Search Results (63)

We present an advanced restrictively preconditioned biconjugate gradient-stabilized (RPBiCGSTAB) algorithm specifically designed to improve the convergence speed of Krylov subspace methods for nonlinear systems characterized by a structured 5-by-5 block configuration. This configuration frequently arises from cell-centered finite difference discretizations employed in solving image deblurring problems governed by mean curvature dynamics. The RPBiCGSTAB method is crafted to exploit this block structure, thereby optimizing both computational efficiency and convergence behavior in complex image processing tasks. Analyzing the spectral characteristics of preconditioned matrices often reveals a beneficial distribution of eigenvalues, which plays a critical role in accelerating the convergence of the RPBiCGSTAB algorithm. Furthermore, our numerical experiments validate the computational efficiency and practical applicability of the method in addressing nonlinear systems commonly encountered in image deblurring. Our analysis also extends to the spectral properties of the preconditioned matrices, noting a pronounced clustering of eigenvalues around 1, which contributes to enhanced stability and convergence performance.Through numerical simulations that focus on mean curvature-driven image deblurring, we highlight the superior performance of the RPBiCGSTAB method in comparison to other techniques in this specialized field. Full article

Due to the coupling of DC and AC components, the ion flow field of HVDC and HVAC transmission lines in the same corridor or even the same tower is complex and time-dependent. In order to effectively analyze the ground-level electric field of hybrid transmission lines, the Krylov subspace methods with pre-conditioning treatment are used to solve the discretization equations. By optimizing the coefficient matrix, the calculation efficiency of the iterative process of the electric field in the time domain is greatly increased. Based on the limit of electric field, radio interference and audible noise applied in China, the main factor influencing the design of hybrid transmission lines is determined in terms of electromagnetic environment. After the ground-level electric field of transmission lines with different configurations is analyzed, the minimum height and corridor width of double-circuit 500 kV HVAC lines and one-circuit ±800 kV HVDC lines in the same corridor are obtained. The research provides valuable practical recommendations for optimal tower configurations, minimum heights, and corridor widths under various electromagnetic constraints. Full article

As semiconductor technologies continue to scale aggressively, electromigration (EM) has become critical in modern VLSI design. Since traditional EM assessment methods fail to accurately capture the complex behavior of multi-segment interconnects, recent physics-based models have been developed to provide a more accurate representation of EM-induced stress evolution. However, numerical methods for these models result in large-scale systems, which are computationally expensive and impractical for complex interconnect structures. Model order reduction (MOR) has emerged as a key enabler for scalable EM analysis, with moment-matching (MM) techniques offering a favorable balance between efficiency and accuracy. However, conventional Krylov-based approaches often suffer from limited frequency resolution or high computational cost. Although the extended Krylov subspace (EKS) improves frequency coverage, its symmetric structure introduces significant overhead in large-scale scenarios. This work introduces a novel MOR technique based on the asymmetric extended Krylov subspace (AEKS), which improves upon the conventional EKS by incorporating a sparsity-aware and computationally efficient projection strategy. The proposed AEKS-based moment-matching framework dynamically adapts the Krylov subspace construction according to matrix sparsity, significantly reducing runtime without sacrificing accuracy. Experimental evaluation on IBM power grid benchmarks demonstrates the high accuracy of our method in both frequency-domain and transient EM simulations. The proposed approach delivers substantial runtime improvements of up to 15× over full-order simulations and 100× over COMSOL, while maintaining relative errors below 0.5%, even under time-varying current inputs. Full article

Model order reduction (MOR) is crucial for efficiently simulating large-scale RLCk models extracted from modern integrated circuits. Among MOR methods, balanced truncation offers strong theoretical error bounds but is computationally intensive and does not preserve passivity. In contrast, moment matching (MM) techniques are widely adopted in industrial tools due to their computational efficiency and ability to preserve passivity in RLCk models. Typically, MM approaches based on the rational Krylov subspace (RKS) are employed to produce reduced-order models (ROMs). However, the quality of the reduction is influenced by the selection of the number of moments and expansion points, which can be challenging to determine. This underlines the need for advanced strategies and reliable convergence criteria to adaptively control the reduction process and ensure accurate ROMs. This article introduces a frequency-aware multi-point MM (MPMM) method that adaptively constructs an RKS by closely monitoring the ROM transfer function. The proposed approach features automatic expansion point selection, local and global convergence criteria, and efficient implementation techniques. Compared to an established MM technique, MPMM achieves up to 16.3× smaller ROMs for the same accuracy, over 99.18% reduction in large-scale benchmarks, and up to 4× faster runtime. These advantages establish MPMM as a strong candidate for integration into industrial parasitic extraction tools. Full article

The most popular iterative methods for solving nonsymmetric linear systems are Krylov methods. Recently, an optimal Quasi-ORthogonal (Q-OR) method was introduced, which yields the same residual norms as the Generalized Minimum Residual (GMRES) method, provided GMRES is not stagnating. In this paper, we study how to introduce matrix sketching in this algorithm. It allows us to reduce the dimension of the problem in one of the main steps of the algorithm. Full article

Tempered fractional diffusion equations constitute a critical class of partial differential equations with broad applications across multiple physical domains. In this paper, the Crank–Nicolson method and the tempered weighted and shifted Grünwald formula are used to discretize the tempered fractional diffusion equations. The discretized system has the structure of the sum of the identity matrix and a diagonal matrix multiplied by a symmetric positive definite (SPD) Toeplitz matrix. For the discretized system, we propose a structure approximation-based preconditioning method. The structure approximation lies in two aspects: the inverse approximation based on the row-by-row strategy and the SPD Toeplitz approximation by the $\tau$ matrix. The proposed preconditioning method can be efficiently implemented using the discrete sine transform (DST). In spectral analysis, it is found that the eigenvalues of the preconditioned coefficient matrix are clustered around 1, ensuring fast convergence of Krylov subspace methods with the new preconditioner. Numerical experiments demonstrate the effectiveness of the proposed preconditioner. Full article

For topology optimization problems under harmonic excitation in a frequency band, a large number of displacement and adjoint displacement vectors for different frequencies need to be computed. This leads to an unbearable computational cost, especially for large-scale problems. An effective approach, the Second-Order Arnoldi (SOAR) method, effectively solves the response and adjoint equations by projecting the original model to a reduced order model. The SOAR method generalizes the well-known Krylov subspace in a specified frequency point and can give accurate solutions for the frequencies near the specified point by using only a few basis vectors. However, for a wide frequency band, more expansion points are needed to obtain the required accuracy. This brings up the question of how many points are needed for an arbitrary frequency band. The traditional reduced order method improves the accuracy by uniformly increasing the expansion points. However, this leads to the redundancy of expansion points, as some frequency bands require more expansion points while others only need a few. In this paper, a bisection-based adaptive SOAR method (ASOAR), in which the points are added adaptively based on a local error estimation function, is developed to solve this problem. In this way, the optimal number and position of expansion points are adaptively determined, which avoids the insufficient efficiency or accuracy caused by too many or too few points in the traditional strategy where the expansion points are uniformly distributed. Compared to the SOAR, the ASOAR can deal with wide low/mid-frequency bands both for response and adjoint equations with high precision and efficiency. Numerical examples show the validation and effectiveness of the proposed method. Full article

After nearly two decades of development, transient electromagnetic (TEM) 3D forward modeling technology has significantly improved both numerical precision and computational efficiency, primarily through advancements in mesh generation and the optimization of linear equation solvers. However, the dominant approach still relies on direct solvers, which require substantial memory and complicate the modeling of electromagnetic responses in large-scale models. This paper proposes a new method for solving large-scale TEM responses, building on previous studies. The TEM response is expressed as a matrix exponential function with an analytic initial field for a step-off source, which can be efficiently solved using the Shift-and-Invert Krylov (SAI-Krylov) subspace method. The Arnoldi algorithm is used to construct the orthogonal basis for the Krylov subspace, and the preconditioned conjugate gradient (PCG) method is applied to solve large-scale linear equations. The paper further explores how dividing the off-time and optimizing parameters for each time interval can enhance computational efficiency. The numerical results show that this parameter optimization strategy reduces the iteration count of the PCG method, improving efficiency by a factor of 5 compared to conventional iterative methods. Additionally, the proposed method outperforms direct solvers for large-scale model calculations. Conventional approaches require numerous matrix factorizations and thousands of back-substitutions, whereas the proposed method only solves about 300 linear equations. The accuracy of the approach is validated using 1D and 3D models, and the propagation characteristics of the TEM field are studied in large-scale models. Full article

The thermal runaway propagation (TRP) model of energy storage batteries can provide solutions for the safety protection of energy storage systems. Traditional TRP models are solved using the finite element method, which can significantly consume computational resources and time due to the large number of elements and nodes involved. To ensure solution accuracy and improve computational efficiency, this paper transforms the heat transfer problem in finite element calculations into a state-space equation form based on the reduced-order theory of linear time-invariant (LTI) systems; a simplified method is proposed to solve the heat flow changes in the battery TRP process, which is simple, stable, and computationally efficient. This study focuses on a four-cell 100 Ah lithium iron phosphate battery module, and module experiments are conducted to analyze the TRP characteristics of the battery. A reduced-order model (ROM) of module TRP is established based on the Arnoldi method for Krylov subspace, and a comparison of simulation efficiency is conducted with the finite element model (FEM). Finally, energy flow calculations are performed based on experimental and simulation data to obtain the energy flow rule during TRP process. The results show that the ROM achieves good accuracy with critical feature errors within 10%. Compared to the FEM, the simulation duration is reduced by 40%. The model can greatly improve the calculation efficiency while predicting the three-dimensional temperature distribution of the battery. This work facilitates the efficient computation of TRP simulations for energy storage batteries and the design of safety protection for energy storage battery systems. Full article

This paper describes a Krylov subspace iterative method designed for solving linear systems of equations with a large, symmetric, nonsingular, and indefinite matrix. This method is tailored to enable the evaluation of error estimates for the computed iterates. The availability of error estimates makes it possible to terminate the iterative process when the estimated error is smaller than a user-specified tolerance. The error estimates are calculated by leveraging the relationship between the iterates and Gauss-type quadrature rules. Computed examples illustrate the performance of the iterative method and the error estimates. Full article

Least-squares reverse time migration (LSRTM) is an advanced seismic imaging technique that reconstructs subsurface models by minimizing the residuals between simulated and observed data. Mathematically, the LSRTM inversion of the sub-surface reflectivity is a large-scale, highly ill-posed sparse inverse problem, where conventional inversion methods typically lead to poor imaging quality. In this study, we propose a regularized LSRTM method based on the flexible Krylov subspace inversion framework. Through the strategy of the Krylov subspace projection, a basis set for the projection solution is generated, and then the inversion of a large ill-posed problem is expressed as the small matrix optimization problem. With flexible preconditioning, the proposed method could solve the sparse regularization LSRTM, like with the Tikhonov regularization style. Sparse penalization solution is implemented by decomposing it into a set of Tikhonov penalization problems with iterative reweighted norm, and then the flexible Golub–Kahan process is employed to solve the regularization problem in a low-dimensional subspace, thereby finally obtaining a sparse projection solution. Numerical tests on the Valley model and the Salt model validate that the LSRTM based on Krylov subspace method can effectively address the sparse inversion problem of subsurface reflectivity and produce higher-quality imaging results. Full article

For solving the continuous Sylvester equation, a class of Hermitian and skew-Hermitian based multiplicative splitting iteration methods is presented. We consider two symmetric positive definite splittings for each coefficient matrix of the continuous Sylvester equations, and it can be equivalently written as two multiplicative splitting matrix equations. When both coefficient matrices in the continuous Sylvester equation are (non-symmetric) positive semi-definite, and at least one of them is positive definite, we can choose Hermitian and skew-Hermitian (HS) splittings of matrices A and B in the first equation, and the splitting of the Jacobi iterations for matrices A and B in the second equation in the multiplicative splitting iteration method. Convergence conditions of this method are studied in depth, and numerical experiments show the efficiency of this method. Moreover, by numerical computation, we show that multiplicative splitting can be used as a splitting preconditioner and induce accurate, robust and effective preconditioned Krylov subspace iteration methods for solving the continuous Sylvester equation. Full article

In structural stochastic dynamic analysis, the consideration of the randomness in the physical parameters of the structure necessitates the establishment of numerous stochastic finite element models and the subsequent computation of their corresponding vibration modes. When the complete analysis is applied to calculate the vibration modes for each sample of the stochastic finite element model, a substantial computational expense is incurred. To enhance computational efficiency, this work presents an extended subspace iteration method aimed at rapidly determining the vibration modal parameters of statically indeterminate structures. The essence of this proposed method revolves around efficiently constructing reduced basis vectors during the subspace iteration process, utilizing flexibility disassembly perturbation and the Krylov subspace. This extended subspace iteration method proves particularly advantageous for the modal analysis of finite element models that incorporate a multitude of random variables. The proposed modal random analysis method has been validated using both a truss structure and a beam structure. The results demonstrate that the proposed method achieves substantial savings in computational time. Specifically, for the truss structure, the calculation time of the proposed method is approximately 1.2% and 65% of that required by the comprehensive analysis method and the combined approximation method, respectively. For the beam structure, on average, the computational time of the proposed method is roughly 2.1% of a full analysis and approximately 48.2% of the Ritz vector method’s time requirement. Compared to existing stochastic modal analysis algorithms, the proposed method offers improved computational accuracy and efficiency, particularly in scenarios involving high-discreteness random analyses. Full article

A reduced-dimension robust Capon beamforming method using Krylov subspace techniques (RDRCB) is a diagonal loading algorithm with low complexity, fast convergence and strong anti-interference ability. The diagonal loading level of RDRCB is known to become invalid if the initial value of the Newton iteration method is incorrect and the Hessel matrix is non-positive definite. To improve the robustness of RDRCB, an improved RDRCB (IRDRCB) was proposed in this study. We analyzed the variation in the loading factor with the eigenvalues of the reduced-dimensional covariance matrix and derived the upper and lower boundaries of the diagonal loading level; the diagonal loading level of the IRDRCB was kept within the bounds mentioned above. The computer simulation results show that the IRDRCB can effectively solve the problems of a sharp decline in the signal-to-noise ratio gain and an invalid diagonal loading level. The experimental results demonstrate that the interference noise of the IRDRCB is 3~5 dB higher than that of conventional adaptive beamforming, and the computational complexity is reduced by 15% to 20% compared with that of the RCB method. Full article

The study of matrix functions is highly significant and has important applications in control theory, quantum mechanics, signal processing, and machine learning. Previous work has mainly focused on how to use the Krylov-type method to efficiently calculate matrix functions $f\left(A\right)$$\mathit{\beta }$ and ${\mathit{\beta }}^{T}f\left(A\right)$$\mathit{\beta }$ when A is symmetric. In this paper, we mainly illustrate the convergence using the polynomial approximation theory for the case where A is symmetric positive definite. Numerical results illustrate the effectiveness of our theoretical results. Full article