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.

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.

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.