, the effect of porosity of the barrier is examined. While the pressure plots display no distinct trend based on Reynolds number, the porosity has a marked effect on the flowfield structure and endwall pressure, with the pressure increasing as porosity decreases until a maximum value is reached.