In this paper, we combine a new extrapolation formula with a cascadic multigrid method to solve variational inequality problems with nonlinear source terms. We construct a new extrapolation modulus-based cascadic multigrid method and a new extrapolation modulus-based economical cascadic multigrid method. Based on a new formula for controlling the number of iterations, the convergence is analyzed. Numerical experiments demonstrate the effectiveness of the proposed methods.

This paper introduces two algorithms based on the conjugate residual method and variable $s$-step for solving linear systems with non-square coefficient matrices. The introduced algorithms demonstrate their effectiveness through numerical comparisons with other methods, specifically CGNE and CGNR, to solve non-square linear systems. The approach aims to enhance the efficiency of solving problems related to control systems and imaging underground layers, particularly in the context of seismic tomography.

The model order reduction and data assimilation techniques have been extensively applied to construct surrogate models as core models for digital twins in different industries. However, in most applications, the online phases rely on a single offline basis construction within the parameter domain, which is computationally expensive and often overlooks local parameter variations due to anisotropic behavior. Consequently, this poses challenges for real-time monitoring where parameters vary dynamically. To address this, we introduce a relative deviation score of reduced basis coefficients, represented by $\mathcal{W},$ to assess the reconstruction capability of basis functions under current parameters. Additionally, we provide a quantitative threshold $\overline{\mathcal{W}}$ for $\mathcal{W}.$ Using the reconstruction of a two-dimensional reactor core field distribution as a case study, we demonstrate that if $\mathcal{W}$ remains below the threshold $\overline{\mathcal{W}}=1,$ the average reconstruction error can be maintained within the same order of magnitude as the fundamental error limit $ε (κ ≤ 10).$ Furthermore, we compare $\mathcal{W}$ with the traditional one-time reduced basis selection method and the metric $C_{R2F}$ [F. Bai and Y. Wang, SN Applied Sciences, 2 (2020), p. 2165], confirming the lower computational cost and enhanced robustness of our proposed algorithm.

This paper is concerned about the implicit-explicit (IMEX) methods for a class of dissipative wave systems with time-varying velocity feedbacks and nonlinear potential energies, equipped with different boundary conditions. Firstly, we approximate the problems by using the Hochbruck-Leibold IMEX method, which is a second-order scheme for the problems when the damping terms are time-independent. However, rigors analysis shows that the error rate declines from second to first order due to the nonautonomous dampings. To recover the convergence order, we propose a modified IMEX scheme and apply it to the nonautonomous wave equations with a kinetic boundary condition. Our numerical experiments demonstrate that the modified scheme can not only achieve second-order accuracy but also improve the computational efficiency.

In this paper, we propose a generalized eigenproblem algorithm, called spectral shifted inverse power method (SIPM) for computing tensor generalized eigenvalue. The SIPM method is developed to overcome the limitations of existing approaches, such as the generalized eigenproblem adaptive power, tensor Noda iteration, modified tensor Noda iteration, and generalized Newton-Noda iteration. These methods often suffer from slow convergence, sensitivity to initial conditions, and computational inefficiency. First, we express SIPM as a fixed point iteration form and establish the connection between the fixed points and generalized eigenvectors of symmetric tensors. Moreover, we introduce a shift power method that further enhances SIPM. Next, we provide a technique for selecting the optimal starting point. Finally, we present numerical results that confirm the effectiveness of our method in solving the tensor generalized eigenvalue problem more efficiently than other methods.

In this paper, we propose and analyze a second-order accurate numerical scheme for the Poisson-Nernst-Planck equations. The proposed scheme combines a novel second-order temporal discretization with the centered finite difference method in space. By using a Taylor expansion for the logarithmic term in the chemical potential, this second-order accurate numerical scheme is able to preserve original energy dissipation. Based on the gradient flow formulation, the resulting scheme guarantees several crucial physical properties: mass conservation, positivity of ionic concentration, preservation of original energy dissipation, and steady states preservation. Remarkably, these properties are ensured without any restriction on the time step size. Furthermore, an optimal rate convergence estimate is provided for the proposed numerical scheme. Due to the non-constant mobility and the nonlinear and singular properties of the logarithm terms, a higher-order asymptotic expansion and a combination of rough and refined error estimation techniques are introduced to accomplish this analysis. A few numerical tests are provided to validate our theoretical claims.

This paper is concerned with devising an efficient numerical method for the piezoelectric equations in an unbounded domain, which plays a fundamental role in design and analysis of microacoustic devices with piezoelectric substrate. We make use of the perfectly matched layer method to transform the underlying problem as a surrogate in a bounded domain, which is further solved by a sparse wavelet element method. The latter method can be viewed as a combination of a wavelet element method and a sparse grid method. The numerical results are performed to show the proposed method is very efficient and outperforms the usual finite element method. It can be naturally extended to two/three dimensional problems in an unbounded domain whose boundary consists of line segments or rectangles parallel to coordinate lines or planes.

We present a modified version of the PRESB preconditioner for two-by-two block systems of linear equations with the coefficient matrix

where $F∈\mathbb{C}^{n×n}$ is Hermitian positive definite and $G ∈\mathbb{C}^{n×n}$ is positive semidefinite. Spectral analysis of the preconditioned matrix is analyzed. In each iteration of a Krylov subspace method, like GMRES, for solving the preconditioned system in conjunction with proposed preconditioner two subsystems with Hermitian positive definite coefficient matrix should be solved which can be accomplished exactly using the Cholesky factorization or inexactly utilizing the conjugate gradient method. Application of the proposed preconditioner to the systems arising from finite element discretization of PDE-constrained optimization problems is presented. Numerical results are given to demonstrate the efficiency of the preconditioner. Our theoretical and numerical results show that the proposed preconditioner is efficient when the norm of the skew-Hermitian part of $G$ is small.

An extrapolation method is proposed for the numerical computation of hypersingular integrals with oscillatory kernels. The oscillatory integral is reformulated as the weighted integral of a Hadamard finite part, which is subsequently approximated using the weighted trapezoidal rule. The asymptotic expansion of the error function is derived, and both the convergence order and the posterior error of the algorithm are analyzed. Numerical examples verify the theoretical results and demonstrate the validity of the proposed method.

We propose a novel diffusion-based generative model for solving linear stochastic diffusion equations, while enabling uncertainty quantification (UQ). By embedding the governing physical laws into the generative process, our approach establishes a mapping between control parameters and solutions across the latent space of the diffusion model. This ensures that the generated solutions satisfy the underlying physical constraints. Additionally, our method overcomes the limitation of conventional diffusion models, which struggle to generate accurate solutions for new control terms, and achieves superior accuracy compared to traditional data-driven operator learning techniques. Furthermore, by sampling different noise realizations and analyzing variations in the generated solutions, we efficiently capture solution diversity, enabling simultaneous prediction and comprehensive UQ. Experimental results demonstrate that our method outperforms deep operator networks and variational inference-based deep operator networks in both accuracy and confidence estimation.

We found that no convergence to the correct solution can happen when a popular method is applied to discretize the derivative appearing in the objective function for optimization problems with singularly perturbed ODE constraints. The non-convergence mentioned above can occur even if the error bound of the numerical solution of the state equation has nothing to do with the small parameter. We disclose that the underlying reason for non-convergence to the correct solution is an inaccurate derivative calculation in the objective function for a model problem, which is solvable mathematically. To ensure correct convergence regardless of the small parameter, we propose an entirely exponential-type scheme for solving the optimization problem, in which an exponential-type scheme is used for the derivative in the objective function, together with an exponential-type finite difference scheme for the state equation. Both theoretical analysis and numerical experiments can verify the correct convergence of E2S in solving the singularly perturbed equation-constrained optimization problem.