Some Inverse Problems Of Fractional Partial Differential Equations

In this doctoral dissertation,we mainly consider several inverse problems re-lated to fractional diffusion equations.In the framework of Tikhonov regularization and Bayesian inference,we consider the inverse problems for the time fractional d-iffusion equations,the multi-term time fractional diffusion models in heterogeneous media and the space-time nonlocal diffusion equations.Firstly,we consider the influence of the multi-parameter regularization method on inverse problems.In order to overcome the ill posedness of inverse problems and identify the unknown parameters with different properties,we apply the L~2+BV method to an inverse problem of recovering the reaction coefficient based on time fractional diffusion equations.We prove that the forward operator is continuous with respect to the model inputs.In the framework of Tikhonov regularization,we analyze the existence and stability of the solution to the regularized variational problem and the convergence.A few numerical results are presented for the inverse problem in time fractional diffusion models to confirm the theoretic analysis and the efficacy of different regularization methods.Secondly,we consider an implicit sampling method for hierarchical Bayesian inverse problems in fractional multiscale diffusion equations.The most widely used approach for sampling from posterior distribution is Markov chain Monte Carlo(MCMC).However,the samples generated by MCMC are usually strongly corre-lated,which may lead to a small size of effective samples.The implicit sampling method can generate independent samples and then capture inherent non-Gaussian features of the posterior.In implicit sampling,the posterior samples are generated by a map and distribution around the maximum a posterior(MAP)point.For practical applications,some parameters in prior information are often unknown.The hierarchical Bayesian formulation can be used to estimate the MAP point and the unknown parameters in prior density,simultaneously.It is applied to the inverse problems of multi-term time fractional diffusion models.For the above models in heterogeneous media,there may exist some multiple scales and high contrast feature in the diffusion field.Direct simulating the multiscale models may be very computationally expensive.However,Bayesian inversion requires to sim-ulate the forward model for many times.To effectively capture the heterogeneity feature,we introduce a mixed generalized multiscale finite element method(mixed GMs FEM),which aims at dividing the computation into offline and online steps.Then a reduced computational model is constructed to substantially speed up the Bayesian inversion.Some few numerical examples are carried out to effectively recover different types of unknown inputs.Finally,we consider a variational Bayesian method to identify the reaction coefficient for space-time nonlocal diffusion equations using nonlocal averaged flux data.In order to show that the posterior measure is well-defined,we rigorous-ly prove that the forward operator is continuous with respect to the unknown parameters.The prior distribution plays an important role on the Bayesian in-verse problems.The most commonly used Gaussian prior is usually applied to identify model inputs with smooth property.Thus,more sophisticated prior dis-tribution should be adopted if the inversion target has some oscillation features such as smooth oscillation,nonsmooth oscillation and discontinuous oscillation.Then gradient-based prior is applied to capture the oscillation features in the re-action field.The posterior measure is shown to be continuous with respect to the measurement data in the sense of Hellinger distance.To accurately explore the posterior density using uncorrelated samples,an efficient variational Bayesian method is used to estimate the reaction coefficient in the nonlocal models.A few numerical results are presented to illustrate the practicability of the proposed approach.