In this paper, we propose an integral-averaged interpolation operator I-tau in a bounded domain Omega c 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 III(tau)uII(0,Omega) <= IIuII(0,Omega). 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.
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.
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.
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.
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), Pk-1(e), [Pk+1(T)](d)) elements with dimensions of space d = 2, 3 are employed and the numerical examples are tested to confirm the theory.
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.
In this article, an alpha -th (0 < alpha < 1) order time -fractional reaction -diffusion equation with variably diffusion coefficient and initial weak singularity is considered. Combined with the fast L 1 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 alpha -robust fractional Gronwall inequality, we rigorously prove the alpha -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.
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.
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.
Two mathematical models in the context of boundary value problems are proposed for the geometric design of letters in Times Roman font.We adopt radial basis function meshless collocation method for numerically solving the two proposed mathematical models in 2D and 3D.In this paper, Bézier curves play an important role in the design of the letters.Three examples with simply and multiply-connected domains in 2D and 3D are presented to demonstrate the visual effect of the letters in Times Roman font.
Deep learning methods have achieved great success in solving partial differential equations (PDEs), where the loss is often defined as an integral. The accuracy and efficiency of these algorithms depend greatly on the quadrature method. We propose to apply quasi-Monte Carlo (QMC) methods to the Deep Ritz Method (DRM) for solving the Neumann problems for the Poisson equation and the static Schrödinger equation. For error estimation, we decompose the error of using the deep learning algorithm to solve PDEs into the generalization error, the approximation error and the training error. We establish the upper bounds and prove that QMC-based DRM achieves an asymptotically smaller error bound than DRM. Numerical experiments show that the proposed method converges faster in all cases and the variances of the gradient estimators of randomized QMC-based DRM are much smaller than those of DRM, which illustrates the superiority of QMC in deep learning over MC.
We propose an alternating direction method of multipliers (ADMM) to solve an optimization problem stemming from inverse lithography. The objective functional of the optimization problem includes three terms: the misfit between the imaging on wafer and the target pattern, the penalty term which ensures the mask is binary and the total variation regularization term. By variable splitting, we introduce an augmented Lagrangian for the original objective functional. In the framework of ADMM method, the optimization problem is divided into several subproblems. Each of the subproblems can be solved efficiently. We give the convergence analysis of the proposed method. Specially, instead of solving the subproblem concerning sigmoid, we solve directly the threshold truncation imaging function which can be solved analytically. We also provide many numerical examples to illustrate the effectiveness of the method.
In this work, we consider a combined finite element method for fully coupled nonlinear thermo-poroelastic model problems. The mixed finite element (MFE) method is used for the pressure, the characteristics finite element (CFE) method is used for the temperature, and the Galerkin finite element (GFE) method is used for the elastic displacement. The semi-discrete and fully discrete finite element schemes are established and the stability of this method is presented. We derive error estimates for the pressure, temperature and displacement. Several numerical examples are presented to confirm the accuracy of the method.
A novel dynamical model with fixed-time convergence is presented to solve the system of absolute value equations (AVEs). Under a mild condition, it is proved that the solution of the proposed dynamical system converges to the solution of the AVEs. Moreover, in contrast to the existing inversion-free dynamical system (C. Chen et al., Appl. Numer. Math. 168 (2021), 170-181), a conservative settling-time of the proposed method is given. Numerical simulations illustrate the effectiveness of the new method.
Phase-field models are widely used in studying multiphase flow dynamics. Given the complexity and strong nonlinearity, designing accurate, efficient, and stable numerical algorithms to solve these models has been an active research field for decades. This paper proposes a novel numerical scheme to solve a highly cited and used phase field hydrodynamic model for simulating ternary phase fluid flows. The main novelty is the introduction of a supplementary variable to reformulate the original problem into a constrained optimization problem. This reformulation leads to several advantages for our proposed numerical algorithms compared with many existing numerical techniques for solving this model. First, the developed schemes allow more straightforward calculations for the hydrodynamic phase-field models by solving a few decoupled Helmholtz or Poisson-type systems with a constant precomputable coefficient matrix, remarkably reducing the computational cost. Secondly, the numerical schemes can maintain mass conservation and energy dissipation at the discrete level. Additionally, the developed scheme based on the second-order backward difference formula respects the original energy dissipation law that differs from many existing schemes, such as the IEQ, SAV, and Lagrange multiplier approaches for which a modified energy dissipation law is respected. Furthermore, rigorous proof of energy stability and practical implementation strategies are provided. We conduct adequate 2D and 3D numerical tests to demonstrate the proposed schemes' accuracy and effectiveness.
Character models have enormous applications in industry. Efficient creation of detailed character models is an important topic. This paper proposes a new and easy-to-use technique to quickly create detailed character models from sketches. The proposed technique consists of two main components: primitive deformer and shape generators. With this technique, 2D silhouette contours of a character model are drawn or extracted from an image or sketch. Then, proper geometric primitives are selected and aligned with the corresponding 2D silhouette contours. After that, a primitive deformer is used to create a base mesh and three shape generators are used to add 3D details to the base mesh. The primitive deformer and three shape generators are developed from ODE-driven deformations. The primitive deformer deforms the aligned geometric primitives to exactly match the 2D silhouette contours in one view plane and obtains a base mesh of a character model consisting of deformed primitives. The shape generators are used to add 3D details to the base mesh by creating local 3D models. The experimental results demonstrate that the new technique can quickly create detailed 3D character models from sketches with few manual operations. The new technique is physics-based and easy to learn and use.
A three-level linearized difference scheme for solving the Fisher equation is firstly proposed in this work. It has the good property of discrete conservative energy. By the discrete energy analysis and mathematical induction method, it is proved to be uniquely solvable and unconditionally convergent with the second-order accuracy in both time and space. Then another three-level linearized compact difference scheme is derived along with its discrete energy conservation law, unique solvability and unconditional convergence of order two in time and four in space. The resultant schemes preserve the maximum bound principle. The analysis techniques for convergence used in this paper also work for the Euler scheme, the Crank-Nicolson scheme and others. Numerical experiments are carried out to verify the computational efficiency, conservative law and the maximum bound principle of the proposed difference schemes.
In [Dai et al, Multi. Model. Simul., 2020], a structure-preserving gradient flow method was proposed for the ground state calculation in Kohn-Sham density functional theory, based on which a linearized method was developed in [Hu, et al, EAJAM, accepted] for further improving the numerical efficiency. In this paper, a complete convergence analysis is delivered for such a linearized method for the all-electron Kohn-Sham model. Temporally, the convergence, the asymptotic stability, as well as the structure-preserving property of the linearized numerical scheme in the method is discussed following previous works, while spatially, the convergence of the h-adaptive mesh method is demonstrated following [Chen et al, Multi. Model. Simul., 2014], with a key study on the boundedness of the Kohn-Sham potential for the all-electron Kohn-Sham model. Numerical examples confirm the theoretical results very well.
