As is known, there exist numerous alternating direction implicit (ADI) schemes for the two-dimensional linear time fractional partial differential equations (PDEs). However, if the ADI schemes for linear problems combined with local linearization techniques are applied to solve nonlinear problems, the stability and convergence of the methods are often not clear. In this paper, two ADI schemes are developed for solving the two-dimensional time fractional nonlinear super-diffusion equations based on their equivalent partial integro-differential equations. In these two schemes, the standard second-order central difference approximation is used for the spatial discretization, and the classical first-order approximation is applied to discretize the Riemann-Liouville fractional integral in time. The solvability, unconditional stability and $L_2$ norm convergence of the proposed ADI schemes are proved rigorously. The convergence order of the schemes is $O(τ + h^2_x + h^2_y)$, where $τ$ is the temporal mesh size, $h_x$ and $h_y$ are spatial mesh sizes in the $x$ and $y$ directions, respectively. Finally, numerical experiments are carried out to support the theoretical results and demonstrate the performances of two ADI schemes.

An $hp$ version of interface penalty finite element method ($hp$-IPFEM) is proposed to solve the elliptic interface problems in two and three dimensions on unfitted meshes. Error estimates in broken $H^1$ norm, which are optimal with respect to $h$ and suboptimal with respect to $p$ by half an order of $p$, are derived. Both symmetric and non-symmetric IPFEM are considered. Error estimates in $L^2$ norm are proved by the duality argument. All the estimates are independent of the location of the interface relative to the meshes. Numerical examples are provided to illustrate the performance of the method. This paper is adapted from the work originally post on arXiv.com by the same authors (arXiv:1007.2893v1).

Local Fourier analysis (LFA) is a useful tool in predicting the convergence factors of geometric multigrid methods (GMG). As is well known, on rectangular domains with periodic boundary conditions this analysis gives the exact convergence factors of such methods. When other boundary conditions are considered, however, this analysis was judged as been heuristic, with limited capabilities in predicting multigrid convergence rates. In this work, using the Fourier method, we extend these results by proving that such analysis yields the exact convergence factors for a wider class of problems, some of which cannot be handled by the traditional rigorous Fourier analysis.

We present LBW-Net, an efficient optimization based method for quantization and training of the low bit-width convolutional neural networks (CNNs). Specifically, we quantize the weights to zero or powers of 2 by minimizing the Euclidean distance between full-precision weights and quantized weights during backpropagation (weight learning). We characterize the combinatorial nature of the low bit-width quantization problem. For 2-bit (ternary) CNNs, the quantization of $N$ weights can be done by an exact formula in $O$($N$ log $N$) complexity. When the bit-width is 3 and above, we further propose a semi-analytical thresholding scheme with a single free parameter for quantization that is computationally inexpensive. The free parameter is further determined by network retraining and object detection tests. The LBW-Net has several desirable advantages over full-precision CNNs, including considerable memory savings, energy efficiency, and faster deployment. Our experiments on PASCAL VOC dataset [5] show that compared with its 32-bit floating-point counterpart, the performance of the 6-bit LBW-Net is nearly lossless in the object detection tasks, and can even do better in real world visual scenes, while empirically enjoying more than 4× faster deployment.

In this paper, we analyze and provide numerical experiments for a moving finite element method applied to convection-dominated, time-dependent partial differential equations. We follow a method of lines approach and utilize an underlying tensor-product finite element space that permits the mesh to evolve continuously in time and undergo discontinuous reconfigurations at discrete time steps. We employ the TR-BDF2 method as the time integrator for piecewise quadratic tensor-product spaces, and provide an almost symmetric error estimate for the procedure. Our numerical results validate the efficacy of these moving finite elements.

In this paper, we consider the scattering problem of time-harmonic electromagnetic waves from an infinite cylinder having an open arc $Γ$ and a bounded domain $D$ in $\mathbb{R}$² as cross section. We focus on the inverse scattering problem, that is, to reconstruct the shape of $Γ$ and $D$ from the far-field pattern by using the factorization method. Through establishing a mixed reciprocity relation, we prove that the scatters $Γ$ and $D$ can be uniquely determined by the far-field pattern. Furthermore, the mathematical basis is given to explain that the factorization method is feasible to our problem. At the end of this paper, we give some numerical examples to show the efficaciousness of the algorithms.

In recent decades, the fractional Fourier transform as well as the linear canonical transform became very efficient tools in a variety of approximation and signal processing applications. There are many literatures on sampling expansions of interpolation type for band-limited functions in the sense of these transforms. However, rigorous studies on convergence or error analysis are rare. It is our aim in this paper to establish sampling expansions of interpolation type for band-limited functions and to investigate their convergence and error analysis. In particular, we introduce rigorous error estimates for the truncation error and both amplitude and jitter-time errors.

This paper considers a class of discontinuous Galerkin method, which is constructed by Wong-Zakai approximation with the orthonormal Fourier basis, for numerically solving nonautonomous Stratonovich stochastic delay differential equations. We prove that the discontinuous Galerkin scheme is strongly convergent, globally stable and analogously asymptotically stable in mean square sense. In addition, this method can be easily extended to solve nonautonomous Stratonovich stochastic pantograph differential equations. Numerical tests indicate that the method has first-order and half-order strong mean square convergence, when the diffusion term is without delay and with delay, respectively.