Linyu Peng

Linyu Peng, Peter E. Hydon

The difference variational bicomplex, which is the natural setting for systems of difference equations, is constructed and used to examine the geometric and algebraic properties of various systems. Exactness of the bicomplex gives a coordinate-free setting for finite difference variational problems, Euler--Lagrange equations and Noether's theorem. We also examine the connection between the condition for the existence of a Hamiltonian and the multisymplecticity of systems of partial difference equations. Furthermore, we define difference multimomentum maps of multisymplectic systems, which yield their conservation laws. To conclude, we adapt the variational bicomplex to multisymplectic integrators on a mesh that is logically rectangular. By scaling horizontal forms and difference operators according to the local step sizes, all of the results derived earlier can be applied, whether or not the mesh is uniform.

Linyu Peng

This paper mainly contributes to the extension of Noether's theorem to differential-difference equations. For that purpose, we first investigate the prolongation formula for continuous symmetries, which makes a characteristic representation possible. The relations of symmetries, conservation laws and the Fréchet derivative are also investigated. For non-variational equations, since Noether's theorem is now available, the self-adjointness method is adapted to the computation of conservation laws for differential-difference equations. A couple of differential-difference equations are investigated as illustrative examples, including the Toda lattice and semi-discretisations of the Korteweg-de Vries (KdV) equation. In particular, the Volterra equation is taken as a running example.

Weizhi Gao, Zhichao Hou, Junqi Yin et al.

Diffusion models have emerged as powerful generative models, but their high computation cost in iterative sampling remains a significant bottleneck. In this work, we present an in-depth and insightful study of state-of-the-art acceleration techniques for diffusion models, including caching and quantization, revealing their limitations in computation error and generation quality. To break these limits, this work introduces Modulated Diffusion (MoDiff), an innovative, rigorous, and principled framework that accelerates generative modeling through modulated quantization and error compensation. MoDiff not only inherents the advantages of existing caching and quantization methods but also serves as a general framework to accelerate all diffusion models. The advantages of MoDiff are supported by solid theoretical insight and analysis. In addition, extensive experiments on CIFAR-10 and LSUN demonstrate that MoDiff significant reduces activation quantization from 8 bits to 3 bits without performance degradation in post-training quantization (PTQ). Our code implementation is available at https://github.com/WeizhiGao/MoDiff.

Yusuke Ono, Simone Fiori, Linyu Peng

The Euler--Poincaré equations, firstly introduced by Henri Poincaré in 1901, arise from the application of Lagrangian mechanics to systems on Lie groups that exhibit symmetries, particularly in the contexts of classical mechanics and fluid dynamics. These equations have been extended to various settings, such as semidirect products, advected parameters, and field theory, and have been widely applied to mechanics and physics. In this paper, we introduce the discrete Euler--Poincaré reduction for discrete Lagrangian systems on Lie groups with advected parameters and additional dynamics, utilizing the group difference map technique. Specifically, the group difference map is defined using either the Cayley transform or the matrix exponential. The continuous and discrete Kelvin--Noether theorems are extended accordingly, that account for Kelvin--Noether quantities of the corresponding continuous and discrete Euler--Poincaré equations. As an application, we show both continuous and discrete Euler--Poincaré formulations about the dynamics of underwater vehicles, followed by numerical simulations. Numerical results illustrate the scheme's ability to preserve geometric properties over extended time intervals, highlighting its potential for practical applications in the control and navigation of underwater vehicles, as well as in other domains.

Xiaomin Duan, Huafei Sun, Linyu Peng

In this paper, the Riemannian gradient algorithm and the natural gradient algorithm are applied to solve descent direction problems on the manifold of positive definite Hermitian matrices, where the geodesic distance is considered as the cost function. The first proposed problem is control for positive definite Hermitian matrix systems whose outputs only depend on their inputs. The geodesic distance is adopted as the difference of the output matrix and the target matrix. The controller to adjust the input is obtained such that the output matrix is as close as possible to the target matrix. We show the trajectory of the control input on the manifold using the Riemannian gradient algorithm. The second application is to compute the Karcher mean of a finite set of given Toeplitz positive definite Hermitian matrices, which is defined as the minimizer of the sum of geodesic distances. To obtain more efficient iterative algorithm compared with traditional ones, a natural gradient algorithm is proposed to compute the Karcher mean. Illustrative simulations are provided to show the computational behavior of the proposed algorithms.