ASKIT: Approximate Skeletonization Kernel-Independent Treecode in High Dimensions

We present a fast algorithm for kernel summation problems in high-dimensions. These problems appear in computational physics, numerical approximation, non-parametric statistics, and machine learning. In our context, the sums depend on a kernel function that is a pair potential defined on a dataset of points in a high-dimensional Euclidean space. A direct evaluation of the sum scales quadratically with the number of points. Fast kernel summation methods can reduce this cost to linear complexity, but the constants involved do not scale well with the dimensionality of the dataset. The main algorithmic components of fast kernel summation algorithms are the separation of the kernel sum between near and far field (which is the basis for pruning) and the efficient and accurate approximation of the far field. We introduce novel methods for pruning and approximating the far field. Our far field approximation requires only kernel evaluations and does not use analytic expansions. Pruning is not done using bounding boxes but rather combinatorially using a sparsified nearest-neighbor graph of the input. The time complexity of our algorithm depends linearly on the ambient dimension. The error in the algorithm depends on the low-rank approximability of the far field, which in turn depends on the kernel function and on the intrinsic dimensionality of the distribution of the points. The error of the far field approximation does not depend on the ambient dimension. We present the new algorithm along with experimental results that demonstrate its performance. We report results for Gaussian kernel sums for 100 million points in 64 dimensions, for one million points in 1000 dimensions, and for problems in which the Gaussian kernel has a variable bandwidth. To the best of our knowledge, all of these experiments are impossible or prohibitively expensive with existing fast kernel summation methods.Comment: 22 pages, 6 figure

A surface vorticity method for wake–body interactions

The objective of this dissertation research is to develop a surface vorticity method for simulating high Reynolds number incompressible aerodynamic flows with strong unsteady interactions between wakes and lifting bodies. Examples of these types of flows include rotors in hover, propeller/wing installations, and impingement of vortex cores shed from wing strakes or flaps on downstream surfaces. Although higher-order panel codes provide good representation of potential flow around lifting bodies, their treatment of wakes is inadequate for our purpose. In the absence of significant boundary layer separation, the vorticity in these flows concentrates into thin shear layers. Therefore, vortex sheets are a natural mathematical representation of these flows. We leverage and extend rigorous methods from the vortex methods literature to model a wake as a free vortex sheet discretized as a triangulation of panels with linearly varying surface vorticity. The vorticity evolution equation is solved approximately by maintaining constant circulation along each half-edge in the triangulation, an approach that generalizes current methods for constant-strength elements. The vortex sheet is regularized with a smoothing parameter which provides an apparent thickness that mimics the limited viscous mixing in high Reynolds number flow. An adaptive paneling algorithm is implemented to maintain the desired level of detail as the wake triangulation stretches and deforms. The induced velocities from the wake vortex sheet are computed with a treecode implemented on a graphics processing unit (GPU) to allow computations with millions of panels. Lifting bodies are modeled with bound vortex sheets that are also triangulated with linear strength panels. These higher-order vorticity elements provide accurate velocity predictions on and near the surface, allowing for high resolution streamline tracing. Surface vorticity is determined by enforcing flow tangency constraints at each triangle centroid, zero circulation around each panel perimeter, and the unsteady pressure matching Kutta condition. These constraints result in an overdetermined system that is solved in a least squares formulation. Thus, our method is a second-order surface vorticity boundary element method that combines both solid bodies and wakes in a rigorous and consistent manner. The results of the method are shown to compare favorably to wind tunnel experimental results, including wake profiles, for a rectangular wing in a steady freestream, and for a horizontal axis wind turbine. Finally, we demonstrate the capabilities of our method in the context of strong wake–body interactions by simulating two flying wing aircraft in close formation, with the wake from the leading aircraft impacting the tailing aircraft.Ph.D