Adaptive Ensemble Kalman Inversion Algorithm And Its Application To Bayesian Inverse Problems

With the continuous development of modern science and technology,a large number of Bayesian inverse problem models described by partial differential equations have appeared in important fields such as medical imaging,energy exploration,numerical weather prediction,national defense and military.In recent years,the introduction of particle-based methods provides a new idea for solving inverse problems.Particle-based methods are gradient-free optimization iterative methods.The most representative is the ensemble Kalman inversion algorithm,but the traditional ensemble Kalman inversion algorithm has the property of invariant subspace,the performance of the algorithm depends on the selection of the initial particle set,and the phenomenon of model collapse is prone to occur.Therefore,in order to overcome the above difficulties,this paper will design a new adaptive ensemble Kalman inversion algorithm combined with the particle-based algorithm.Specifically,the construction process of the classical ensemble Kalman inversion algorithm is reviewed first,and the statistical linearization technique is introduced.Then two proposed adaptive update algorithms are presented.In the implementation process of the adaptive algorithm,we first introduce hyperparameters into the prior covariance matrix,use the properties of the Bayesian hierarchical model and conjugate priors,and use the maximum a posteriori estimator of the posterior distribution as the value of the hyperparameters.Update standard.On this basis,in order to make the model closer to the real situation,a more universal adaptive update algorithm is considered.We introduce hyperparameters into the prior covariance matrix and observation noise matrix respectively,and obtain a more general adaptive ensemble Kalman inversion algorithm.The addition of the adaptive algorithm enables the regularization parameters in the traditional particle-based method to be adaptively selected,which reduces the difficulty and randomness of artificially setting parameters,and alleviates the phenomenon of model collapse to a certain extent.Most importantly,the adaptive algorithm proposed in this paper can accurately invert the noise level,improving the standard of early stopping of traditional particle-based methods.In order to verify the accuracy and stability of our newly proposed algorithm,we tested three different models in numerical experiments,and compared the adaptive ensemble Kalman inversion algorithm proposed in this paper with the traditional ensemble Kalman inversion algorithm.Numerical experiments show that the algorithm can converge faster while maintaining numerical accuracy.