空间分数阶扩散方程能有效地描述众多科学领域中的反常扩散现象。而大多数FDE难以得到解析解,需通过建立离散格式以获得高精度的数值解。数值求解FDE通常归结为线性方程组的求解,而预处理技术则是加速迭代求解的关键。近年来,基于不同...空间分数阶扩散方程能有效地描述众多科学领域中的反常扩散现象。而大多数FDE难以得到解析解,需通过建立离散格式以获得高精度的数值解。数值求解FDE通常归结为线性方程组的求解,而预处理技术则是加速迭代求解的关键。近年来,基于不同离散格式下系数矩阵的结构和性质,学者们研究了高效的预处理方法,显著地降低了计算成本。针对数值求解空间分数阶扩散方程问题,本文整理和分析了方程不同形式下的离散情形和预处理方法,并为预处理进一步的研究提供思路参考。Spatial fractional diffusion equations effectively describe anomalous diffusion phenomena in various scientific fields. However, most FDEs are difficult to solve analytically, necessitating the establishment of discrete schemes to obtain high-precision numerical solutions. Numerical solutions of FDEs typically reduce to solving linear systems, where preconditioning techniques are crucial for accelerating iterative solvers. In recent years, scholars have investigated efficient preconditioning methods based on the structure and properties of coefficient matrices under different discretization schemes, significantly reducing computational costs. For the numerical solution of spatial fractional diffusion equations, this paper organizes and analyzes the discrete scenarios and preconditioning methods for various forms of spatial fractional diffusion equations, providing insights and references for further research in preconditioning.展开更多
Analysis and design of linear periodic control systems are closely related to the periodic matrix equations.The biconjugate residual method(BCR for short)have been introduced by Vespucci and Broyden for efficiently so...Analysis and design of linear periodic control systems are closely related to the periodic matrix equations.The biconjugate residual method(BCR for short)have been introduced by Vespucci and Broyden for efficiently solving linear systems Aα=b.The objective of this paper is to provide one new iterative algorithm based on BCR method to find the symmetric periodic solutions of linear periodic matrix equations.This kind of periodic matrix equations has not been dealt with yet.This iterative method is guaranteed to converge in a finite number of steps in the absence of round-off errors.Some numerical results are performed to illustrate the efficiency and feasibility of new method.展开更多
Engineering tests can yield inaccurate data due to instrument errors,human factors,and environmental interference,introducing uncertainty in numerical model updating.This study employs the probability-box(p-box)method...Engineering tests can yield inaccurate data due to instrument errors,human factors,and environmental interference,introducing uncertainty in numerical model updating.This study employs the probability-box(p-box)method for representing observational uncertainty and develops a two-step approximate Bayesian computation(ABC)framework using time-series data.Within the ABC framework,Euclidean and Bhattacharyya distances are employed as uncertainty quantification metrics to delineate approximate likelihood functions in the initial and subsequent steps,respectively.A novel variational Bayesian Monte Carlo method is introduced to efficiently apply the ABC framework amidst observational uncertainty,resulting in rapid convergence and accurate parameter estimation with minimal iterations.The efficacy of the proposed updating strategy is validated by its application to a shear frame model excited by seismic wave and an aviation pump force sensor for thermal output analysis.The results affirm the efficiency,robustness,and practical applicability of the proposed method.展开更多
The harmonic balance method(HBM)has been widely applied to get the periodic solution of nonlinear systems,however,its convergence rate as well as computation efficiency is dramatically degraded when the system is high...The harmonic balance method(HBM)has been widely applied to get the periodic solution of nonlinear systems,however,its convergence rate as well as computation efficiency is dramatically degraded when the system is highly non-smooth,e.g.,discontinuous.In order to accelerate the convergence,an enriched HBM is developed in this paper where the non-smooth Bernoulli bases are additionally introduced to enrich the conventional Fourier bases.The basic idea behind is that the convergence rate of the HB solution,as a truncated Fourier series,can be improved if the smoothness of the solution becomes finer.Along this line,using non-smooth Bernoulli bases can compensate the highly non-smooth part of the solution and then,the smoothness of the residual part for Fourier approximation is improved so as to achieve accelerated convergence.Numerical examples are conducted on systems with non-smooth restoring and/or external forces.The results confirm that the proposed enriched HBM indeed increases the convergence rate and the increase becomes more significant if more non-smooth bases are used.展开更多
文摘空间分数阶扩散方程能有效地描述众多科学领域中的反常扩散现象。而大多数FDE难以得到解析解,需通过建立离散格式以获得高精度的数值解。数值求解FDE通常归结为线性方程组的求解,而预处理技术则是加速迭代求解的关键。近年来,基于不同离散格式下系数矩阵的结构和性质,学者们研究了高效的预处理方法,显著地降低了计算成本。针对数值求解空间分数阶扩散方程问题,本文整理和分析了方程不同形式下的离散情形和预处理方法,并为预处理进一步的研究提供思路参考。Spatial fractional diffusion equations effectively describe anomalous diffusion phenomena in various scientific fields. However, most FDEs are difficult to solve analytically, necessitating the establishment of discrete schemes to obtain high-precision numerical solutions. Numerical solutions of FDEs typically reduce to solving linear systems, where preconditioning techniques are crucial for accelerating iterative solvers. In recent years, scholars have investigated efficient preconditioning methods based on the structure and properties of coefficient matrices under different discretization schemes, significantly reducing computational costs. For the numerical solution of spatial fractional diffusion equations, this paper organizes and analyzes the discrete scenarios and preconditioning methods for various forms of spatial fractional diffusion equations, providing insights and references for further research in preconditioning.
基金Supported by NSFC (No.12371378)NSF of Fujian Province (Nos.2024J01980,2023J01955)。
文摘Analysis and design of linear periodic control systems are closely related to the periodic matrix equations.The biconjugate residual method(BCR for short)have been introduced by Vespucci and Broyden for efficiently solving linear systems Aα=b.The objective of this paper is to provide one new iterative algorithm based on BCR method to find the symmetric periodic solutions of linear periodic matrix equations.This kind of periodic matrix equations has not been dealt with yet.This iterative method is guaranteed to converge in a finite number of steps in the absence of round-off errors.Some numerical results are performed to illustrate the efficiency and feasibility of new method.
基金supported by the National Natural Science Foundation of China(Grant No.U23B20105).
文摘Engineering tests can yield inaccurate data due to instrument errors,human factors,and environmental interference,introducing uncertainty in numerical model updating.This study employs the probability-box(p-box)method for representing observational uncertainty and develops a two-step approximate Bayesian computation(ABC)framework using time-series data.Within the ABC framework,Euclidean and Bhattacharyya distances are employed as uncertainty quantification metrics to delineate approximate likelihood functions in the initial and subsequent steps,respectively.A novel variational Bayesian Monte Carlo method is introduced to efficiently apply the ABC framework amidst observational uncertainty,resulting in rapid convergence and accurate parameter estimation with minimal iterations.The efficacy of the proposed updating strategy is validated by its application to a shear frame model excited by seismic wave and an aviation pump force sensor for thermal output analysis.The results affirm the efficiency,robustness,and practical applicability of the proposed method.
基金supported by the National Natural Science Foundation of China (Grant No. 12372028)the National Key Research and Development Program of China (Grant No. 2020YFC2201101)the Guangdong Basic and Applied Basic Research Foundation (Grant No.2022A1515011809)。
文摘The harmonic balance method(HBM)has been widely applied to get the periodic solution of nonlinear systems,however,its convergence rate as well as computation efficiency is dramatically degraded when the system is highly non-smooth,e.g.,discontinuous.In order to accelerate the convergence,an enriched HBM is developed in this paper where the non-smooth Bernoulli bases are additionally introduced to enrich the conventional Fourier bases.The basic idea behind is that the convergence rate of the HB solution,as a truncated Fourier series,can be improved if the smoothness of the solution becomes finer.Along this line,using non-smooth Bernoulli bases can compensate the highly non-smooth part of the solution and then,the smoothness of the residual part for Fourier approximation is improved so as to achieve accelerated convergence.Numerical examples are conducted on systems with non-smooth restoring and/or external forces.The results confirm that the proposed enriched HBM indeed increases the convergence rate and the increase becomes more significant if more non-smooth bases are used.
