利用贝叶斯推理估计二维含源对流扩散方程参数

Application of Bayesian Inference to Estimate the Parameters in 2D Convection-diffusion Equation with Source

为了克服观测数据的不确定性给参数反演带来的困难,利用贝叶斯推理建立了二维含源对流扩散方程参数估计的数学模型。通过贝叶斯定理,获得了模型参数的后验分布,从而获得反问题的解。对于多参数反演问题,基于数值解计算得到的参数后验分布很难直观地表现出来,采用马尔科夫链蒙特卡罗方法对参数的后验分布进行采样,获得了扩散系数和降解系数的估计值。研究了观测点位置对计算结果的影响;同时研究了似然函数的形式对估计结果的影响,结果表明在异常值可能出现时采用Laplace分布型的似然函数可以获得稳健估计。对不同观测点数目下的估计值进行了对比,认为对于二维稳态对流扩散方程的双参数估计问题,至少需要两个观测点才有可能得到合理的解。

To overcome the parameter inversion＇ s difficulty from the measurement uncertainty, mathematical model based on Bayesian inference was set up to estimate the coefficients in 2D convection-diffusion equation with source. By Bayes＇ theorem, the posterior distribution of model parameters was obtained, which is the solution of the inverse problem. While for multi-parameter inversion problem, the posterior distribution obtained by numerical computation is hard to express easily, Markov chain Monte Carlo method is used to explore the posterior distribution to get the estimation. The effect of the observation position on inversion results was investigated and the form of the likelihood function＇ s effect was also studied. It indicates that the Laplacian distribution can lead to a robust estimation. Comparing the estimates under different number of measurement points, it can be concluded that at least two points are needed to ensure reasonable solution for the two parameters estimation problem of 2D steady convection diffusion equation.