The Research On The Key Problems Of The Approximate Bayesian Computational Inverse Problem With Uncertainty

Currently,numerical simulation has been widely used in science and engineering area.The accuracy of the numerical simulation mainly depends on the model parameters.In science and engineering problem,these model parameters commonly are identified by the inverse problem which is defined as identifying the model parameters with given system response.Compared with the forward problem which is defined as solving system response given model parameters,the inverse problem is more complicated.The model parameters for the system response are not unique or the small perturbation of system response will lead to large modification of model response.However,in practice,the system response is inevitably polluted by the uncertainty like measurement error.This will lead to inaccurate model parameters of the inverse problem.In order to address this issue,the Bayesian method which can integrate known information from the prior distribution and the experimental information from the likelihood function and can essentially address the ill-posed problem has been widely used.However,the likelihood function is always intractable in practice due to either computationally prohibitive or analytically unavailable.In this study,to circumvent the intractable likelihood function,an approximate Bayesian computation(ABC)is extended to the practice engineering problem.In order to improve the efficiency and make it feasible,we have done:In the ABC,the high dimensional observations are reduced to their low dimensional summary statistics.The selection of the summary statistics is significant to the performance of approximate Bayesian computation.However,the traditional summary statistics selection method is complicated,unfeasible and time-consuming.In order to address this issue,we proposed two method: the first one is proper orthogonal decomposition(POD)based ABC;the second one is auto-encoder based ABC.This study proves that these two methods can reduce the high dimension of observations to rather low dimension of vector with almost no information loss.Secondly,in order to improve efficiency and reduce the number of samples of the sampling method,we analysis the sampling methods comprehensively and propose two sampling methods.With analysis,we figure out the correlation between the computational cost and tolerance.Based on this correlation,we propose a more efficient and feasible non-parametric population Monte Carlo(NPMC)method.In addition,we expand the nested sampling method to ABC and propose adaptive nested sampling method.Thirdly,in order to further reduce computational cost,we propose reanalysis based ABC and neural network based ABC.Thus,the forward problem of the samples can be solved efficiently with these methods instead of the time-consuming numerical simulations.At last,we expand the ABC to two ill-posed inverse problems firstly.The first one is the classical ill-posed heat conduction inverse problem.In it,the observations are the temperature field solved with given boundary condition.We aim to infer the boundary condition with this temperature.With it we can demonstrate the efficiency and accuracy of the proposed method since the true value of model parameter is given.The second one is a real engineering problem.It aims to identify the material parameter of the formed advance high strength steel with the ill-posed indention test.With the validation test,we can demonstrate the performance of ABC in real engineering problem.