Yushun Wang

Chaolong Jiang, Wenjun Cai, Yushun Wang et al.

In this paper, a novel sixth order energy-conserved method is proposed for solving the three-dimensional time-domain Maxwell's equations. The new scheme preserves five discrete energy conservation laws, three momentum conservation laws, symplectic conservation law as well as two divergence-free properties and is proved to be unconditionally stable, non-dissipative. An optimal error estimate is established based on the energy method, which shows that the proposed method is of sixth order accuracy in time and spectral accuracy in space in discrete $L^{2}$-norm. The constant in the error estimate is proved to be only $O(T)$. Furthermore, the numerical dispersion relation is analyzed in detail and a fast solver is presented to solve the resulting discrete linear equations efficiently. Numerical results are addressed to verify our theoretical analysis.

Yu Xing, Shuang Liang, Lingzhi Sui et al.

The convolutional neural network (CNN) has become a state-of-the-art method for several artificial intelligence domains in recent years. The increasingly complex CNN models are both computation-bound and I/O-bound. FPGA-based accelerators driven by custom instruction set architecture (ISA) achieve a balance between generality and efficiency, but there is much on them left to be optimized. We propose the full-stack compiler DNNVM, which is an integration of optimizers for graphs, loops and data layouts, and an assembler, a runtime supporter and a validation environment. The DNNVM works in the context of deep learning frameworks and transforms CNN models into the directed acyclic graph: XGraph. Based on XGraph, we transform the optimization challenges for both the data layout and pipeline into graph-level problems. DNNVM enumerates all potentially profitable fusion opportunities by a heuristic subgraph isomorphism algorithm to leverage pipeline and data layout optimizations, and searches for the best choice of execution strategies of the whole computing graph. On the Xilinx ZU2 @330 MHz and ZU9 @330 MHz, we achieve equivalently state-of-the-art performance on our benchmarks by naïve implementations without optimizations, and the throughput is further improved up to 1.26x by leveraging heterogeneous optimizations in DNNVM. Finally, with ZU9 @330 MHz, we achieve state-of-the-art performance for VGG and ResNet50. We achieve a throughput of 2.82 TOPs/s and an energy efficiency of 123.7 GOPs/s/W for VGG. Additionally, we achieve 1.38 TOPs/s for ResNet50 and 1.41 TOPs/s for GoogleNet.

Qi Hong, Yushun Wang, Yuezheng Gong

In this paper, two novel linear-implicit and momentum-preserving Fourier pseudo-spectral schemes are proposed and analyzed for the regularized long-wave equation. The numerical methods are based on the blend of the Fourier pseudo-spectral method in space and the linear-implicit Crank-Nicolson method or the leap-frog scheme in time. The two fully discrete linear schemes are shown to possess the discrete momentum conservation law, and the linear systems resulting from the schemes are proved uniquely solvable. Due to the momentum conservative property of the proposed schemes, the Fourier pseudo-spectral solution is proved to be bounded in the discrete $L^{\infty}$ norm. Then by using the standard energy method, both the linear-implicit Crank-Nicolson momentum-preserving scheme and the linear-implicit leap-frog momentum-preserving scheme are shown to have the accuracy of $\mathcal{O}(τ^2+N^{-r})$ in the discrete $L^{\infty}$ norm without any restrictions on the grid ratio, where $N$ is the number of nodes and $τ$ is the time step size. Numerical examples are carried out to verify the correction of the theory analysis and the efficiency of the proposed schemes.

Wenjun Cai, Chaolong Jiang, Yushun Wang

This paper presents two kinds of strategies to construct structure-preserving algorithms with homogeneous Neumann boundary conditions for the sine-Gordon equation, while most existing structure-preserving algorithms are only valid for zero or periodic boundary conditions. The first strategy is based on the conventional second-order central difference quotient but with a cell-centered grid, while the other is established on the regular grid but incorporated with summation by parts (SBP) operators. Both the methodologies can provide conservative semi-discretizations with different forms of Hamiltonian structures and the discrete energy. However, utilizing the existing SBP formulas, schemes obtained by the second strategy can directly achieve higher-order accuracy while it is not obvious for schemes based on the cell-centered grid to make accuracy improved easily. Further combining the symplectic Runge-Kutta method and the scalar auxiliary variable (SAV) approach, we construct symplectic integrators and linearly implicit energy-preserving schemes for the two-dimensional sine-Gordon equation, respectively. Extensive numerical experiments demonstrate their effectiveness with the homogeneous Neumann boundary conditions.

Wenjun Cai, Haochen Li, Yushun Wang

The classic second-order average vector field (AVF) method can exactly preserve the energy for Hamiltonian ordinary differential equations and partial differential equations. However, the AVF method inevitably leads to fully-implicit nonlinear algebraic equations for general nonlinear systems. To address this drawback and maintain the desired energy-preserving property, a first-order partitioned AVF method is proposed which first divides the variables into groups and then applies the AVF method step by step. In conjunction with its adjoint method we present the partitioned AVF composition method and plus method respectively to improve its accuracy to second order. Concrete schemes for two classic model equations are constructed with semi-implicit, linear-implicit properties that make considerable lower cost than the original AVF method. Furthermore, additional conservative property can be generated besides the conventional energy preservation for specific problems. Numerical verification of these schemes further conforms our results.

Haochen Li, Yushun Wang

In this paper, we apply the Boole discrete line integral to solve the Lorentz force system which is written as a non-canonical Hamiltonian system. The method is exactly energy-conserving for polynomial Hamiltonians of degree $ν\leq 4$. In any other case, the energy can also be conserved approximatively. With comparison to well-used Boris method, numerical experiments are presented to demonstrate the energy-preserving property of the method.

Haochen Li, Yushun Wang, Mengzhao Qin

In this paper, based on the theory of rooted trees and B-series, we propose the concrete formulas of the substitution law for the trees of order =5. With the help of the new substitution law, we derive a B-series integrator extending the averaged vector field (AVF) method to high order. The new integrator turns out to be of order six and exactly preserves energy for Hamiltonian systems. Numerical experiments are presented to demonstrate the accuracy and the energy-preserving property of the sixth order AVF method.