Solving diffusion problems requires numerical methods able to capture the heterogeneity over complex geometries and are robust in terms of positivity preserving, nonlinearity, and radiation diffusion. Current deep learning methods, although mesh-free, encounter difficulties in achieving convergence and exhibit low accuracy when confronted with these specific issues. In this paper, we develop a novel method to overcome these issues based on the recently proposed random feature method (RFM). Our contributions include: (1) for anisotropic and discontinuous coefficient problems, we rewrite a diffusion problem into a first-order system and construct the corresponding loss function and approximation spaces; (2) to avoid negative solutions, we employ the square function as the activation function to enforce the positivity and the trust-region least-square solver to solve the corresponding optimization problem; (3) for the radiation diffusion problem, we enrich the approximation space of random feature functions with the heat kernel. Various numerical experiments show that the current method outperforms the standard RFM as well as deep learning methods in terms of accuracy, efficiency, and positivity preserving.

We propose a new non-convex first-order variational model for the image super-resolution problem. The model employs a recently developed regularizer that has proven to be effective in image restoration. Due to this regularizer, the salient feature of our model lies in the fact it can construct sharp edges in those generated super-resolution images from lower-resolution ones. Moreover, it also helps suppress the staircase effect. The maximum-minimum principle is proved, which indicates that there is no need to impose hard constraints on the objective function. Alternating direction method of multipliers with spectral penalty selection is utilized to minimize the associated functional. Cartoon and real gray and color images are tested to demonstrate the features of our model to show the comparison with state-of-the-art image super-resolution techniques.

Up to now, COVID-19 caused by SARS-CoV-2 is still widely spreading. Most patients have a good prognosis, with some critically ill patients dying. This paper considers the SEQIR COVID-19 model with standard incidence. Based on the characteristics of the model, we study the content of threshold behavior in deterministic and stochastic systems. We can first perform dimensionality reduction on the model due to the fact that the reduced model has the same stability as the equilibrium point of the original model. We first express the local stability of boundary equilibrium points for deterministic system after dimension reduction with the method of Lyapunov functions. After considering the perturbation of logarithmic Ornstein-Uhlenbeck processes, we study the existence and uniqueness of positive solutions. Subsequently, the critical value $R^s_0$ related to the basic regeneration number $R_0$ was obtained. And then, the conditions of $R^s_0$ about the persistence and extinction of the disease is in-depth researched, it is a critical condition. When $R^s_0 < 1,$ the disease tends to become extinct, while when $R^s_0 > 1,$ the system exhibits a stationary distribution. And the density function near the positive equilibrium point is described in detail. Finally, our conclusions are well supported through numerical simulation.

A high-order arbitrary Lagrangian-Eulerian (ALE) oscillation-free discontinuous Galerkin scheme for one-dimensional compressible multi-material flows is proposed. It couples a conservative equation in the volume-fraction model and the Euler equations of the fluid mixture dynamics. The mesh velocity is obtained by an adaptive mesh method, which automatically concentrates mesh nodes near the regions with large gradients. This greatly reduce numerical dissipation at material interfaces. Besides, the resolution of the solution at material interfaces can be also improved. To control the oscillations, we add damping terms into the weak formulation of the system. Neither parameter in new terms has to be artificially adjusted, and the difficulties with discontinuous solutions and complexities of designing limiters can be avoided. The scheme can be efficiently applied to compressible multi-material flows with essentially non-oscillatory property and its steps are more concise than in indirect ALE methods. Although here we only consider a third-order scheme, the method can be extended to any other order by choosing suitable basis functions. Examples demonstrate the third-order accuracy and good performance of the scheme for one-dimensional compressible multi-material flows.

This paper focuses on the randomized Milstein scheme for approximating solutions to stochastic Volterra integral equations with weakly singular kernels and non-differentiable drift coefficients. An essential component of the error analysis involves the utilization of randomized quadrature rule for stochastic integral to avoid the Taylor expansion in the drift coefficient. Finally, we implement the simulation of multiple singular stochastic integral in the numerical experiment by applying the Riemann-Stieltjes integral.

In this paper, we effectively solve the inverse source problem of the fractional Poisson equation using a Monte Carlo sampling-based PINN method (MC-fPINN). We construct two neural networks $u_{NN} (x;θ)$ and $f_{NN}(x;ψ)$ to approximate the solution $u^∗ (x)$ and the forcing term $f^∗(x)$ of the fractional Poisson equation. To optimize these networks, we use the Monte Carlo sampling method and define a new loss function combining the measurement data and underlying physical model. Meanwhile, we present a comprehensive error analysis for this method, along with a prior rule to select the appropriate parameters of neural networks. Numerical examples demonstrate the great accuracy and robustness of the method in solving high-dimensional problems up to 10D, with various fractional orders and noise levels of the measurement data ranging from 1% to 10%.

A high-order compact finite difference scheme for solving natural convection problems using velocity-vorticity formulation of the incompressible Navier-Stokes equations is presented. The basic idea of the method is to regard all controlling equations as the Poisson-type. We construct a fourth-order finite difference scheme for the velocity-vorticity equation based on the nine-point stencils for each Poisson-type equation. Next we give an example with an exact solution to verify that the scheme has the fourth-order accuracy. Finally, numerical solutions for the model problem of natural convection in a square heating cavity are presented to show the reliability and effectiveness of this method.

We consider numerical discretizations for nonlinear Klein-Gordon/wave equations in a rotating frame. Due to the strong centrifugal forces in the model, non-proper spatial discretizations of the rotating terms (under finite difference or finite element) would lead to numerical instability that cannot be overcome by standard time averages. We identify a class of boundary-stable type finite difference discretizations. Based on it, we propose several stable and accurate finite difference time-domain schemes with discrete conservation laws. Extensive numerical experiments and simulations are done to understand the significance of the model and the proposed schemes.

A state of the art technology is employed to investigate the local ultraconvergence properties of a quadratic rectangular element for the Poisson equation. The proposed method combine advantages of a novel interpolation postprocessing operator $\overline{P}^6_{6h,m} R^∗_h ,$ the Richardson extrapolation technique, and properties of a discrete Green’s function. The local ultraconvergence of the post-processed gradient of the finite element solution is derived with the order $\mathcal{O}(h^6 |{\rm ln}h|^2).$ A numerical example shows a good agreement with the theoretical findings.