对流-扩散方程源项识别反问题的MCMC方法

MCMC method on an inverse problem of source term identification for convection-diffusion equation

给出了利用马尔科夫链蒙特卡罗(MCMC)方法求解对流-扩散方程源项识别反问题的一种新方法。该方法把源项识别反问题视为贝叶斯估计问题,然后用MCMC法求解。首先利用贝叶斯公式,导出了多点源中源强和位置等未知参数分布规律的后验概率密度函数;接着以后验概率分布为目标分布采用自适应Metropolis算法构造Markov链;然后截取收敛的链序列,利用后验均值方法估计出源项中的未知参数。数值试验结果表明,该方法具有精度高,收敛速度快且易于计算机实现等优点。

A new approach based on Markov Chain Monte Carlo（MCMC） method for source term identification of convection-diffusion equation was proposed. It views the inverse problem of source term identification as the problem of Bayesian estimation resolved by MCMC algorithm. Firstly, the posterior probability density function for unknown parameters of multiple point sources was deduced with the Bayesian formula. Secondly, taking the posterior probability as the target distribution, the Adaptive Metropolis algorithm was used to construct the Markov Chains of unknown parameters. And the converged samples were used to estimate the unknown parameters of source term. The results of numerical experiments show that the method has many virtues, such as high accuracy, quick convergent speed and easy to program and implement with computer.