Computational Inverse Method Research For Inverse Problem Considering Uncertain Model Parameters

Uncertain inverse problem exists widely in engineering practice.For example,the seismic detection,the damage identification and the traffic accident reconstruction inevitably involve uncertainties in the reverse modeling process due to changes in the service environment,physical parameters difficult to define and lack of cognition.Giving full consideration to the influence of the uncertainty of the model parameters on the inverse parameters,so as to a more objective evaluation of the results can be obtained.Therefore,it is of great theoretical and practical value to study the inverse method considering uncertainties of the model parameter for uncertain inverse problem.When the uncertainty occurs in the system model,the traditional methods to solve such inverse problems often involve a double nesting problem including the uncertainty propagation and inverse calculation,leading to low efficiency.In order to solve this problem,this paper uses the probability to measure the uncertainty parameter and its effect on the inverse result,and conduct systematic research based on the point estimation method,the sparse grid integral method and the similarity system approximation method for the solving efficiency and accuracy of such problems.Aiming at the inverse problem whose system model has low degree of non-linearity and dimension of uncertain parameters is low,the inverse method based on the point estimation and the maximum entropy principle is studied.Compared with the traditional method,the solving efficiency is improved.Aiming at the inverse problem whose system model has high degree of non-linearity and dimension of uncertain parameters is high,the inverse method based on the sparse grid integral method is studied.In the premise of ensuring the solving efficiency of the,the accuracy of the uncertainty inverse process is further improved.Aiming at the deterministic reverse processes involved in the uncertain inverse problem,an efficient inverse method based on the similarity system approximation is studied,which can reduce the number of deterministic inverse processed and further improve solving efficiency of uncertain inverse problem.The content of this paper will specifically focus on the following three aspects:(1)Considering the probability distribution of the uncertain parameters in the system model and aiming at the inverse problem whose system model has low degree of non-linearity and dimension of uncertain parameters is low,the inverse method based on the point estimation and the maximum entropy principle is studied.Firstly,according to the high-order moments of the uncertain model parameters,the distribution characteristics of the uncertain model parameters are represented by several discrete probability concentration points.Then,the deterministic inverse processes are conducted at these concentration points and the corresponding probability concentration points of identified parameters are obtained.Thus,the uncertain inverse problem is transformed into several deterministic inverse problems,which greatly simplifies the complexity of the traditional double nesting algorithm structure.Finally,the probability concentration points of identified parameters are transformed into the mean,standard deviation,skewness and kurtosis by combining the weight values assigned by each probability concentration point.The maximum entropy principle is used to further obtain the probability density function of the identified parameters.(2)Aiming at the inverse problem whose system model has high degree of non-linearity and dimension of uncertain parameters is high,the inverse method based on the sparse grid integral method is studied.In the process of uncertain inverse propagation,instead of the point estimation method,the sparse grid integral method is used as the new moment propagation method to obtain the mean,standard deviation,skewness and kurtosis of the identified parameters.The sparse grid integration method uses more reasonable probability concentration points than point estimation method,and makes full use of the distribution characteristics of known random variables,showing greater adaptability in dealing with the uncertain inverse problem.The inverse algorithm based on sparse grid integration method employs the algorithm structure similar to the point estimation method.The number of deterministic inverse processes increases compared with the point estimation method,resulting in a slight reduction in efficiency and a higher inverse accuracy.It can be better applied to the inverse problem in which the non-linear degree of the system model is high or the dimension of uncertain parameters is high.(3)In order to further improve the computational efficiency of uncertain inverse problem,an uncertain inverse method based on similarity system approximation is proposed.Firstly,the uncertain propagation method based on the moment propagation is adopted to obtain the configuration points of the known random parameters related to the deterministic inverse process.The known model parameter mean configuration point is regard as the initial configuration point to be calculated,and the initial inverse point is obtained accordingly.Secondly,a similar system is obtained by Taylor's first order expansion at the joint point of the new configuration point and the initial inverse point.By calculating the gradient matrix at the point,the approximate relation between the new inverse point corresponding to the new configuration point and the initial inverse point is obtained.After approaching the new inversion point by using the similar system,transform the position of the initial configuration point,try to ensure that the space distance between the new configuration point and the initial configuration point is as small as possible,and gradually get all the inverse points corresponding to the configuration points.Finally,combined with the weight distribution method in the moment propagation method,the approximate inverse points are transformed into the mean,standard deviation,skewness and kurtosis of the identified parameters.Then the probability distribution of identified parameters is obtained by the maximum entropy principle.The method can obtain the probability distribution of identified parameters by only one time of deterministic inverse process,which further improves the computational efficiency of uncertain inverse problem.