Simplicial Ricci Flow

We construct a discrete form of Hamilton's Ricci flow (RF) equations for a d-dimensional piecewise flat simplicial geometry, S. These new algebraic equations are derived using the discrete formulation of Einstein's theory of general relativity known as Regge calculus. A Regge-Ricci flow (RRF) equation is naturally associated to each edge, L, of a simplicial lattice. In defining this equation, we find it convenient to utilize both the simplicial lattice, S, and its circumcentric dual lattice, S*. In particular, the RRF equation associated to L is naturally defined on a d-dimensional hybrid block connecting $\ell$ with its (d-1)-dimensional circumcentric dual cell, L*. We show that this equation is expressed as the proportionality between (1) the simplicial Ricci tensor, Rc_L, associated with the edge L in S, and (2) a certain volume weighted average of the fractional rate of change of the edges, lambda in L*, of the circumcentric dual lattice, S*, that are in the dual of L. The inherent orthogonality between elements of S and their duals in S* provide a simple geometric representation of Hamilton's RF equations. In this paper we utilize the well established theories of Regge calculus, or equivalently discrete exterior calculus, to construct these equations. We solve these equations for a few illustrative examples.Comment: 34 pages, 10 figures, minor revisions, DOI included: Commun. Math. Phy

A thermal lattice Boltzmann model for micro/nano-flows

The dynamic behavior of charged micro and nanofluids plays a crucial role in a large variety of industrial and biological processes. Such dynamic behavior is characterized by the simultaneous occurrence of several competing mechanisms, such as electrostatic interactions, viscous dissipation and hydrodynamic effects, often taking place in complex geometries. This paper focuses on a thermal lattice Boltzmann model for micro/nano-flows

Learning Convolutional Networks for Content-weighted Image Compression

Lossy image compression is generally formulated as a joint rate-distortion optimization to learn encoder, quantizer, and decoder. However, the quantizer is non-differentiable, and discrete entropy estimation usually is required for rate control. These make it very challenging to develop a convolutional network (CNN)-based image compression system. In this paper, motivated by that the local information content is spatially variant in an image, we suggest that the bit rate of the different parts of the image should be adapted to local content. And the content aware bit rate is allocated under the guidance of a content-weighted importance map. Thus, the sum of the importance map can serve as a continuous alternative of discrete entropy estimation to control compression rate. And binarizer is adopted to quantize the output of encoder due to the binarization scheme is also directly defined by the importance map. Furthermore, a proxy function is introduced for binary operation in backward propagation to make it differentiable. Therefore, the encoder, decoder, binarizer and importance map can be jointly optimized in an end-to-end manner by using a subset of the ImageNet database. In low bit rate image compression, experiments show that our system significantly outperforms JPEG and JPEG 2000 by structural similarity (SSIM) index, and can produce the much better visual result with sharp edges, rich textures, and fewer artifacts