In this paper, Aitken’s extrapolation normally applied to convergent fixed point iteration is extended to extrapolate the solution of a divergent iteration. In addition, higher order Aitken extrapolation is introduce...In this paper, Aitken’s extrapolation normally applied to convergent fixed point iteration is extended to extrapolate the solution of a divergent iteration. In addition, higher order Aitken extrapolation is introduced that enables successive decomposition of high Eigen values of the iteration matrix to enable convergence. While extrapolation of a convergent fixed point iteration using a geometric series sum is a known form of Aitken acceleration, it is shown that in this paper, the same formula can be used to estimate the solution of sets of linear equations from diverging Gauss-Seidel iterations. In both convergent and divergent iterations, the ratios of differences among the consecutive values of iteration eventually form a convergent (divergent) series with a factor equal to the largest Eigen value of the iteration matrix. Higher order Aitken extrapolation is shown to eliminate the influence of dominant Eigen values of the iteration matrix in successive order until the iteration is determined by the lowest possible Eigen values. For the convergent part of the Gauss-Seidel iteration, further acceleration is made possible by coupling of the extrapolation technique with the successive over relaxation (SOR) method. Application examples from both convergent and divergent iterations have been provided. Coupling of the extrapolation with the SOR technique is also illustrated for a steady state two dimensional heat flow problem which was solved using MATLAB programming.展开更多
为设计高效稳定的演化算法,将方程求根的不动点迭代思想引入到优化领域,通过将演化算法的寻优过程看作为在迭代框架下方程不动点的逐步显示化过程,设计出一种基于数学模型的演化新算法,即不动点演化算法(fixed point evolution algorith...为设计高效稳定的演化算法,将方程求根的不动点迭代思想引入到优化领域,通过将演化算法的寻优过程看作为在迭代框架下方程不动点的逐步显示化过程,设计出一种基于数学模型的演化新算法,即不动点演化算法(fixed point evolution algorithm,FPEA).该算法的繁殖算子是由Aitken加速的不动点迭代模型导出的二次多项式,其整体框架继承传统演化算法(如差分演化算法)基于种群的迭代模式.试验结果表明:在基准函数集CEC2014、CEC2019上,本文算法的最优值平均排名在所有比较算法中排名第1;在4个工程约束设计问题上,FPEA与CSA、GPE等多个算法相比,能以较少的计算开销获得最高的求解精度.展开更多
文摘In this paper, Aitken’s extrapolation normally applied to convergent fixed point iteration is extended to extrapolate the solution of a divergent iteration. In addition, higher order Aitken extrapolation is introduced that enables successive decomposition of high Eigen values of the iteration matrix to enable convergence. While extrapolation of a convergent fixed point iteration using a geometric series sum is a known form of Aitken acceleration, it is shown that in this paper, the same formula can be used to estimate the solution of sets of linear equations from diverging Gauss-Seidel iterations. In both convergent and divergent iterations, the ratios of differences among the consecutive values of iteration eventually form a convergent (divergent) series with a factor equal to the largest Eigen value of the iteration matrix. Higher order Aitken extrapolation is shown to eliminate the influence of dominant Eigen values of the iteration matrix in successive order until the iteration is determined by the lowest possible Eigen values. For the convergent part of the Gauss-Seidel iteration, further acceleration is made possible by coupling of the extrapolation technique with the successive over relaxation (SOR) method. Application examples from both convergent and divergent iterations have been provided. Coupling of the extrapolation with the SOR technique is also illustrated for a steady state two dimensional heat flow problem which was solved using MATLAB programming.
文摘为设计高效稳定的演化算法,将方程求根的不动点迭代思想引入到优化领域,通过将演化算法的寻优过程看作为在迭代框架下方程不动点的逐步显示化过程,设计出一种基于数学模型的演化新算法,即不动点演化算法(fixed point evolution algorithm,FPEA).该算法的繁殖算子是由Aitken加速的不动点迭代模型导出的二次多项式,其整体框架继承传统演化算法(如差分演化算法)基于种群的迭代模式.试验结果表明:在基准函数集CEC2014、CEC2019上,本文算法的最优值平均排名在所有比较算法中排名第1;在4个工程约束设计问题上,FPEA与CSA、GPE等多个算法相比,能以较少的计算开销获得最高的求解精度.
