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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 (spsADMM) 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.

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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.

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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. Numerous numerical experiments demonstrate that the current method outperforms the standard RFM as well as deep learning methods in terms of accuracy, efficiency, and positivity preserving.

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This paper focuses on the randomized Milstein scheme for approximating solutions to stochastic Volterra integral equations with weakly singular kernels, whose drift coefficients are non-differentiable. 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.

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Symplecticity is a significant property of stochastic Hamiltonian systems, and the symplectic methods are very attractive. Compared with the classical non-exponential Runge-Kutta methods, the exponential Runge-Kutta methods are more suitable for stiff problems. Therefore, the focus of this paper is on constructing stochastic symplectic exponential Runge-Kutta (SSERK) integrators for semilinear stochastic differential equations (SDEs) driven by multiplicative noise. The first is to establish the symplectic conditions of stochastic exponential Runge-Kutta (SERK) methods. It can be found that when the stiffness matrix is 0, these conditions will degenerate into the symplectic conditions of classical stochastic Runge-Kutta methods. Based on this idea, we construct a class of SSERK integrators with remarkable properties of structure-preservation. In addition, we verify the existence of the quadratic first integral of the stochastic Hamiltonian system and investigate the connection between preserving both the quadratic first integral and the symplectic structure by the SERK methods. Numerical experiments demonstrate a better structure-preserving ability and a higher accuracy of the SSERK integrators in solving the considered semilinear SDEs than the corresponding stochastic symplectic Runge-Kutta integrators. Excitingly, the SSERK integrators perform well when applied to the temporal discretization of stochastic nonlinear Schrödinger equation.

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Stomata are important for plants as nodes linking the transpiration process and photosynthesis process. Accurate regulation of stomatal aperture balances water loss and carbon-dioxide uptake. In a recent work, a mathematical model is developed for seed-plant stomata, which exhibits rich dynamical behaviors and can semi-quantitatively explain responses of the stomata to environmental changes. In this work, we numerically study the dynamical behaviors of the stomatal model using phase diagrams with respect to a few important parameters. The phase diagrams are helpful for further validation of the model and experimental measurements of model parameters.

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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. Moreover, our scheme. 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.

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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.

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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.

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A discontinuous Galerkin (DG) method on Bakhvalov-type ($B$-type) meshes for singularly perturbed Volterra integro-differential equations (SPVIDEs) is proposed. We derive abstract error bounds of the DG method for the SPVIEDs in the $L^2$-norm. It is shown that the approximate solution generated by the DG method on $B$-type meshes has optimal convergence rate $k+1$ in the $L^2$-norm, when using the piecewise polynomial space of degree $k.$ Numerical simulations demonstrate the validity of the theoretical results.

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In this paper, we effectively solve the inverse source problem of the fractional Poisson equation using MC-fPINNs. 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%.

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The matrix rank approximation has shown high effectiveness in the matrix rank minimization (MRM) problem, which aims to recover the underlying low-rank structure from the observed matrix by imposing the rank constraint. The nuclear norm, serving as a convex surrogate of matrix rank, is employed in the MRM problem by shrinking singular values of the observed entry. However, this substitution treats each singular value equally, which is virtually $ℓ_1$-norm penalty of the singular value vector. Theoretically, the rank function of the matrix can be considered as $ℓ_0$-norm of its singular values. Consequently, minimizing the nuclear norm frequently results in biased solutions in various applications. In this article, we first propose a novel nonconvex rank approximation, named tight and flexible rank (TFR) approximation, to describe rank function effectively. Specifically, the TFR approximation can more tightly approach the rank function and exhibit greater flexibility in handling diverse singular values, as compared to existing nonconvex rank approximations. Furthermore, we apply TFR approximation to matrix completion and develop a solving algorithm with guaranteed convergence based on the framework of proximal alternating minimization. Extensive experiments reveal that the proposed matrix completion model with TFR approximation outperforms several existing state-of-the-art convex and nonconvex methods.

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In this paper, we propose the optimal production transport model, which is an extension of the classical optimal transport model. We observe in economics, the production of the factories can always be adjusted within a certain range, while the classical optimal transport does not take this situation into account. Therefore, differing from the classical optimal transport, one of the marginals is allowed to vary within a certain range in our proposed model. To address this, we introduce a multiple relaxation optimal production transport model and propose the generalized alternating Sinkhorn algorithms, inspired by the Sinkhorn algorithm and the double regularization method. By incorporating multiple relaxation variables and multiple regularization terms, the inequality and capacity constraints in the optimal production transport model are naturally satisfied. Alternating iteration algorithms are derived based on the duality of the regularized model. We also provide a theoretical analysis to guarantee the convergence of our proposed algorithms. Numerical results indicate significant advantages in terms of accuracy and efficiency. Furthermore, we apply the optimal production transport model to the coal production and transport problem. Numerical simulation demonstrates that our proposed model can save the production and transport cost by 13.17%.

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Subdiffusion equations with distributed-order fractional derivatives describe important physical phenomena. In this paper, we consider an inverse space-dependent source term problem governed by a distributed order time-fractional diffusion equation using final time data. Based on the series expression of the solution, the inverse source problem can be transformed into a first kind of Fredholm integral equation. The existence, uniqueness and a conditional stability of the considered inverse problem are established. Building upon this foundation, a generalized quasi-boundary value regularization method is proposed to solve the inverse source problem, and then we prove the well-posedness of the regularized problem. Further, we provide the convergence rates of the regularized solution by employing an a priori and an a posteriori regularization parameter choice rule. Numerical examples in one-dimensional and two-dimensional cases are provided to validate the effectiveness of the proposed method.

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A molecular beam epitaxial (MBE) model with slope selection consisting of a fourth order Ginzburg-Landau double well potential, is derived from the variation of the free energy. One challenge in constructing the local structure preserving algorithms (SPAs) for the MBE model with slope selection is how to properly discretize the equation in space and time simultaneously in order to preserve the local structure at the discrete level. To resolve this issue, we employ the local energy dissipation property and the energy quadratization techniques. One novelty is that all nonlinear terms are treated semi-explicitly. The other novelty is that we introduce proper intermediate variables to make the space operators act on one single term, which is one crucial step in constructing local SPAs. We then develop two local energy dissipation preserving schemes and show rigorously the local energy dissipation property of the two schemes. Under suitable boundary conditions, such as periodic boundary conditions, the algorithms can preserve not only mass but also global energy dissipation property. Numerical experiments confirm the second-order accuracy and show the excellent performance of the schemes proposed.

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This paper concerns the mathematical analyses of the diffusion model in machine learning. The drift term of the backward sampling process is represented as a conditional expectation involving the data distribution and the forward diffusion. The training process aims to find such a drift function by minimizing the mean-squared residue related to the conditional expectation. Using small-time approximations of the Green’s function of the forward diffusion, we show that the analytical mean drift function in DDPM and the score function in SGM asymptotically blow up in the final stages of the sampling process for singular data distributions such as those concentrated on lower-dimensional manifolds, and are therefore difficult to approximate by a network. To overcome this difficulty, we derive a new target function and associated loss, which remains bounded even for singular data distributions. We validate the theoretical findings with several numerical examples.

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