A Fourier pseudo-spectral method that conserves mass and energy is developed for a two-dimensional nonlinear Schrödinger equation. By establishing the equivalence between the semi-norm in the Fourier pseudo-spectral method and that in the finite difference method, we are able to extend the result in Ref.  to prove that the optimal rate of convergence of the new method is in the order of O(N −r + τ 2 ) in the discrete L 2 norm without any restrictions on the grid ratio, where N is the number of modes used in the spectral method and τ is the time step size. A fast solver is then applied to the discrete nonlinear equation system to speed up the numerical computation for the high order method. Numerical examples are presented to show the efficiency and accuracy of the new method.
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.
In this paper, we discuss the concept of local structure-preserving algorithms (SPAs) for partial differential equations, which are the natural generalization of the corresponding global SPAs. Local SPAs for the problems with proper boundary conditions are global SPAs, but the inverse is not necessarily valid. The concept of the local SPAs can explain the difference between different SPAs and provide a basic theory for analyzing and constructing high performance SPAs. Furthermore, it enlarges the applicable scopes of SPAs. We also discuss the application and the construction of local SPAs and derive several new SPAs for the nonlinear Klein-Gordon equation.
In this paper, we derive a multi-symplectic Fourier pseudospectral scheme for the Kawahara equation with special attention to the relationship between the spectral differentiation matrix and discrete Fourier transform. The relationship is crucial for implementing the scheme efficiently. By using the relationship, we can apply the Fast Fourier transform to solve the Kawahara equation. The effectiveness of the proposed methods will be demonstrated by a number of numerical examples. The numerical results also confirm that the global energy and momentum are well preserved.
We propose two schemes [AVF(2) and AVF(4)] for Maxwell's equations, by discretizing the Hamiltonian formulation with Fourier pseudospectral method for spatial discretization and average vector field method for time integration. Both AVF(2) and AVF(4) hold the two Hamiltonian energies automatically, while being energy-, momentum-and divergence-preserving, unconditionally stable, non-dissipative and spectral accurate. Rigorous error estimates are obtained for the proposed schemes. The numerical dispersion relations are also investigated. Numerical experiments support well the theoretical analysis results. The proposed schemes are valid for the regular domain, but invalid for the domain with complex geometries.
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 computationbound 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.
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