Research On Inverse Problems Of Environmental Hydraulics By Bayesian Inference

With the development of the solution for direct problems (prediction problems), the related research on inverse problems are receieving more and more attention. The main difficulty of the solution for inverse problems of environmental hydraulics lies in the ill-posed, while it, especially for uniqueness, comes from the uncertainties of the water environmental systems. There are many uncertainties factors in the water environmental systems, so how to face directly and solve these uncertainties is a problem in dire need. In the view of probability theory, a Bayesian statistics method-Bayesian inference is adopted to set up the inverse models of the environmental hydraulics. The model parameters, measurement data, prior information and the solution of the inverse probles are all expressed in the probability language, thus the uncertainty problems of the inverse enviormental hydraulics research is solved conveniently. In the aspect of theory research, the standard sampling method for Markov chain Monte Carlo method (MCMC) in Bayesian inference are investigated in this work. In order to increase the exploration capability for posterior parameters space (solution space) of Markov chain, an algorithm named "Dynamic Multi-chains Metropolis-Hastings, DMMH" based on the classical Metropolis-Hastings is advanved. Since multi-chains exploration mechanics is used, the algorithm solves the weakness of single chain exploration well. At the same time, the number of the chains can be changed timely according to the current exploration status, so the computation time is less than the traditional multi-chains exploration. The basic mathematic model of the diect problems of environmental hydraulics is the convection-diffusion equation. In the work, finite volume method and finite element method are the solution methods. On finite volume method, the flux limter method with high resolution and better stability is investigated, which is suitable for the solution of inverse problems of environmeantal hydraulics which needs large number of direct simulation. An improved limited flux named M7 is advanced, and the numerical computation indicates that M7 limiter increase the computation resolution and the stability of explicit difference.On application research, Bayesian inference is used to solve two kinds of inverse problems, that is, parameters estimation problems of environmental hydraulics and pollution source identification problems. Much atterntion are given to the parameters estimation problems, since the soure identification can be treated as parameters estimation problems. There are various kinds of mathematics models of environmental hydraulics, to verify the rationality and reliability of the inversion models based on Bayesian inference, kinds of models such as one dimensional model, two dimensional model, steady model, unsteady model, constant coefficients model, variational coefficients model, model with source and model without source are all investigated. Lots of computation examples indicate the inversion solutions based on Bayesian inference and MCMC sampling are informative, which can give the posterior parameters distribution, and accurate as well. As to the inversion's reliability and estimate's preicision, the effect of measurement's position, the outliers and the number of measurement position are the mainly cared, and the ununiqueness of the inverse problems are investigated. On the aspect of pollutant source identification, singe point source and multi points cases are investigated, and the position inversion and the intensity inversion base on MCMC Bayesian inference both can get the good identification precision.The optimization methods are the mainstream in the solution of inverse problems of environmental hydraulics, while it and the Bayesian approach are different from each other and share some common things. So in this work the optimization method (genetic algorithm) for the inverse problems of environmental hydralics is also introduced, and a "FVM-HGA" parameter identification method is advanced, which has the strong global and local search ability and its inversion precision is high and convergence is fast. Its shortcoming lies in that it can not give the parameter distribution like the Bayesian inference.Since MCMC can be used for parameter identification itself, the parameter optimization problem of environmental hydraulics is studied as well, and the author gives out a new parameter identification method based on Bayesian method with the likelihood function as the convergence rule, named L-MCMC. Computational case indicates L-MCMC method can reduce the inversion burden greatly.All in all, for the environmental system inverse problem such as environmental hydraulics, Bayesian inference with MCMC sampling is a powerful tool, which can give the distribution of the model's parameters and the "best" estimate as the optimization method and has the important research value and application foreground.