We introduce and analyze the integral state constrained optimal control problem governed by a space-time fractional diffusion equation in this paper. The optimality conditions are first derived to consider the problem, then spectral Galerkin discretization of the problem is constructed. On the basis of some properties of projection operators and auxiliary equations, a priori error analysis of spectral Galerkin approximation is established rigorously. Moreover, reliable a posteriori error analysis is also investigated for state variable, adjoint state variable, control variable, and Lagrange multiplier in detail. The error estimates theoretical finding denotes that it can obtain spectral accuracy on time direction and space direction under suitable regularity assumptions.

Abstract In this paper, we focus on the mixed covolume approximation for a class of linear elliptic optimal control problems, and devise a two-grid algorithm. The state and costate are approximated using the lowest-order Raviart–Thomas mixed finite elements, while the control is discretized by means of piecewise constant functions. First, the mixed covolume approximation of the optimal control problems is constructed via the discretize-then-optimize approach. Second, the a priori error estimates for the state variables, costate variables and the control variable are established. Finally, a two-grid algorithm is formulated, and its convergence is analysed.

The purpose of this paper is to investigate the superconvergence of collocation methods for fractional integro-differential equations (FIDEs) with weakly singular kernels and Caputo derivative of order 0 < alpha < 1. First, the initial value problem of FIDEs is reformulated as a weakly singular Volterra integral equation (VIE), and the existence, uniqueness, and regularity of the exact solution for the original FIDE are obtained with the help of the resolvent theory of VIEs, and it is shown that the singularity of the exact solution is governed by the Caputo derivative, not the weakly singular kernel. Next, the piecewise polynomial collocation method is employed to solve the reformulated VIE numerically, and the optimal convergence order of the collocation solution is obtained on graded meshes. In order to improve the numerical accuracy, two types of postprocessing techniques are used - one is the classical iterated technique for VIEs and another one is the interpolation postprocessing technique. The superconvergence is thoroughly investigated and the optimal superconvergence orders are obtained for both of these two postprocessing techniques. Compared to the classical iterated collocation method, the interpolation postprocessing method has a lower calculation cost. The theoretical results are illustrated by numerical experiments.

In this paper, we present a generalized Jacobi spectral Galerkin method for fractional Volterra integro-differential equations (FVIDEs). The basis functions of the proposed method are generalized Jacobi functions, which serve as natural basis functions for appropriately designed spectral methods for FVIDEs. We establish a convergence analysis of the generalized Jacobi spectral Galerkin method under reasonable assumptions. Numerical experiments are provided to demonstrate the effectiveness of the proposed method.

In this paper, an hp spectral element approximation for distributed optimal control problem governed by an elliptic equation is investigated, whose objective functional does not include the control variable. And the constraint set on control variable is stated with L2-norm. Optimality condition of the continuous and discretized systems are deduced. In order to solve the equivalent systems with high accuracy, hp spectral element method is employed to discretize the constrained optimal control systems. Based on the property of some interpolation operators, a posteriori error estimates are also established by using some properties of some interpolation operators carefully. Finally, a projection gradient algorithm and a numerical example are provided, which confirm our analytical results. Such estimators guarantee the construction of reliable adaptive methods for optimal control problems.

As a novel computing paradigm, multi-access edge computing (MEC) empowers resource-constrained mobile terminals with robust and versatile computational capabilities. However, the development of precise offloading strategies, which determine the execution location for tasks, remains a prominent and unresolved challenge for MEC. The real-life MEC environment is characterized by its non-ideal and uncertain nature. In progressively sophisticated applications, intricate dependency relationships among their tasks continuously intensify. These factors significantly amplify the challenges associated with developing efficient offloading strategies. In this pa per, we study the dependent task offloading problem for MEC under uncertainty with joint consideration of concurrent multi-path transmission, task caching, and privacy protection. Firstly, we employ stochastic numbers to effectively characterize uncertainties and formulate the task offloading problem as a constrained stochastic joint optimization model. We establish an uncertain joint optimization model to formulate the problem of de pendent task offloading in MEC under uncertainty and prove its NP-hardness. The established model focuses on optimizing the joint strategy to achieve the objectives of reducing latency, decreasing energy consumption, and enhancing task utility. Secondly, we decompose the model into two chance-constrained sub-problems and propose a hybrid two-layer offloading algorithm that combines an improved genetic algorithm, Monte Carlo simulation, and artificial neural networks to solve the joint strategy. The proposed algorithm leverages the collaborative synergy between inner and outer layers to solve the joint strategy. The experimental results demonstrate that the established model exhibits significant effectiveness in reducing the objective value, achieving a 34.53% decrease in the objective value compared to local execution. Our offloading algorithm surpasses existing algorithms and attains an average reduction of at least 20.34% in the objective value.

Rapid and accurate State-of-Health (SOH) prediction for lithium-ion batteries is critical to guaranteeing battery safety and reliability in diverse application scenarios. To address the challenges in SOH prediction, including the difficulty in capturing strong nonlinear characteristics and the limited generalization capability of single models, this paper presents an enhanced ensemble prediction framework via the integration of Conditional Variational Autoencoders (CVAE) and Gated Recurrent Units (GRU). First, a multi-model prediction ensemble consisting of ten heterogeneous base learners is established to enable multi-perspective modeling of battery degradation features. Subsequently, the outputs of these base learners are fed as conditional inputs into a nonlinear fusion module. In this module, the CVAE characterizes latent probability distributions and mitigates noise interference, whereas the GRU captures the temporal dynamics and dependencies inherent in capacity degradation, thereby realizing the collaborative modeling of nonlinear features and temporal information. Based on the above, an output layer optimization mechanism is devised using signal space orthogonal projection theory. This mechanism aligns the prediction subspace with the genuine SOH subspace, improving the stability of weight distribution and reducing model bias. Five-fold cross-validation experiments conducted on the public MIT and XJTU datasets verify that the proposed framework achieves significant performance improvements over multiple mainstream methods on both datasets, with the root mean square error and mean absolute percentage error reduced by up to approximately 20% and 10.3%, respectively. In summary, the proposed method exhibits superior performance in prediction accuracy, generalization capability and stability, which validates its effectiveness and application potential for SOH prediction under complex degradation scenarios.

In this paper, we introduce the Nonconforming Virtual Element Method (NVEM) to solve nonlinear coupled prey-predator equations on general polygonal meshes, employing a linearized variable two-step backward differentiation formula (BDF2) for time discretization. The analysis of boundedness and error estimates for the time discrete system is conducted using discrete orthogonal convolution kernels and discrete complementary convolution kernels. Then, the time-space error splitting technique and the projection operator are integrated to establish the L infinity-norm boundedness of the fully discrete solution, independent of any grid ratio conditions, thereby naturally deriving the unconditionally optimal error estimate. The analytical method presented herein is not restricted to the NVEM and can be readily extended to other numerical techniques. Finally, the theoretical results are validated through numerical examples, demonstrating the scheme's effectiveness across various grid configurations.

In this paper, we present and analyze Newton linearized Virtual Element Methods (VEMs) for nonlinear general parabolic problems on arbitrary polygonal meshes in two-dimensions. We employ the Crank-Nicolson algorithm for temporal discretization and the VEM for spatial discretization. New linearized VEMs based on Newton iteration for the nonlinear parabolic problems are investigated. The linearized methods we proposed achieve second-order convergence accuracy in time direction and require only one single starting step. Our analysis demonstrates the boundedness of fully-discrete VEM solutions in L∞ norm and establishes optimal error estimates for the algorithm without any restrictions on the time and space mesh sizes. Numerical examples are provided to validate our theoretical results.

This paper presents a family of fully discrete energy-conserving virtual element methods for solving the coupled nonlinear Klein-Gordon equation on arbitrary polygonal meshes. The proposed method employs a scalar auxiliary variable formulation and discretizes the time variable using the Crank-Nicolson scheme, while ensuring energy conservation at the discrete level. By applying the Schaefer's fixed point theorem, we have proven the existence, uniqueness, and convergence of numerical solutions. Furthermore, we derive the optimal error estimates of order O(a2 + hr) in H1-norm independent of the grid-ratio condition, where a denotes the time step, h is the mesh size and r represents the degree of monomial. Compared with traditional theoretical analysis techniques, our approach avoid the use of commonly employed temporal-spatial splitting techniques and tedious mathematical induction methods. Instead, by estimating error difference quotient in the H-1-norm and analyzing the interdependence of errors in coupled systems, this difficulty is effectively addressed. Numerical experiments are provided to validate the effectiveness of the proposed theoretical analysis and to confirm the energy conservation in long-term simulations.

In this paper, we focus on a class of next-generation neural field models with periodic solutions and propose an efficient numerical scheme that combines a third-order Runge-Kutta (RK3) method with a high-accuracy spectral method. The scheme fully utilizes the exponential convergence advantage of the spectral method in spatial discretization, while incorporating the excellent stability and accuracy of the RK3 method in time evolution. This allows for the accurate and stable solution of nonlocal nonlinear integro-differential equations. Through rigorous theoretical analysis and systematic numerical experiments, we not only validate the high accuracy and excellent convergence of the scheme but also successfully reproduce the typical space-time dynamical patterns supported by the model, including stable space-time periodic solutions and their sensitive dependence on key parameters.

A nonconforming virtual element scheme for the Allen-Cahn equation on polygonal meshes is developed. The numerical scheme employs a variable-time-step two-step backward differentiation formula (BDF2) for temporal discretization and a lowest-order nonconforming virtual element method in space. With the help of discrete orthogonal convolution (DOC) and discrete complementary convolution (DCC) kernels, we obtain the error estimates of temporal semi-discrete system under a mild time-step ratio $0

This paper is dedicated to studying an inverted finite element method (IFEM) for elliptic optimal control problems (OCPs) in cylindrical unbounded domains. Initially, weighted Sobolev spaces are introduced for the cylindrical unbounded domains, and the properties of these spaces are further investigated. Subsequently, the IFEM, specifically designed for the weighted Sobolev spaces, is proposed. Based on the node interpolation operator, we rigorously derive interpolation error estimates from the weighted Sobolev spaces to the inverted finite element space. In addition, these results are used for elliptic OCPs in cylindrical unbounded domains for the first time, and an error estimate of the IFEM is obtained successfully. Finally, the correctness of our findings is verified through two numerical examples, fully demonstrating the effectiveness and accuracy of the IFEM in tackling elliptic OCPs in cylindrical unbounded domains.

Transport equations with distributed-order memory kernels arise naturally in modeling anomalous transport and multi-scale memory effects in complex media. Such models capture a wide spectrum of relaxation behaviors and pose substantial challenges for numerical approximation, particularly in higher dimensions. In this work, we propose a fully discrete scheme for the twodimensional transport equation with distributed-order memory kernel. The method employs a Gr & uuml;nwald-type approximation for the distributed-order derivative in time, combined with a discontinuous Galerkin method on triangular meshes for spatial discretization. We rigorously establish the unconditional stability and convergence of the scheme. A set of numerical experiments is presented to confirm the accuracy, robustness, and flexibility of the proposed method.

This article investigates both a priori and a posteriori error estimates for a pressure-robust and divergence-free virtual element method to approximate the incompressible Brinkman problem on polygonal meshes. The exactly divergence-free property of virtual space preserves the mass-conservation of the system. By extending the lowest-order Raviart–Thomas element to polygonal meshes, we construct a divergence-preserving reconstructor for the discretization of the right-hand side. A rigorous a priori error analysis is developed, showing that the velocity error is independent of both the continuous pressure and the viscosity. Taking advantage of the virtual element method’s ability to handle more general polygonal meshes, we design an adaptive mesh refinement approach and construct a residual-type a posteriori error indicator. This indicator is proven to provide global upper and local lower bounds for the discretization error. Finally, some numerical experiments demonstrate the robustness, accuracy, reliability and efficiency of the method.

In this paper, we present the first stabilization-free virtual element method (SFVEM) for solving semilinear elliptic problems in both two and three dimensions. By employing high-order polynomial projections, we reconstruct a discrete bilinear form without any stabilization term, which still maintains coercivity. The well-posedness of the discrete scheme is established, and optimal H1- and L2-error estimates are derived for the 2D problem. Moreover, we construct a lowest-order, stabilization-free virtual element scheme for the 3D case. Numerical experiments exhibit optimal convergence and demonstrate superior robustness under anisotropic diffusion.

This study proposes a high-precision finite element method (FEM) for nonlinear two-dimensional space-time-fractional diffusion equations, combining the Galerkin spatial discretization with the L2- 1_σ temporal scheme. By rigorously analyzing unconditional stability and deriving error estimates, the method achieves spatial convergence of order 2 and temporal convergence of order 3-α , where α∈ (0,1) , validated through numerical experiments under diverse initial conditions. Compared to existing works focusing on linear systems or single-term fractional dynamics, this research innovatively extends the L2- 1_σ scheme to complex nonlinear space-time-coupled problems, demonstrating its capability to simultaneously handle nonlinearities and fractional derivatives while maintaining computational efficiency and geometric adaptability.

Second-order time discrete schemes involving the parameter θ are proposed and analyzed, in combination with the finite element method (FEM), for the numerical approximation of the solution to a neural field model governed by nonlocal integro-differential equations incorporating both dendritic fibers and somatic layers The spatial discretization employs FEM, while numerical integration handles the nonlocal interactions. The stability of the second-order θ schemes within the FEM framework is studied. A priori error estimates in the L^2 norm and the H_ξ ^1 norm are also derived. The theoretical convergence rates predicted by the error analysis are validated through numerical experiments, confirming the effectiveness and feasibility of the proposed schemes.

In this paper, we study an optimal control problem with point values of the state in the objective functional. The state and adjoint state are approximated by a hybridized discontinuous Galerkin (HDG) method, and the control is discretized by the variational discretization concept. With the help of the error estimates of Green's function and Oswald interpolation, reliable and efficient a posteriori error estimates for the errors in the control, state and adjoint state variables are obtained. Several numerical examples are provided to show the performance of the obtained a posteriori error estimators.

In this scholarly article, we analyze the results of error estimates and phenomena of superconvergence associated with the mixed covolume approximation method, which is applied to a particular class of linear elliptic optimal control problems. The control variable is discretized using piecewise constant functions. Additionally, the state and the costate variables are both approximated using the lowest-order Raviart-Thomas (RT0) mixed finite element method. First, mixed covolume approximation of optimal control problems is constructed. Second, "a priori error estimations" for each variable are computed. Third, a superconvergence result is established, and it is proved that there exists a second-order superconvergence relationship between the centroid interpolation of variable u and its numerical solution. Finally, two carefully designed numerical examples are presented to validate the reliability of the theoretical findings, providing concrete evidence to corroborate the above results and strengthen the coherence of the study's conclusions.