Dhavide A. Aruliah

Geometric applications of the Bezout matrix in the Lagrange basis

Using a new formulation of the Bézout matrix, we construct bivariate matrix polynomials expressed in a tensor-product Lagrange basis. We use these matrix polynomials to solve common tasks in computer-aided geometric design. For example, we show that these bivariate polynomials can serve as stable and efficient implicit representations of plane curves for a variety of curve intersection problems.

Block LU factors of generalized companion matrix pencils

We present formulas for computations involving companion matrix pencils as may arise in considering polynomial eigenvalue problems. In particular, we provide explicit companion matrix pencils for matrix polynomials expressed in a variety of polynomial bases including monomial, orthogonal, Newton, Lagrange, and Bernstein/Bezier bases. Additionally, we give a pair of explicit LU factors associated with each pencil and a prescription for block pivoting when required.

Algorithm 956: PAMPAC, A Parallel Adaptive Method for Pseudo-Arclength Continuation

Pseudo-arclength continuation is a well-established method for generating a numerical curve approximating the solution of an underdetermined system of nonlinear equations. It is an inherently sequential predictor-corrector method in which new approximate solutions are extrapolated from previously converged results and then iteratively refined. Convergence of the iterative corrections is guaranteed only for sufficiently small prediction steps. In high-dimensional systems, corrector steps are extremely costly to compute and the prediction step length must be adapted carefully to avoid failed steps or unnecessarily slow progress. We describe a parallel method for adapting the step length employing several predictor-corrector sequences of different step lengths computed concurrently. In addition, the algorithm permits intermediate results of correction sequences that have not converged to seed new predictions. This strategy results in an aggressive optimization of the step length at the cost of redundancy in the concurrent computation. We present two examples of convoluted solution curves of high-dimensional systems showing that speed-up by a factor of two can be attained on a multicore CPU while a factor of three is attainable on a small cluster.

Stereo Reconstruction of Droplet Flight Trajectories

We developed a new method for extracting 3D flight trajectories of droplets using high-speed stereo capture. We noticed that traditional multi-camera tracking techniques fare poorly on our problem, in part due to the fact that all droplets have very similar shapes, sizes and appearances. Our method uses local motion models to track individual droplets in each frame. 2D tracks are used to learn a global, non-linear motion model, which in turn can be used to estimate the 3D locations of individual droplets even when these are not visible in any camera. We have evaluated the proposed method on both synthetic and real data and our method is able to reconstruct 3D flight trajectories of hundreds of droplets. The proposed technique solves for both the 3D trajectory of a droplet and its motion model concomitantly, and we have found it to be superior to 3D reconstruction via triangulation. Furthermore, the learned global motion model allows us to relax the simultaneity assumptions of stereo camera systems. Our results suggest that, even when full stereo information is available, our unsynchronized reconstruction using the global motion model can significantly improve the 3D estimation accuracy.

A Parallel Adaptive Method for Pseudo-arclength Continuation

Pseudo-arclength continuation is a well-established method for constructing a numerical curve comprising solutions of a system of nonlinear equations. In many complicated high-dimensional systems, the corrector steps within pseudo-arclength continuation are extremely costly to compute; as a result, the step-length of the preceding prediction step must be adapted carefully to avoid prohibitively many failed steps. We describe the essence of a parallel method for adapting the step-length of pseudo-arclength continuation. Our method employs several predictor-corrector sequences with differing step-lengths running concurrently on distinct processors. Our parallel framework permits intermediate results of correction sequences that have not yet converged to seed new predictor-corrector sequences with various step-lengths; the goal is to amortize the cost of corrector steps to make further progress along the underlying numerical curve. Results from numerical experiments suggest a three-fold speedup is attainable when the continuation curve sought has great topological complexity and the corrector steps require significant processor time.

Computing the topology of a real algebraic plane curve whose equation is not directly available

We present a collection of methods and tools for computing the topology of real algebraic plane curves de .ned by bivariate polynomial equations that are known at certain values or easy to evaluate, but whose explicit description is not available.The principal techniques used are the reduction of the computation of the real roots of the discriminant to a sparse generalized eigenvalue problem,the use of the structure of the nullspace of the classical Bezoutian, and its description in terms of the Lagrange Basis.

Companion matrix pencils for hermite interpolants

This abstract describes new methods for solving polynomial problems where the polynomials are expressed as (generalized) Hermite interpolants; that is, where the polynomials are given by values and by derivatives at certain nodes. We consider here the mixed case where at some nodes, only values are known, whereas at others, both values and derivatives are known. We neither consider the case where derivatives higher than the 1st are known, nor the Birkhoff case where some data is ‘missing’. In the full paper, we will give the general case for higher derivatives; we leave the Birkhoff case for a future paper. For example, suppose that we know that a polynomial has the values p(tm) = pm, p(tm+1/2) = pm+1/2, and p(tm+1) = pm+1, for distinct nodes τ1 = tm, τ2 = tm+1/2, τ3 = tm+1, and suppose further that we also know the derivatives p′(tm) and p′(tm+1), which we denote p ′ m and p ′ m+1. This can occur naturally in the context of the numerical solution of initial value problems for ordinary differential equations, for example. At the beginning of a numerical step, t = tm, the value is known and the derivative is calculated by a function evaluation, so we know pm and p ′ m. At the end of the step, t = tm+1, another function evaluation is carried out in order to continue the marching process and so we know p′m+1 as well as pm+1. However, it is not necessarily the case that derivatives are known in the interior of the interval (tm, tm+1), although for graphical and other purposes the values of the polynomial p(t) interpolating the numerical

Numerical parameterization of affine varieties using

In the present work, we extend the standard idea of numerical parameterization (i.e., parameterization by the numerical solution of initial-value problems (IVPs) for ordinary differential equations (ODEs) to affine varieties in ℂ;n for n≥2. We use these results with an efficient implementation in Maple to explore the use of numerical parameterization for the visualization of Riemann surfaces.