The Calderόn formulas have been recently utilized in the process of constructing valid boundary integral equation systems which possess highly favorable spectral properties. This work is devoted to studying the theoretical properties of elastodynamic Calderόn formulas which provide us with a solid basis for the design of fast boundary integral equation methods solving elastic wave problems defined on a close- or open-surface in two dimensions. For the closed-surface case, it is proved that the Calderόn formula is a Fredholm operator of second-kind except for certain circumstances. For the open-surface case, we investigate weighted integral operators instead of the original integral operators which are resulted from dealing with edge singularities of potentials corresponding to the elastic scattering problems by open-surfaces, and show that the Calderόn formula is a compact perturbation of a bounded and invertible operator whose spectrum enjoys the same accumulation points as the Calderόn formula in the closed-surface case.

We propose a relaxed alternating minimization algorithm for solving two-block separable convex minimization problems with linear equality constraints, where one block in the objective functions is strongly convex. We prove that the proposed algorithm converges to the optimal primal-dual solution of the original problem. Furthermore, the convergence rates of the proposed algorithm in both ergodic and nonergodic senses have also been studied. We apply the proposed algorithm to solve several composite convex minimization problems arising in image denoising and evaluate the numerical performance of the proposed algorithm on a novel image denoising model. Numerical results for both artificial and real noisy images demonstrate the efficiency and effectiveness of the proposed algorithm.

The Chan-Vese (CV) model is a classic region-based method in image segmentation. However, its piecewise constant assumption does not always hold for practical applications. Many improvements have been proposed but the issue is still far from well solved. In this work, we propose an unsupervised image segmentation approach that integrates the CV model with deep neural networks, which significantly improves the original CV model’s segmentation accuracy. Our basic idea is to apply a deep neural network that maps the image into a latent space to alleviate the violation of the piecewise constant assumption in image space. We formulate this idea under the classic Bayesian framework by approximating the likelihood with an evidence lower bound (ELBO) term while keeping the prior term in the CV model. Thus, our model only needs the input image itself and does not require pre-training from external datasets. Moreover, we extend the idea to multi-phase case and dataset based unsupervised image segmentation. Extensive experiments validate the effectiveness of our model and show that the proposed method is noticeably better than other unsupervised segmentation approaches.

Gradient descent yields zero training loss in polynomial time for deep neural networks despite non-convex nature of the objective function. The behavior of network in the infinite width limit trained by gradient descent can be described by the Neural Tangent Kernel (NTK) introduced in [25]. In this paper, we study dynamics of the NTK for finite width Deep Residual Network (ResNet) using the neural tangent hierarchy (NTH) proposed in [24]. For a ResNet with smooth and Lipschitz activation function, we reduce the requirement on the layer width $m$ with respect to the number of training samples $n$ from quartic to cubic. Our analysis suggests strongly that the particular skip-connection structure of ResNet is the main reason for its triumph over fully-connected network.

Deep Ritz methods (DRM) have been proven numerically to be efficient in solving partial differential equations. In this paper, we present a convergence rate in $H^1$ norm for deep Ritz methods for Laplace equations with Dirichlet boundary condition, where the error depends on the depth and width in the deep neural networks and the number of samples explicitly. Further we can properly choose the depth and width in the deep neural networks in terms of the number of training samples. The main idea of the proof is to decompose the total error of DRM into three parts, that is approximation error, statistical error and the error caused by the boundary penalty. We bound the approximation error in $H^1$ norm with ${\rm ReLU}^2$ networks and control the statistical error via Rademacher complexity. In particular, we derive the bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm with ${\rm ReLU}^2$ network, which is of immense independent interest. We also analyze the error inducing by the boundary penalty method and give a prior rule for tuning the penalty parameter.

In this paper, we formulate a special epidemic dynamic model to describe the transmission of COVID-19 in Algeria. We derive the threshold parameter control reproduction number $(\mathcal{R}^0_c),$ and present the effective control reproduction number $(\mathcal{R}_c(t))$ as a real-time index for evaluating the epidemic under different control strategies. Due to the limitation of the reported data, we redefine the number of accumulative confirmed cases with diagnostic shadow and then use the processed data to do the optimal numerical simulations. According to the control measures, we divide the whole research period into six stages. And then the corresponding medical resource estimations and the average effective control reproduction numbers for each stage are given. Meanwhile, we use the parameter values which are obtained from the optimal numerical simulations to forecast the whole epidemic tendency under different control strategies.

In this paper, we first propose a concept of Weyl almost periodic random processes on time scales, including the concepts of Weyl almost periodic random processes in $p$-th mean and Weyl almost periodic random processes in distribution. Then, using the Banach fixed point theorem, time scale calculus theory and inequality techniques, the existence and stability of Weyl almost periodic solutions for Clifford-valued stochastic high-order Hopfield neural networks with time-varying delays on time scales are studied. Even when the system we consider degenerates into a real-valued system, our results are new. Finally, a numerical example is given to illustrate the feasibility of our theoretical results.