We propose a novel first-order non-convex model for the fusion of infrared and visible images. It maintains thermal radiation information by ensuring that the fused image has similar pixel intensities as the infrared image, and it preserves the appearance information, including the edges and texture of the source images, by enforcing similar gray gradients and pixel intensities as the visible image. Our model could effectively reduce the staircase effect and enhance the preservation of sharp edges. The maximum-minimum principle of the model with Neumann boundary condition is discussed and the existence of a minimizer of our model in $W^{1,2} (Ω)$ is also proved. We employ the augmented Lagrangian method (ALM) to design a fast algorithm to minimize the proposed model and establish the convergence analysis of the proposed algorithm. Numerical experiments are conducted to showcase the distinctive features of the model and to provide a comparison with other image fusion techniques.

Physics-Informed Neural Networks (PINNs) encounter challenges in dealing with imbalanced training losses, especially when there are sample points with extremely high losses. This can make the optimization process unstable, making it challenging to find the correct descent direction during training. In this paper, we propose a progressive learning approach based on anomaly points awareness to improve the optimization process of PINNs. Our approach comprises two primary steps: the awareness of anomaly data points and the update of training set. Anomaly points are identified by utilizing an upper bound calculated from the mean and standard deviation of the feedforward losses of all training data. In the absence of anomalies, the parameters of the PINN are optimized using the default training data; however, once anomalies are detected, a progressive exclusion method aligned with the network learning pattern is introduced to exclude potentially unfavorable data points from the training set. In addition, intermittent detection is employed, rather than performing anomaly detection in each iteration, to balance performance and efficiency. Extensive experimental results demonstrate that the proposed method leads to substantial improvement in approximation accuracy when solving typical benchmark partial differential equations. The code is accessible at https://github.com/JcLimath/Anomaly-Aware-PINN.

Radial basis function generated finite-difference (RBF-FD) methods have recently gained popularity due to their flexibility with irregular node distributions. However, the convergence theories in the literature, when applied to nonuniform node distributions, require shrinking fill distance and do not take advantage of areas with high data density. Non-adaptive approach using same stencil size and degree of appended polynomial will have higher local accuracy at high density region, but has no effect on the overall order of convergence and could be a waste of computational power. This work proposes an adaptive RBF-FD method that utilizes the local data density to achieve a desirable order accuracy. By performing polynomial refinement and using adaptive stencil size based on data density, the adaptive RBF-FD method yields differentiation matrices with higher sparsity while achieving the same user-specified convergence order for nonuniform point distributions. This allows the method to better leverage regions with higher node density, maintaining both accuracy and efficiency compared to standard non-adaptive RBF-FD methods.

In this paper, we primarily investigate the existence, dependence and optimal control results related to solutions for a system of hemivariational inequalities pertaining to a non-stationary Navier-Stokes equation coupled with an evolution equation of temperature field. The boundary conditions for both the velocity field and temperature field incorporate the generalized Clarke gradient. The existence and uniqueness of the weak solution are established by utilizing the Banach fixed point theorem in conjunction with certain results pertaining to hemivariational inequalities. The finite element method is used to discretize the system of hemivariational inequalities and error bounds are derived. Ultimately, a result confirming the existence of a solution to an optimal control problem for the system of hemivariational inequalities is elucidated.

Remote sensing images (RSIs) encompass abundant spatial and spectral/temporal information, finding wide applications in various domains. However, during image acquisition and transmission, RSI often encounter noise interference, which adversely affects the accuracy of subsequent applications. To address this issue, this paper proposes a novel non-local fully connected tensor network (NLFCTN) decomposition algorithm for denoising RSI, aiming to fully exploit their global correlation and non-local self-similarity (NSS) characteristics. FCTN, as a recently developed tensor decomposition technique, exhibits remarkable capability in capturing global correlations and minimizing information loss. In addition, we introduce an efficient algorithm based on proximal alternating minimization (PAM) to efficiently solve the model and prove the convergence. The effectiveness of the proposed method is validated through denoising experiments on both simulated and real RSI data, employing objective evaluation metrics and subjective visual assessments. The results of the experiment show that the proposed method outperforms other RSI denoising techniques in terms of denoising performance.

The aim of this paper is to solve the Hamilton-Jacobi-Bellman (HJB) quasi-variational inequalities arising in regime switching utility maximization with optimal stopping. The HJB quasi-variational inequalities are penalized into the HJB equations and the convergence of the viscosity solution of the penalized HJB equations to that of the HJB variational inequalities is proved. The finite difference methods with iteration policy are used to solve the penalized HJB equations and the convergence is proved. The approach is implemented via numerical examples and the figures for the exercise boundaries and optimal strategies with sample paths are sketched.

In this article, an $α$-th $(0 < α < 1)$ order time-fractional reaction-diffusion equation with variably diffusion coefficient and initial weak singularity is considered. Combined with the fast $L1$ time-stepping method on graded temporal meshes, we develop and analyze a fourth-order compact block-centered finite difference (BCFD) method. By utilizing the discrete complementary convolution kernels and the $α$-robust fractional Grönwall inequality, we rigorously prove the $α$-robust unconditional stability of the developed fourth-order compact BCFD method whether for positive or negative reaction terms. Optimal sharp error estimates for both the primal variable and its flux are simultaneously derived and carefully analyzed. Finally, numerical examples are given to validate the efficiency and accuracy of the developed method.

In this work, we consider Richardson extrapolation of the Euler scheme for backward stochastic differential equations (BSDEs). First, applying the Adomian decomposition to the nonlinear generator of BSDEs, we introduce a new system of BSDEs. Then we theoretically prove that the solution of the Euler scheme for BSDEs admits an asymptotic expansion, in which the coefficients in the expansions are the solutions of the system. Based on the expansion, we propose Richardson extrapolation algorithms for solving BSDEs. Finally, some numerical tests are carried out to verify our theoretical conclusions and to show the stability, efficiency and high accuracy of the algorithms.

In this paper, we propose an integral-averaged interpolation operator $I_\tau$ in a bounded domain $Ω ⊂ \mathbb{R}^n$ by using $Q_1$-element. The interpolation coefficient is defined by the average integral value of the interpolation function $u$ on the interval formed by the midpoints of the neighboring elements. The operator $I_\tau$ reduces the regularity requirement for the function $u$ while maintaining standard convergence. Moreover, it possesses an important property of $||I_\tau u||_{0,Ω} ≤ ||u||_{0,Ω}.$ We conduct stability analysis and error estimation for the operator $I\tau.$ Finally, we present several numerical examples to test the efficiency and high accuracy of the operator.

This paper presents error analysis of a stabilizer free weak Galerkin finite element method (SFWG-FEM) for second-order elliptic equations with low regularity solutions. The standard error analysis of SFWG-FEM requires additional regularity on solutions, such as $H^2$-regularity for the second-order convergence. However, if the solutions are in $H^{1+s}$ with $0 < s < 1,$ numerical experiments show that the SFWG-FEM is also effective and stable with the $(1+s)$-order convergence rate, so we develop a theoretical analysis for it. We introduce a standard $H^2$ finite element approximation for the elliptic problem, and then we apply the SFWG-FEM to approach this smooth approximating finite element solution. Finally, we establish the error analysis for SFWG-FEM with low regularity in both discrete $H^1$-norm and standard $L^2$-norm. The $(P_k(T ), P_{k−1}(e), [P_{k+1}(T)]^d)$ elements with dimensions of space $d = 2, 3$ are employed and the numerical examples are tested to confirm the theory.

We present a rigorous analysis of the convergence rate of the deep mixed residual method (MIM) when applied to a linear elliptic equation with different types of boundary conditions. The MIM has been proposed to solve high-order partial differential equations in high dimensions. Our analysis shows that MIM outperforms deep Ritz method and deep Galerkin method for weak solution in the Dirichlet case due to its ability to enforce the boundary condition. However, for the Neumann and Robin cases, MIM demonstrates similar performance to the other methods. Our results provide valuable insights into the strengths of MIM and its comparative performance in solving linear elliptic equations with different boundary conditions.