Guanghui Zheng

Xiaoyan Song, Guanghui Zheng, Lijian Jiang

In this paper, we present an inverse problem of identifying the reaction coefficient for time fractional diffusion equations in two dimensional spaces by using boundary Neumann data. It is proved that the forward operator is continuous with respect to the unknown parameter. Because the inverse problem is often ill-posed, regularization strategies are imposed on the least fit-to-data functional to overcome the stability issue. There may exist various kinds of functions to reconstruct. It is crucial to choose a suitable regularization method. We present a multi-parameter regularization $L^{2}+BV$ method for the inverse problem. This can extend the applicability for reconstructing the unknown functions. Rigorous analysis is carried out for the inverse problem. In particular, we analyze the existence and stability of regularized variational problem and the convergence. To reduce the dimension in the inversion for numerical simulation, the unknown coefficient is represented by a suitable set of basis functions based on a priori information. A few numerical examples are presented for the inverse problem in time fractional diffusion equations to confirm the theoretic analysis and the efficacy of the different regularization methods.

Xiaoyan Song, Lijian Jiang, Guanghui Zheng

This paper presents an improved implicit sampling method for hierarchical Bayesian inverse problems. A widely used approach for sampling posterior distribution is based on Markov chain Monte Carlo (MCMC). However, the samples generated by MCMC are usually strongly correlated. This may lead to a small size of effective samples from a long Markov chain and the resultant posterior estimate may be inaccurate. An implicit sampling method proposed in [11] can generate independent samples and capture some inherent non-Gaussian features of the posterior based on the weights of samples. In the implicit sampling method, the posterior samples are generated by constructing a map and distribute around the MAP point. However, the weights of implicit sampling in previous works may cause excessive concentration of samples and lead to ensemble collapse. To overcome this issue, we propose a new weight formulation and make resampling based on the new weights. In practice, some parameters in prior density are often unknown and a hierarchical Bayesian inference is necessary for posterior exploration. To this end, the hierarchical Bayesian formulation is used to estimate the MAP point and integrated in the implicit sampling framework. Compared to conventional implicit sampling, the proposed implicit sampling method can significantly improve the posterior estimator and the applicability for high dimensional inverse problems. The improved implicit sampling method is applied to the Bayesian inverse problems of multi-term time fractional diffusion models in heterogeneous media. To effectively capture the heterogeneity effect, we present a mixed generalized multiscale finite element method (mixed GMsFEM) to solve the time fractional diffusion models in a coarse grid, which can substantially speed up the Bayesian inversion.

Fuchen Chen, Lijian Jiang, Guanghui Zheng

In this paper, we present a dimension reduction method to reduce the dimension of parameter space and state space and efficiently solve inverse problems. To this end, proper orthogonal decomposition (POD) and radial basis function (RBF) are combined to represent the solution of forward model with a form of variable separation. This POD-RBF method can be used to efficiently evaluate the model's output. A gradient regularization method is presented to solve the inverse problem with fast convergence. A generalized cross validation method is suggested to select the regularization parameter and differential step size for the gradient computation. Because the regularization method needs many model's evaluations. This is desirable for POD-RBF method. Thus, the POD-RBF method is integrated with the gradient regularization method to provide an efficient approach to solve inverse problems. We focus on the coefficient inversion of diffusion equations using the proposed approach. Based on different types of measurement data and different basis functions for coefficients, we present a few numerical examples for the coefficient inversion. The numerical results show that accurate reconstruction for the coefficient can be achieved efficiently.