A Note on Symplectic, Multisymplectic Scheme in Finite Element Method

We find that with uniform mesh, the numerical schemes derived from finite element method can keep a preserved symplectic structure in one-dimensional case and a preserved multisymplectic structure in two-dimentional case in certain discrete version respectively. These results are in fact the intrinsic reason that the numerical experiments indicate that such finite element algorithms are accurate in practice.Comment: 7 pages, 3 figure

Dual-Branch Temperature Scaling Calibration for Long-Tailed Recognition

The calibration for deep neural networks is currently receiving widespread attention and research. Miscalibration usually leads to overconfidence of the model. While, under the condition of long-tailed distribution of data, the problem of miscalibration is more prominent due to the different confidence levels of samples in minority and majority categories, and it will result in more serious overconfidence. To address this problem, some current research have designed diverse temperature coefficients for different categories based on temperature scaling (TS) method. However, in the case of rare samples in minority classes, the temperature coefficient is not generalizable, and there is a large difference between the temperature coefficients of the training set and the validation set. To solve this challenge, this paper proposes a dual-branch temperature scaling calibration model (Dual-TS), which considers the diversities in temperature parameters of different categories and the non-generalizability of temperature parameters for rare samples in minority classes simultaneously. Moreover, we noticed that the traditional calibration evaluation metric, Excepted Calibration Error (ECE), gives a higher weight to low-confidence samples in the minority classes, which leads to inaccurate evaluation of model calibration. Therefore, we also propose Equal Sample Bin Excepted Calibration Error (Esbin-ECE) as a new calibration evaluation metric. Through experiments, we demonstrate that our model yields state-of-the-art in both traditional ECE and Esbin-ECE metrics

Image Denoising via Nonlinear Hybrid Diffusion

A nonlinear anisotropic hybrid diffusion equation is discussed for image denoising, which is a combination of mean curvature smoothing and Gaussian heat diffusion. First, we propose a new edge detection indicator, that is, the diffusivity function. Based on this diffusivity function, the new diffusion is nonlinear anisotropic and forward-backward. Unlike the Perona-Malik (PM) diffusion, the new forward-backward diffusion is adjustable and under control. Then, the existence, uniqueness, and long-time behavior of the new regularization equation of the model are established. Finally, using the explicit difference scheme (PM scheme) and implicit difference scheme (AOS scheme), we do numerical experiments for different images, respectively. Experimental results illustrate the effectiveness of the new model with respect to other known models