Over the last two decades, several fast, robust, and high-order accurate methods have been developed for solving the Poisson equation in complicated geometry using potential theory. In this approach, rather than discretizing the partial differential equation itself, one first evaluates a volume integral to account for the source distribution within the domain, followed by solving […]

Despite the considerable success in machine learning of Bregman proximal-type algorithms, such as mirror descent,  a critical question remains: Can existing stationarity measures, often based on Bregman divergence, reliably distinguish between stationary and non-stationary points?  In this paper, we present a groundbreaking finding:  All existing stationarity measures necessarily imply the existence of spurious stationary points. […]

TBA

We present and analyze a new randomized algorithm called rand_cholQR for computing tall-and-skinny QR factorizations. Using one or two random sketch matrices, it is  proved that with high probability, its orthogonality error is bounded by a constant of the order of unit roundoff for any numerically full-rank matrix. An evaluation of the performance of rand_cholQR on a […]