基于BFGS算法的概率积分模型的参数反演

BFGS method based inversion of parameters in probability integral model

概率积分模型是描述煤炭开采引起地表变形规律的重要数学模型,通常采用传统优化算法和智能优化算法,对概率积分模型中的未知参数进行反演,以确定明确的地表变形特征。为了更有效地反演概率积分模型中的未知参数,引入最优化方法,提出一种基于BFGS算法的概率积分模型中参数反演方法,该方法利用相邻两迭代点之间的位移和目标函数的一阶导数构建近似Hesse矩阵,然后确定牛顿方程产生的搜索方向,通过非精确线搜索完成迭代过程,从而获得参数的最优解。仿真实验结果表明:新算法能够有效反演出全部参数,且在不同的误差水平下,新算法的参数反演精度均明显高于模矢法(传统优化算法)和遗传算法(智能优化算法);同时,在不同的粗差水平和数据缺失情况下,新算法的参数反演精度仍明显优于其他两种算法,一定程度上说明了引入新算法进行概率积分模型中参数反演的可行性和优越性。工程实例中,反演结果和下沉值的拟合曲线图进一步验证了新算法的可靠性,但工程实例中拟合单位权中误差明显大于仿真实验,其主要原因是矿区较大的地表起伏造成较大的概率积分模型误差,进而影响参数反演的精度。因此,对于具体的工程案例,除了采用有效的参数反演方法,还需要根据实际情况制定合理的反演方案,以达到更好的参数反演效果。

Probability integral model is an important mathematical model to describe the surface deformation law caused by coal mining.Usually the traditional optimization algorithm and intelligent optimization algorithm are used to invert the unknown parameters in the probability integral model to determine the explicit surface defor-mation characteristics.In this paper,the authors propose to adopt the optimization methods to perform the inversion of the parameters.Specifically,the BFGS method is used as an example of a widely used unconstrained op-timization method.Based on the BFGS method,the authors can utilize the displacement between two adjacent iteration points to generate the search direction by Newton equation.On this basis,the iterative process is completed through inexact line search to obtain the optimal parameters.In the experimental part,the authors employ the pattern search method and genetic algorithm as the representatives of the traditional optimization algorithm and intelligent optimization algorithm.The simulation results show that the BFGS method can effectively invert all parameters,and the parameter inversion accuracy of BFGS method is significantly higher than the pattern search method and genetic algorithm under different error levels.In addition,the parameter inversion accuracy of BFGS algorithm is still better than the other two algorithms under different gross error levels and data missing conditions,which,to some extent,illustrates the feasibility and superiority of introducing BFGS method for parameter inversion in probability integral model.In the engineering case,the fitting curves of the inversion results and the sinking values further verify the reliability and accuracy of the new algorithm.However,the unit weight error in the case is significantly larger than that in the simulation experiment.Taking the actual factors of the case into consideration,the main reason for the large difference of the unit weight error is the large surface fluctuations in the mining area,which causes the larger model error of the probability integral method.Therefore,for the specific engineering case,in addition to the effective parameter inversion method,it is necessary to determine a reasonable inversion scheme according to the actual situation,in order to achieve better parameter inversion effect.