This paper provides a brief exploration of the basic principles and applications of mathematical formalization, with a focus on the formal language Lean and its application in mathematical optimization. We first review the development background of mathematical formalization, explain the construction principles of the Lean language and its correctness guarantee mechanisms, and introduce the role of the theorem library Mathlib4 in Lean. By comparing natural language with formalized expressions, we illustrate the advantages of using formalization to verify mathematics, emphasizing its important role in the accurate verification of mathematical theories. In the field of mathematical optimization, this paper discusses the current progress of formalizing mathematical optimization theory, using formalized examples of classic theorems such as the quadratic upper bound lemma, and further highlights the characteristics and advantages of formalized mathematics. Additionally, we explore the formalization goals in operations research and investigate the technologies of automated formalization and automated theorem proving, analyzing the potential and challenges of automation tools in the mathematical formalization process. Finally, we summarize the current state of research in mathematical formalization, give suggestions for further advancing the field, and discuss the significant role of formalization in the development of applied mathematical theory.

This paper focuses on intrinsic finite elements, exploring their applications in numerical partial differential equations and their potential connections to discrete differential geometry and topological data analysis. Driven by the numerical discretization that preserves the mathematical and physical structures of continuous problems, the paper briefly reviews the development of Finite Element Exterior Calculus (FEEC). Through the canonical discretization of the classical de Rham complex and the BGG complex, an extended finite element periodic table for form-valued differential forms is proposed, covering Whitney forms, distributional finite elements, Regge finite elements, and Hessian and div div complexes, providing a unified tool for the numerical solution of tensor problems. The paper further analyzes the potential of intrinsic finite elements in interdisciplinary applications, including Riemann-Cartan geometry, generalized continua, and gravitational wave computations.

In this paper, we consider a special kind of matrix combined by a skew-symmetric tridiagonal matrix and a periodic skew-symmetric tridiagonal matrix. It is referred to as a generalized periodic skew-symmetric tridiagonal matrix. An inverse eigenvalue problem for constructing thus a matrix is studied, that is, the matrix is reconstructed from three prescribed balanced sets and a positive number. Firstly, the matrix can be transformed into a generalized periodic symmetric tridiagonal matrix by unitary similarity, and then the relationships between the eigenvalues of this matrix and those of its leading and trailing principal submatrices are analyzed. Two aspects are respectively discussed about whether the spectra of these two principal submatrices have a disjoint set and the parity of the order of the above trailing principal submatrix. Then, the necessary and sufficient conditions for the existence of solutions to the inverse eigenvalue problems in various cases are provided, and the largest number of solutions and reconstruction algorithms are determined. Finally, the effectiveness of the algorithm is validated by two numerical examples.

A compact difference scheme for the two-dimensional Sobolev equation with fourth-order accuracy in space is derived, and the convergence of the compact difference scheme is proved. The difference scheme is rewritten in vector form, a reduced-order high-order compact difference scheme is constructed using the Proper Orthogonal Decomposition (POD) method, and error estimate for the approximate solution is also provided. Numerical example is presented to calculate the numerical error, spatial convergence order and temporal convergence order of both the compact difference scheme and the reduced-order compact difference scheme, verifying that the experimental results are consistent with the theoretical analysis. Furthermore, by comparing the CPU computation time before and after dimension reduction, the superiority of applying the POD method for dimension reduction in compact schemes is demonstrated.

This thesis proposes a class of space-time mixed finite element method for time fractional diffusion equations involving Riemann-Liouville derivatives of order α ∈ (0, 1). The spatial discretization employs m(m ≥ 0) order Raviart-Thomas (RT) finite elements, while the temporal discretization uses piecewise r(r ≥ 0) degree discontinuous Galerkin (DG) finite elements. To address the solution singularity near t = 0, graded meshes are used in the time direction. The well-posedness of the fully discrete scheme is analyzed. Error estimates are derived in two cases: r = 0, i.e. the temporal discretization uses the piecewise constant DG scheme, and r = 1, i.e. the temporal discretization uses the piecewise linear DG scheme. Numerical experiments are provided.

Single-phase multi-component flow problems are widely existed in oil and gas reservoirs, groundwater pollution, etc., and numerical simulation is of great significance. In this paper, for the traditional IMPEC (Implicit Pressure-Explicit Concentration) method, which does not follow the problem of mass conservation for all components, a physically preserving IMPEC method with mass conservation for each component is used for the time discretization, and the upwind block-centered finite-difference method is used for the spatial discretization of the pressure equation, Darcy’s equation, and the component equations. In this paper, the physics-preserving mass conservation properties of all the constructed components as well as the molar concentrations are rigorously proved under reasonable condition, respectively. Finally, numerical simulations are carried out to demonstrate the validity of the proposed algorithm by means of a single-phase multi-component flow problem.

Vertical nonlinear complementarity problems have wide applications. The design of numerical methods for solving vertical nonlinear complementarity problems has been a hot topic among researchers in recent years. In this paper, a modulus-based synchronous multisplitting iteration method is established by matrix multisplitting and equivalent modulus equation for vertical nonlinear complementarity problem. Some convergence conditions of the proposed method are presented under H-matrix assumption. The convergence domain of the relaxation parameters of the accelerated overrelaxation iteration is obtained. By OpenACC framework, numerical tests are given to show the high parallel computational efficiency of the proposed method.

In recent years, the exponential scalar auxiliary variable (E-SAV) method is very popular for approximating the phase field models. This method is very effective and does not have to assume that the nonlinear function is bounded from below. In the current study, an E-SAV method is proposed for the numerical investigation of a nonlinear wave equation. This scheme with first order accuracy is obtained by using two variables and backward Euler formula. The error of the approximation of the proposed scheme for the nonlinear wave equation is analyzed. To verify the theoretical results, and the effectiveness of the method with other important methods, two numerical experiments are carried out.

For the composite optimization problems widely encountered in machine learning and image processing, the Primal-Dual Fixed Point(PDFP) algorithm is an efficient algorithm. In this paper, we propose an Accelerated Primal-Dual Fixed Point algorithm(APDFP) by combining the PDFP and Nesterov acceleration technique. APDFP can encompass the Accelerated Proximal Alternating Prediction Corrector Algorithm(APAPC) as a special case. Under appropriate conditions, we prove that APDFP has a non-ergodic convergence rate of O(1/N). Furthermore, numerical experiments on the fused lasso problem and computed tomography(CT) image reconstruction verify the effectiveness of the proposed algorithm.

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.

Multidimensional scaling (MDS) is a technique used in multidimensional data analysis that depicts the similarities or relationships between observed objects as distances between points in a lower-dimensional space. By representing high-dimensional data within a lowdimensional framework, MDS preserves the relative distances between data points. This study focuses on developing an efficient numerical algorithm for a specific type of individual differences scaling model, known as O-INDSCAL, within symmetric multidimensional scaling, which accounts for individual differences among observed objects. Initially, leveraging the concept of the alternating least squares algorithm, the multivariable constrained matrix optimization model associated with the O-INDSCAL model is transformed into a fixed-point iteration problem. By thoroughly examining the acceleration principles and implementation processes of various polynomial extrapolation and Anderson acceleration methods in vector sequence acceleration, we have designed a matrix sequence acceleration algorithm tailored to this problem model. Numerical experiments indicate that the proposed acceleration algorithms significantly enhance the convergence speed of sequences generated by fixed-point iterations. Furthermore, when compared with existing continuous-time projection gradient flow algorithms and the first-order and second-order Riemannian algorithms available in the Manopt toolbox for manifold optimization, the proposed algorithms demonstrate notable improvements in iterative efficiency.

In this paper, the Hermite polynomials are used as the hidden layer of the neural network, the initial weights of the Hermite neural network are optimized using genetic algorithm, and the inverse of the error function between the actual output and the desired output of the Hermite neural network optimized by the genetic algorithm is also chosen as the fitness function of the genetic algorithm, to construct a new genetic algorithm-optimized Hermite neural network to solve the Caputo fractal-fractional order Bagley-Torvik differential equation numerical method. The general form of the numerical solution of the Caputo fractal-fractional order Bagley-Torvik differential equation is given in conjunction with the Taylor’s formula at multiple points, and the absolute error and convergence of the algorithm are theoretically investigated. Comparison with existing numerical methods is made and the results show the effectiveness and feasibility of the method in this paper.

In this paper, we propose an adaptive step-size rule three-operator splitting algorithm with an inertia term to solve the nonsmooth DC programming problem. Under suitable assumptions, we prove that the sequence generated by the algorithm converges to a critical point of the problem. Finally, we apply the algorithm for solving the sparse recovery problem, and numerical experiments show the effectiveness and superiority of the new algorithm.

The Cahn-Hilliard-Hele-Shaw (CHHS) model, a Cahn-Hilliard equation coupled with the Darcy equation. It has been widely used to simulate two-phase flow in porous media and tumor growth. For the CHHS model, two energy dissipation schemes based on the Lagrange multiplier method are proposed in this paper. The Backward-Euler and Crank-Nicolson schemes are used in the time direction, and the Fourier spectral method is used in the space direction. Theoretical analysis shows that resulted schemes maintain the original energy dissipation. Finally, various numerical simulations are performed to validate the accuracy and efficiency of the proposed schemes.

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.

An accelerated surrogate hyperplane Kaczmarz method based on two-dimensional search is proposed, which generates a new hyperplane by searching for the optimal weighted vector in a two-dimensional subspace. Theoretical analysis provides the convergence rate of the new method. Numerical experiments demonstrate that the new proposed Kaczmarz method is convergent and outperforms the original method in terms of both the number of iterations and computational time.

In this paper, we propose a nonconforming virtual element method for nonlinear Sobolev equation on polygonal meshes by applying backward Euler scheme. In order to establish the convergence of the method, we construct a novel projection operator based on two discrete trilinear forms and give the corresponding error estimates in the L² norm and broken H¹ semi-norm. Leveraging this projection operator, we prove the optimal convergence of the nonconforming virtual element method in the fully discrete formulation. Finally, several numerical experiments on various polygonal meshes are conducted to confirm the accuracy and optimal convergence of the proposed method.

For solving the large-scale overdetermined linear least-squares problem with nonnegative constraints, we propose two constrained Gauss-Seidel methods, namely constrained greedy randomized Gauss-Seidel method based on greedy criterion and constrained random sampling Gauss-Seidel method based on random sampling. We also build the convergence theories and implement some numerical experiments in this paper. The numerical results demonstrate that the proposed methods significantly outperform the existing methods.

So far, all of the tamed methods are implicit for highly nonlinear stochastic differential equations. For the stochastic differential equation that its drift coefficients can be divided into a linear term and a highly nonlinear term, we present the implicit semi-tamed Euler methods which only require similar computational effort as an explicit method. Under the Khasminskii-type and polynomial growth conditions, the convergence of the presented method is proved. In addition, we study the stability of the method and prove that it can preserve the stability of analysis solutions of an stable system. Finally, we give some numerical examples to verify the theoretical results and show that the stability of the presented methods is better than that of some explicit tamed methods.

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.

A two-level algorithm based on the finite element discretization is proposed for a class of time-dependent Poisson-Nernst-Planck (PNP) equations. This algorithm decouples the PNP equations through a linear finite element approximation, followed by solving the decoupled equations in a quadratic finite element space. Compared with the classical Gummel algorithm based on the finite element discretization, this method accelerates the solution process. The H¹ norm error estimates are established based on the L² norm error estimates of the twolevel finite element solution. Numerical experiment confirms the correctness of the theoretical results and shows the efficiency of the two-level algorithm.

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.

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.

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.

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.

In this paper, a class of arbitrary high-order energy-preserving schemes for the KleinGordon equation is developed. By introducing the quadratic auxiliary variable, the Hamiltonian energy is transformed into the quadratic form, i.e., the energy conservation law is transformed into the quadratic invariants. Then, the original system is reformulated into a new system with quadratic invariants simultaneously. The fully discrete scheme is derived using the Fourier pseudo-spectral method and the symplectic Runge-Kutta methods. The proposed schemes achieve arbitrarily high-order convergence in time, spectral accuracy in space, and can preserve the original energy conservation precisely. The numerical results further substantiate the effectiveness and high-precision convergence of the proposed schemes.

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 mainly discusses the oscillation of numerical solutions for a class of neutral delay differential equations with multiple delays. The original equation is discretized by using the linear θ-method to obtain the corresponding difference equation. By discussing the properties of the solutions of the difference equation, the oscillation properties of the numerical solutions for the original equation are transformed into that of a non-neutral difference equation. According to the relationship between the oscillation of the difference equation and the characteristic roots of the characteristic equation, the oscillation of the numerical solutions is discussed for $0\leq \theta \leq \frac{1}{2}$ and $\frac{1}{2}<\theta \leq1$， respectively. Meanwhile, the properties of non-oscillatory numerical solutions are also investigated. Finally, numerical examples are given to illustrate the conclusions.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.