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The relationship between Lp affine surface area and curvature measures is investigated. As a result, a new representation of the existing notion of Lp affine surface area depending only on curvature measures is derived. Direct proofs of the equivalence between this new representation and those previously known are provided. The proofs show that the new representation is, in a sense, “polar” to that of Lutwak’s and “dual” to that of Schutt & Werner’s.

In this paper, we present a staggered discontinuous Galerkin immersed boundary method (SDGIBM) for the numerical approximation of fluid-structure interaction. The immersed boundary method is used to model the fluid-structure interaction, while the fluid flow is governed by incompressible Navier-Stokes equations. One advantage of using Galerkin method over the finite difference method with immersed boundary method is that we can avoid approximations of the Dirac Delta function. Another key ingredient of our method is that our solver for incompressible Navier-Stokes equations combines the advantages of discontinuous Galerkin methods and staggered meshes, and results in many good properties, namely local and global conservations and pointwise divergence-free velocity field by a local postprocessing technique. Furthermore, energy stability is improved by a skew-symmetric discretization of the convection term. We will present numerical results to show the performance of the method.

Nonconvex reformulations via low-rank factorization for stochastic convex semidefinite optimization problem have attracted arising attention due to their empirical efficiency and scalability. Compared with the original convex formulations, the nonconvex ones typically involve much fewer variables, allowing them to scale to scenarios with millions of variables. However, it opens a new challenge that under what conditions the nonconvex stochastic algorithms may find the population minimizer within the optimal statistical precision despite their empirical success in applications. In this paper, we provide an answer that the stochastic gradient descent (SGD) method can be adapted to solve the nonconvex reformulation of the original convex problem, with a global linear convergence when using a fixed step size, i.e., converging exponentially fast to the population minimizer within an optimal statistical precision in the restricted strongly convex case. If a diminishing step size is adopted, the bad effect caused by the variance of gradients on the optimization error can be eliminated but the rate is dropped to be sublinear. The core of our treatment relies on a novel second-order descent lemma, which is more general than the existing best result in the literature and improves the analysis on both online and batch algorithms. The developed theoretical results and effectiveness of the suggested SGD are also verified by a series of experiments.

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