This paper aims to investigate some recent progress on inverse source problems of timeharmonic wave equations and establish the stability in general cases. For scattering models of deterministic and stochastic wave equations, we show the methodology to obtain stability and summarize existing theoretical and numerical results.

Based on jump-adapted time partition, this paper proposes a jump-adapted split-step backward Euler numerical approximation method for solving a class of nonlinear jump-diffusion problems. Under non-global Lipschitz conditions, by overcoming the main difficulties caused by strong nonlinear coefficients and random time partition in numerical analysis, we establish strong error estimates for the proposed numerical method, and obtain the optimal mean square convergence order. Finally, numerical experiments are provided to validate the theoretical results.

The primal-dual algorithm (PDA) is a full-splitting method that simultaneously obtain solutions to both primal and dual problems, which is a classical approach to solve bilinear saddle point problems. However in the existing PDA, the step size depends on the spectral norm of linear transform or can be estimated by linesearch, which is often overly conservative or requires extra computations of proximal operators or linear transform. In this paper, we present a splitting preconditioned PDA (SP-PDA), by adding proximal terms to the Lagrangian function and introducing a preconditioning strategy by solving linear matrix inversion problem. The proposed method has free step sizes and involves matrix decomposition only once, thus the computational can be burdened for solving linear inverse problems. We establish global iterative convergence and derive an $\mathcal{O}(1/N)$ ergodic convergence rate measured by function value residuals and constraint violations. Finally, numerical experiments on LASSO and matrix game problems demonstrate the efficient of SP-PDA.

The conjugate gradient method is one of the most effective methods to solve large-scale optimization problems. In this paper, three sets of Dai-Liao conjugate condition parameters are provided, with truncated Dai-Liao conjugate parameters, and a restart procedure is set in the search direction. Thus a new Dai-Liao conjugate gradient algorithm is proposed. The search direction generated by the new algorithm satisfies the sufficient descent condition at each iteration without depending on any line search condition. Under the usual assumptions and the weak Wolfe line search condition, the algorithm is strongly convergent. Finally, the new algorithm is applied to solve large-scale unconstrained optimization, image restorations and machine learning. Numerical results show that the proposed algorithm is effective.

Solving large-wavenumber Helmholtz equations with traditional numerical methods faces an inherent trade-off between computational accuracy and efficiency. This paper proposes Frequency-Enhanced High-order ReLU-KAN (FE-HRKAN). It introduces a learnable adaptive frequency modulation mechanism into the existing High-order ReLU-KAN (HRKAN) framework, expanding the input features to a combination of the original variables and parameterized high-frequency oscillatory features. The paper proves HRKAN’s spectral limitations and demonstrates the extended high-frequency expressiveness of FE-HRKAN, ensuring that FE-HRKAN enhances the capability to represent high-frequency oscillations while maintaining the original performance of HRKAN. Experimental results show that in function approximation tasks, FE-HRKAN reduces the L₂ relative error for approximating high-frequency oscillatory functions by two orders of magnitude compared to the baseline HRKAN model, while also reducing the L₂ relative error for approximating non-oscillatory functions by 34%. In solving large-wavenumber Helmholtz equations, FE-HRKAN achieves L₂ relative errors on the order of 10^-3 to 10^-4 across wavenumbers ranging from 5 to 1000, reducing errors by 3 to 4 orders of magnitude compared to HRKAN in large-wavenumber scenarios.

This paper investigates the exponential stability of numerical solutions for stochastic differential equations (SDEs) and explores the necessity of fully implicit methods. Centered around two counterexamples, the limitations of Euler-type methods (including the stochastic theta method and the truncated Euler method) are discussed. Based on the theory of exponential martingales, the almost sure exponential stability conditions of the zero solution for SDEs are improved. It is then proven that the fully implicit Milstein method performs well for these two counterexamples. Numerical experiments validate the conclusions. Specifically, there exist SDEs for which, when considering exponential stability, commonly used Eulertype methods (such as the stochastic theta method and the truncated Euler method) are not applicable, whereas the fully implicit Milstein method remains effective. Thus, fully implicit schemes are essential in the study of exponential stability for numerical solutions of SDEs.

This article designs a Chebyshev-Galerkin spectral method based on a linear combination of Chebyshev polynomials as the basis functions to solve the Schrödinger equation with transparent boundary conditions. The paper rigorously analyzes the convergence of the spectral method. Through the design of numerical experiments, it verifies the high-order convergence of this algorithm. We also compare it with the traditional finite difference method, highlighting the advantages of this algorithm. For potential energy functions of single and double barriers, the quantum tunneling and resonance tunneling phenomena are simulated by calculating the transmission rate variation curves. The algorithm is then applied to simulate the current-voltage characteristics of the resonant tunneling diode, successfully reproducing the negative resistance characteristic of the resonant tunneling diode.

We consider how to solve a class of mathematical program with complementarity constraints (MPCC) where the objective function is a cardinality function. For tackling the cardinality function, we use capped-$ \ell_1 $ function to transform it to a difference-of-convex function, and give a continuous approximation. Then, a proximal penalty method is proposed for finding a weak directional (d)-stationary point of the continuous approximation, which is stronger than Clarke stationary point. The proposed algorithm is a novel combination of penalty method and non-monotonic proximal gradient method. We prove that our algorithm converges to a weak d-stationary point of MPCC under MPCC linear independence constraint qualification. The numerical results demonstrate the effectiveness of the proposed method.

This paper studies a class of nonconvex and nonsmooth two-block optimization problems, where the objective function consists of two nonconvex and nonsmooth separable functions and a smooth coupling function. To address such problems, we propose an improved algorithm based on the inexact inetial proximal gradient method and Nesterov’s acceleration method, namely the proximal alternating linearized minimization algorithm with two distinct extrapolation parameters. Building upon the traditional proximal alternating linearized minimization framework, the algorithm introduces distinct extrapolation parameter sequences to achieve dual extrapolation for one of the variables. Specifically, during each iteration, the algorithm constructs more tractable subproblems by linearizing the coupling function and adding proximal terms based on two different extrapolated points. Theoretically, under certain assumptions, we prove that any limit point of the bounded sequence generated by the algorithm is a critical point of the objective function. Furthermore, when the objective function satisfies the Kurdyka-Lojasiewicz property, we establish the global convergence of the algorithm. Notably, the proposed algorithm allows extrapolation parameters to take negative values, providing new possibilities for enhancing performance. To validate the effectiveness of the proposed algorithm, we apply it to solve the sparse principal component analysis problem. Extensive numerical experiments demonstrate that the proposed algorithm consistently outperforms existing methods in terms of both convergence speed and computational efficiency. Notably, when one of the extrapolation parameters is negative, the algorithm achieves further performance improvements, demonstrating its enhanced flexibility in parameter selection and underscoring its potential for broader applications.

Unit Commitment (UC) is a core challenge in optimizing the operation of power systems. With the rapid expansion of power system scale, Mixed Integer Programming (MIP) methods are facing severe computational challenges. We propose an innovative two-stage column generation algorithm to efficiently solve the large-scale unit commitment. This method is based on Dantzig-Wolfe decomposition and reconstructs the original problem by introducing scheduling strategy variables, achieving effective decoupling between units. For the reconstructed model, we develop a two-stage computational approach: in the first stage, a parallel column generation method is used to solve the Linear Programming (LP) relaxation problem; in the second stage, based on the generated columns, the restricted master problem is solved to obtain high-quality integer feasible solutions. Numerical experiments conducted on large-scale unit commitment test cases involving 1000 to 1500 units show that the proposed method achieves over 2.5×average speedup in solving time compared to the commercial solver CPLEX. The relative optimality gap is around 0.01%, and the solving time is very stable with prominent scalability. This algorithm provides an effective computational tool for practical scheduling optimization of large-scale power systems, with both theoretical and practical significance.

Image restoration is to estimate the clean image from the degraded image, it is a highly ill-posed inverse problem. Regularized methods can mitigate the ill-posedness, which can usually be achieved by minimizing a cost function consisting of a data-fidelity term and a regularization term. In this paper, we consider the multiplicative half-quadratic regularized method for the image restoration problem and employ the Newton method to solve the model. At each Newton iteration step, a linear system of equations with symmetric positive definite coefficient matrix arises. In order to solve the linear system efficiently, we propose a linear Taylor approximation preconditioner for the Schur complement inverse matrix, based on the block triangular decomposition of the coefficient matrix, and the preconditioned conjugate gradient method is applied to solve the linear system. Spectral analysis of the preconditioned matrix reveals that the proposed preconditioner yields a relatively clustered eigenvalue distribution, with some eigenvalues exactly equal to one. Numerical experiments demonstrate that the proposed preconditioner significantly reduces both the number of iterations and the computational time compared to existing methods when solving the system using PCG.

This paper designs a golden ratio type Douglas-Rachford (DR) splitting method for a class of structured inverse variational inequality problems. The proposed method is based on an inexact customized DR splitting method, effectively integrating the golden ratio convex combination coefficients with a strategy for dynamically adjusting the step size parameter. Under general assumption conditions, we prove the global convergence of the new method and further establish the sublinear convergence rate results of the new method. In addition, we apply the new method to solve actual spatial price equilibrium control problems, and the relevant numerical experimental results also verify the effectiveness and superiority of the new method.

The gradient method is one of the simplest and most fundamental computational methods for solving optimization problems. Since all gradient methods find the next iterate along the steepest descent direction, the difference among various gradient methods lies in the choice of step size. The BB step size is one of the most renowned choices for step size in gradient methods. In this paper, by interpreting the BB step as a step size based on onedimensional subspace approximation, we construct new step sizes based on two-dimensional and three-dimensional subspace approximations. These new step sizes possess favorable theoretical properties and are expected to be developed into effective numerical methods.

The compressible miscible displacement in a porous medium is widely used in many computational simulation fields of science and engineering. The mathematical model of the problem is the initial boundary value problem coupled by two parabolic partial differential equations. We apply the mixed finite element method to discrete pressure equation and the characteristic expanded mixed finite element method to discrete concentration equation. Next, we present the proofs of error estimates of the mixed finite element methodcharacteristic expanded mixed finite element method. Finally, the theoretical results are verified through numerical examples.

This paper introduces an extended extragradient algorithm to solve a class of generalized nonsmooth DC problems. We establish the global convergence of the proposed algorithm under appropriate conditions. Our algorithm efficiently exploits the DC structure, and some numerical results demonstrate that it works better than the classical DCA algorithm.

This paper studies a class of structured composite optimization problems, where the objective function is the sum of convex functions, subject to nonconvex equality constraints, with both the objective and constraint functions being continuously differentiable. Such problems have significant applications in electronic design automation, particularly in chiplet placement. Based on the Proximal-Perturbed Lagrangian (P-Lagrangian) method [Oper. Res. Lett. 51, 357–363, 2023], we propose the P-Lagrangian based Alternating Direction Method of Multipliers (PLADMM) for solving nonconvex constrained structured composite optimization problems. Under standard assumptions, we establish the convergence theory for PLADMM, proving that it converges to a KKT point. Numerical experiments demonstrate that PLADMM can effectively solve the MCNC benchmark chiplet placement problem.

The fractional nonlinear Schr$\mathrm{\ddot{o}} $dinger equation is used to describe non-local phenomena in quantum physics and to explore the quantum behavior of remote interactions or time-dependent processes with multiple scales. In this paper, by using the second-order Strang time-splitting Fourier spectral method, the long-time dynamic improved uniform error bound of the spatial fractional nonlinear Schr$\mathrm{\ddot{o}} $dinger equation with small potential energy term is established. Firstly, the equations are semi-discretization by the second-order Strang time splitting method, and then the fully discretization scheme is derived by the spatial Fourier spectral method. Regularity compensation oscillation (RCO) technique is employed to prove the improved uniform error bounds at $O(\varepsilon^2 \tau^2)$ in temporal semi-discretization and $O(h^m+\varepsilon^2\tau^2)$ in full-discretization up to the long-time $T_\varepsilon=T/{\varepsilon^2} (T>0$ fixed), respectively. Finally, some numerical examples are given for convergence test and application analysis, which confirms the error bounds established and verifies the effectiveness of the numerical method presented in this paper.

Randomized double and triple Kaczmarz algorithms are effective stochastic iterative methods for solving the extended normal equations $ A^{\mathsf{T}} A x = A^{\mathsf{T}} b - c $. However, their computational efficiency still has room for improvement. Based on the surrogate hyperplane projection technique, in this paper we propose the residual-based surrogate hyperplane double and triple Kaczmarz (RSHDK/RSHTK) algorithms for solving the extended normal equations. The new algorithms are applicable to arbitrary coefficient matrices $A\in \mathbb{R}^{m\times n}$ and significantly outperform the randomized double and triple Kaczmarz algorithms in terms of both iteration counts and computation time. For both consistent and inconsistent systems, we establish the convergence theories of the new algorithms and demonstrate that their convergence factors are smaller than those of the corresponding standard algorithms. Finally, numerical experiments further validate the effectiveness of the new algorithms.

This work first establishes the well-posedness of solutions for Caputo tempered variableorder fractional stochastic differential equations (CTVO-FSDEs), including the existence, uniqueness and continuous dependence on initial conditions. An Euler-Maruyama scheme is further developed, with a rigorous proof of its strong convergence. Notably, when the fractional order reduces to a constant, our results align with existing theoretical findings in the literature. Finally, numerical simulations are presented to validate the theoretical analysis, demonstrating excellent agreement between computational results and analytical predictions.

For the numerical solution of complex symmetric indefinite linear systems, we apply the minimal residual technique to improve the modified real and imaginary parts (MCRI) iteration method, and propose an iteration method referred to as minimum residual MCRI (MRMCRI) iteration method. Theoretically, the unconditional convergence of the method is proved by spectral theory, and the quasi-optimal iteration parameter is given, which is not affected by the scale or characteristics of the problem. Numerical experiments further validate the efficiency and robustness of the MRMCRI method, especially in solving problems dominated by the imaginary part of the coefficient matrix.

Due to nonlinear schemes, the numerical performance of high-precision schemes is significantly influenced by sensitivity parameters and scale factor values, and they are unable to preserve the steady-state solution of the hyperbolic balance law equation. To address these issues, this paper first introduces a scaling function and constructs a class of scale and translation invariant fifth-order weighted compact nonlinear schemes. Then, through a pre-balancing scheme and source term splitting method, the proposed scheme is applied to the numerical solution of the shallow water equation with bottom topography source terms. Theoretical analysis shows that the new scheme's computational results are scale and translation invariant, and it can accurately preserve the moving water equilibrium of the shallow water equation. Numerical examples verify that the scheme achieves high-order accuracy, possesses good stability and balance, and can precisely capture small disturbances near the steady-state solution.

Stochastic gradient descent algorithm is one of effective algorithms for solving stochastic optimization problems and has received extensive attention from many scholars in recent years. However, selecting an appropriate step size for such algorithms remains a critical area of research. In this paper, we propose a novel adaptive stochastic gradient descent algorithm tailored for stochastic convex optimization problems. Notably, the adaptive step size sequence proposed in this paper is bounded and convergent. Under the assumption of strong convexity, we prove that the sequence generated by the algorithm converges linearly to a neighborhood of the optimal value point. Finally, numerical experiments demonstrate the effectiveness and superiority of the new algorithm.

The Cahn-Hilliard (CH) is an important fourth-order diffusion equation. It was originally used to study the physical phenomenon of mutual diffusion between substances. At the same time, it also has important applications in the evolution of biological populations, riverbed migration, and the field of chemistry. This paper intends to combine the advantages of the low computational cost of the adaptive low-rank splitting method and some fast solving techniques to study the adaptive low-rank approximation algorithm of the CH equation. Firstly, we use the finite difference method to discretize the space of the CH equation and split it into a linear subproblem and a nonlinear subproblem, thereby obtaining a full-rank splitting scheme. Based on this split formulation, we propose an adaptive low-rank solution algorithm for the CH equation in combination with a dynamical low-rank method. Finally, we provide several examples to test the effectiveness of the proposed algorithm in this paper.

A novel hybrid algorithm, termed the finite element method-enhanced neural network (FEM-NN), is proposed for the convection-diffusion equations. In contrast to physicsinformed neural networks (PINNs), in which a large number of collocation points are required for loss minimization, only a small set of mesh nodes is utilized in the proposed method. The finite element stiffness matrix and load vector are incorporated into the loss function of a feedforward neural network, and new FEM-NN algorithms are constructed for both steady-state and unsteady-state problems. For the steady-state case, the hyperbolic tangent function is employed as the activation function, while for the unsteady case, the residuals are derived from the discretized backward Euler scheme, and the loss function is formulated by combining boundary and initial conditions with the residuals, the SiLU and GELU functions are adopted as activation functions, meanwhile, FEM solutions and FEMNN solutions are compared through numerical experiments. It is demonstrated that the FEM-NN algorithm is capable of accurately reconstructing the FEM solution in the steadystate regime, while higher accuracy and stronger generalization capability are exhibited in the unsteady regime. The method proposed in this paper integrates the physical consistency and data-driven adaptability, and it provides a novel numerical approach for convection-diffusion type partial differential equations.

In singularly perturbed ODE-constrained optimization problems, the existence of the nonhomogeneous term increases the difficulty of derivative discretization in the objective function, thereby increasing the difficulty of solving the optimization problem. In this paper, we propose an exponential-type scheme, denoted as ETS-NHE, for discretizing the derivative in the objective function, specifically designed for three classes of singularly perturbed ODEconstrained optimization problems with nonhomogeneous terms. Theoretical analysis and numerical experiments can verify the effectiveness of ETS-NHE and ensure convergence of the optimized numerical solution to the correct solution.

This article deals with asymptotic solution and inversion of fractional order in the fractional SIR epidemic model. The asymptotic solution to the model is derived by the ADM method, and an optimal computable expression of the asymptotic solution is obtained by numerical analysis. Based on the asymptotic solution with an observation of the recovered people at a given time, the inverse order problem is transformed to an algebraic equation, and its uniqueness is obtained by monotonicity of the nonlinear function of the order. Numerical inversions with noisy data are performed to support the uniqueness of the inverse order problem. Parameters identification and data reconstruction are performed by utilizing real data of COVID-19, indicating that fractional-order models have certain advantages in describing complex dynamical epidemiology systems.

A numerical technique for solving fractional-order differential equations based on the seventh-kind Chebyshev wavelets is established. Firstly, the construction of seventh-kind Chebyshev wavelets is based on the corresponding seventh-kind Chebyshev polynomials. The convergence analysis and error estimation of functions expanded in terms of these wavelets are derived by evaluating the coefficients associated with the seventh-kind Chebyshev wavelets. Additionally, we further explore fractional seventh-kind Chebyshev wavelets. Subsequently, we derive the fractional integration formulas for fractional seventh-kind Chebyshev wavelets within the framework of Riemann-Liouville fractional integration using both unit step function and Beta function. By employing these fractional integration formulas alongside an effective collocation method, we discretize the fractional-order differential equation into a system of algebraic equations, from which we obtain the numerical solution to the problem by using Newton iteration method. Several numerical examples demonstrate both the effectiveness and high accuracy of this proposed numerical method.

For the biharmonic wave equation, this paper studies the technique of constructing control fields with active sources by combining potential theory and numerical methods. First, the integral representation for the biharmonic wave equation are derived by using potential theory. Furthermore, by studying the distribution of active sources on the boundary, the control field satisfying the biharmonic wavefield is constructed and applied to the problem of achieving illusion effects. This paper constructs approximate control fields by using numerical methods. In numerical experiments, the goal of wavefield control with a small number of low-order active sources is achieved.

The eigenvalue complementarity problem, as an important branch of linear and nonlinear complementarity problems, holds significant theoretical value and application prospects in fields such as optimization theory, numerical algebra, and engineering mechanics. This paper investigates the generalized quadratic eigenvalue complementarity problem (QEiCP)J defined on the nonnegative cone, establishing sufficient conditions for the existence of its solutions and providing an upper bound estimate for the number of solutions. Within the framework of the semismooth Newton method, three numerical iteration schemes based on the Fischer-Burmeister complementarity function are designed, and the convergence theory for the algorithms is presented. Numerical experiments conducted on various test examples validate the effectiveness of the proposed methods. The results demonstrate that the semismooth Newton algorithm based on the penalized Fischer-Burmeister function exhibits superior performance, particularly for large-scale dense matrix problems, and shows good adaptability and numerical stability in practical tests such as those from the Matrix Market. Furthermore, the algorithm exhibits strong robustness with respect to the choice of the penalty parameter, effectively balancing computational efficiency and accuracy.

This paper presents an efficient numerical method based on the quasi-Newton method and the reproducing kernel method for solving nonlinear singularly perturbed delay differential equations. Firstly, the original problem is decomposed into a regular region problem and a boundary layer problem. Secondly, the regular region problem is solved using the asymptotic expansion method; for the boundary layer problem, the problem interval is first mapped to [0, 1] by means of the variable stretching method. Thirdly, the nonlinear singularly perturbed delay differential equation is converted into a series of linear singularly perturbed delay differential equations via the quasi-Newton method. Lastly, solved using the reproducing kernel method based on the collocation method. Meanwhile, this paper discusses the convergence and stability of the proposed method, and compares the numerical results with those of other numerical methods. The results show that the proposed method can not only provide more accurate approximate solutions but also achieve higher convergence orders.