A fast immersed boundary method for external incompressible viscous flows using lattice Green's functions

A new parallel, computationally efficient immersed boundary method for solving three-dimensional, viscous, incompressible flows on unbounded domains is presented. Immersed surfaces with prescribed motions are generated using the interpolation and regularization operators obtained from the discrete delta function approach of the original (Peskin's) immersed boundary method. Unlike Peskin's method, boundary forces are regarded as Lagrange multipliers that are used to satisfy the no-slip condition. The incompressible Navier-Stokes equations are discretized on an unbounded staggered Cartesian grid and are solved in a finite number of operations using lattice Green's function techniques. These techniques are used to automatically enforce the natural free-space boundary conditions and to implement a novel block-wise adaptive grid that significantly reduces the run-time cost of solutions by limiting operations to grid cells in the immediate vicinity and near-wake region of the immersed surface. These techniques also enable the construction of practical discrete viscous integrating factors that are used in combination with specialized half-explicit Runge-Kutta schemes to accurately and efficiently solve the differential algebraic equations describing the discrete momentum equation, incompressibility constraint, and no-slip constraint. Linear systems of equations resulting from the time integration scheme are efficiently solved using an approximation-free nested projection technique. The algebraic properties of the discrete operators are used to reduce projection steps to simple discrete elliptic problems, e.g. discrete Poisson problems, that are compatible with recent parallel fast multipole methods for difference equations. Numerical experiments on low-aspect-ratio flat plates and spheres at Reynolds numbers up to 3,700 are used to verify the accuracy and physical fidelity of the formulation.Comment: 32 pages, 9 figures; preprint submitted to Journal of Computational Physic

Generalized Filtering Decomposition

This paper introduces a new preconditioning technique that is suitable for matrices arising from the discretization of a system of PDEs on unstructured grids. The preconditioner satisfies a so-called filtering property, which ensures that the input matrix is identical with the preconditioner on a given filtering vector. This vector is chosen to alleviate the effect of low frequency modes on convergence and so decrease or eliminate the plateau which is often observed in the convergence of iterative methods. In particular, the paper presents a general approach that allows to ensure that the filtering condition is satisfied in a matrix decomposition. The input matrix can have an arbitrary sparse structure. Hence, it can be reordered using nested dissection, to allow a parallel computation of the preconditioner and of the iterative process

ThumbNet: One Thumbnail Image Contains All You Need for Recognition

Although deep convolutional neural networks (CNNs) have achieved great success in computer vision tasks, its real-world application is still impeded by its voracious demand of computational resources. Current works mostly seek to compress the network by reducing its parameters or parameter-incurred computation, neglecting the influence of the input image on the system complexity. Based on the fact that input images of a CNN contain substantial redundancy, in this paper, we propose a unified framework, dubbed as ThumbNet, to simultaneously accelerate and compress CNN models by enabling them to infer on one thumbnail image. We provide three effective strategies to train ThumbNet. In doing so, ThumbNet learns an inference network that performs equally well on small images as the original-input network on large images. With ThumbNet, not only do we obtain the thumbnail-input inference network that can drastically reduce computation and memory requirements, but also we obtain an image downscaler that can generate thumbnail images for generic classification tasks. Extensive experiments show the effectiveness of ThumbNet, and demonstrate that the thumbnail-input inference network learned by ThumbNet can adequately retain the accuracy of the original-input network even when the input images are downscaled 16 times