Applied Numerical Mathematics最新文献

New families of composition methods with processing of order 4 and 6 are presented and analyzed. They are specifically designed to be used for the numerical integration of differential equations whose vector field is separated into three or more parts which are explicitly solvable. The new schemes are shown to be more efficient than previous state-of-the-art splitting methods.

Objective image quality assessment involves the use of mathematical models to quantitatively describe image quality. FR-IQA (Full-Reference Image Quality Assessment) methods using reference images are also often used to evaluate image processing and computer vision algorithms. Quality indices often use gradient operators to express relevant visual information, such as edges. Fractional calculus has been applied in the last two decades in various fields such as signal processing, image processing, and pattern recognition. Fractional derivatives are generalizations of integer-order derivatives and can be computed using various operators such as the Riemann-Liouville, Caputo-Fabrizio, and Grünwald-Letnikov operators. In this paper, we propose a modification of the FSIMc image quality index by including fractional derivatives to extract and enhance edges. A study of the usefulness of fractional derivative in the FSIMc model was conducted by assessing Pearson, Spearman and Kendall correlations with MOS scores for images from the TID2013 and KADID-10k databases. Comparison of FD_FSIMc with the classic FSIMc shows an increase of several percent in the correlation coefficients for the modified index. The results obtained are superior to those other known approaches to FR-IQA that use fractional derivatives. The results encourage the use of fractional calculus.

This paper proposes a local discontinuous Galerkin (LDG) method for the stochastic Korteweg-de Vries (KdV) equation with multi-dimensional multiplicative noise. In the mean square sense, we show that the numerical method is $L^2$ stable and it preserves energy conservation and energy dissipation. If the degree of the polynomial is n, the optimal error estimate in the mean square sense can reach as $n+1$. Finally, structure-preserving and convergence are verified by numerical experiments.

It is well known that the implicit central difference discretization for unsteady advection diffusion equation (ADE) suffers from being time-consuming to solve when the advection term dominates. In this paper, we propose an operator-splitting scheme for the unsteady ADE, in which the ADE is firstly discretized by Crank-Nicolson (CN) scheme in time and central difference scheme in space; and then the discrete advection-diffusion problem is split as advection sub-problem and diffusion sub-problem at each time-level. The significance of the new scheme is that these sub-problems can be fast and directly solved within a linearithmic complexity (a linear-times-logarithm complexity) by means of fast sine transforms (FSTs). In particular, the complexity is independent of the dominance of the advection term. Theoretically, we show that proposed scheme is unconditionally stable and of second-order convergence in time and space. Numerical results are reported to show the efficiency of the proposed scheme.

In this paper, we introduce two one-parameter families of quadratic polynomial enrichments designed to enhance the accuracy of the classical Crouzeix–Raviart finite element. These enrichments are realized by using weighted line integrals as enriched linear functionals and quadratic polynomial functions as enrichment functions. To validate the effectiveness of our approach, we conduct numerical experiments that confirm the improvement achieved by the proposed method.

In this paper, based on the framework of the supplementary variable method, we present two classes of high-order, linearized, structure-preserving algorithms for simulating the regularized long-wave equation. The suggested schemes are as accurate and efficient as the recently proposed schemes in Jiang et al. (2022) [20], but share the nice features in two folds: (i) the first type of schemes conserves the original energy conservation, as opposed to a modified quadratic energy in [20]; (ii) the second type of schemes fills the gap of [20] by constructing high-order linear algorithms that preserve both two invariants of mass and momentum. We discretize the SVM systems by employing the algebraically stable Runge-Kutta method together with the prediction-correction technique in time and the Fourier pseudo-spectral method in space. The implementation benefits from solving the optimization problems subject to PDE constraints. Numerical examples and some comparisons are provided to show the effectiveness, accuracy and performance of the proposed schemes.

In this paper, we introduce two accelerated primal-dual methods tailored to address linearly constrained composite convex optimization problems, where the objective function is expressed as the sum of a possibly nondifferentiable function and a differentiable function with Lipschitz continuous gradient. The first method is the accelerated linearized augmented Lagrangian method (ALALM), which permits linearization to the differentiable function; the second method is the accelerated linearized proximal point algorithm (ALPPA), which enables linearization of both the differentiable function and the augmented term. By incorporating adaptive parameters, we demonstrate that ALALM achieves the $O(1/k^2)$ convergence rate and the linear convergence rate under the assumption of convexity and strong convexity, respectively. Additionally, we establish that ALPPA enjoys the $O(1/k)$ convergence rate in convex case and the $O(1/k^2)$ convergence rate in strongly convex case. We provide numerical results to validate the effectiveness of the proposed methods.

Given the Caputo-type fractional differential equation $D^{\alpha }y\left(t\right)=f(t,y(t\left)\right)$ with $\alpha \in (1,2)$, we consider two distinct solutions $y_1,y_2\in C[0,T]$ to this equation subject to different sets of initial conditions. In this framework, we discuss nontrivial upper and lower bounds for the difference $\left|y_1\right(t)-y_2(t\left)\right|$ for $t\in [0,T]$. The main emphasis is on describing how such bounds are related to the differences of the associated initial values.

In this paper we focus on the space of $C^2$ quartic splines on uniform criss-cross triangulations and we propose a method based on weighted essentially non-oscillatory techniques and obtained by modifying classical spline quasi-interpolants in order to approximate piecewise smooth functions avoiding Gibbs phenomenon near discontinuities and, at the same time, maintaining the high-order accuracy in smooth regions. We analyse the convergence properties of the proposed quasi-interpolants and we provide some numerical and graphical tests confirming the theoretical results.

In this paper, we study the mixed virtual element approximation to an elliptic optimal control problem with boundary observations. The objective functional of this type of optimal control problem contains the outward normal derivatives of the state variable on the boundary, which reduces the regularity of solutions to the optimal control problems. We construct the mixed virtual element discrete scheme and derive an a priori error estimate for the optimal control problem based on the variational discretization for the control variable. Numerical experiments are carried out on different meshes to support our theoretical findings.