Efficient algorithms for the computation of optimal quadrature points on Riemannian manifolds

Viac o knihe

The thesis addresses numerical integration, focusing on approximating integrals of continuous functions using quadrature points. It leverages the theory of reproducing kernel Hilbert spaces and explores the worst case quadrature error. Finding optimal quadrature points to minimize this error is complex, particularly as the number of points increases. The research emphasizes efficient computation of these points on the torus T^d, sphere S^d, and rotation group SO(3). A general framework for minimizing the worst case quadrature error on Riemannian manifolds is introduced, treating the error as a function on the product manifold M^N. Optimization techniques such as steepest descent, Newton's method, and conjugate gradient are employed, along with two innovative evaluation approaches for the quadrature error and its derivatives. One approach reduces complexity by interpreting the error as pairwise potential energy, while the other utilizes Fourier transforms to enhance computational efficiency. This results in significant reductions in complexity for polynomial spaces. The methods yield new quadrature formulas for high polynomial degrees on the sphere and rotation group. Additionally, the framework connects worst case quadrature errors to discrepancies, which are crucial for uniform point distributions in high-dimensional integration and applications like image processing and computer graphics, particularly in halftoning technique