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基于贝叶斯推断的地下水流模型非高斯源项场的识别
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Abstract:
地下水模型中非均匀介质的复杂性及观测数据的稀缺性使模型存在不确定性。为了更好地预测模型的输出，需要我们基于有限的观测数据来估计模型中的未知参数，降低不确定性。贝叶斯方法是刻画不完全数据、模型偏差和测量误差带来的不确定性的有效方式，它可以根据现有数据确定参数向量的后验分布。在实际应用中，其主要挑战在于从后验分布中抽样。传统抽样方法随未知参数维数的增加而出现退化现象。本文主要利用最大期望变量选择(EMVS)方法来识别稀疏离散余弦变换(DCT)系数，并对后验分布中的超参数进行自适应更新，以提高问题的求解效率；特别地，利用逆Hessian矩阵来加速朗之万动力学蒙特卡洛马尔科夫链(MCMC)的收敛速度，使用敏感性矩阵构造的简化模型来有效地计算梯度和Hessian矩阵，高效地解决高维不确定性分析和反演问题。基于地下水源项识别高维反演数值实验，验证了反演方法能够得到可靠的参数估计，为提高地下水模拟在实际应用中的可靠性和计算效率提供了新的思路，对后续的地下水资源管理决策制定具有重要意义。
Groundwater model of heterogeneous media and its scarcity of observation data and other related factors makes the existence of uncertainty. In order to better predict the output of the model, we need to estimate the input and parameters of the model based on limited observational data to re-duce the uncertainty. Bayesian method is an effective way to describe the uncertainty caused by incomplete data, model deviation and measurement error. It can determine the posterior distribu-tion of parameter vectors according to the existing data. In practical application, the main challenge lies in sampling. The traditional sampling method degrades with the increase of the dimension of unknown parameters, that is, the convergence is slow. In this paper, an inversion algorithm is pro-posed to identify sparse discrete cosine transform (DCT) coefficients in expectation maximized var-iable selection (EMVS) frame, and adaptively update the hyperparameters to improve the solving efficiency of the problem. In particular, the inverse Hessian is used to accelerate the convergence of Langevin dynamics Monte Carlo Markov Chain (MCMC), and the simplified model is used to compute the gradient and Hessian effectively to solve the high dimensional uncertainty analysis and inver-sion problems efficiently. Based on the high-dimensional inversion numerical experiment of groundwater source item identification, it is verified that the inversion method can obtain reliable parameter estimation, which provides a new idea for improving the reliability and computational efficiency of groundwater simulation in practical application, and has important significance for the subsequent decision-making of groundwater resource management.
| [1] | Kaipio, J. and Somersalo, E. (2006) Statistical and Computational Inverse Problems. Springer Science & Business Media, Berlin. |
| [2] | Tarantola, A. (2005) Inverse Problem Theory and Methods for Model Parameter Estimation. Society for Industrial & Applied Mathematics, Berlin. |
| [3] | Liu, J.S. (2008) Monte Carlo Strategies in Scientific Computing. Springer Science & Business Media, Berlin. |
| [4] | Gamerman, D. and Lopes, H.F. (2006) Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference. CRC Press, New York. https://doi.org/10.1201/9781482296426 |
| [5] | Curtis, A. and Lomax, A. (2001) Tutorial: Prior Information, Sam-pling Distributions, and the Curse of Dimensionality. Acta Chirurgica Italica, 66, 372-698. |
| [6] | Cotter, S.L., Roberts, G.O., Stuart, A.M. and White, D. (2013) MCMC Methods for Functions: Modifying Old Algorithms to Make Them Faster. Statistical Science, 28, 424-446. https://doi.org/10.1214/13-STS421 |
| [7] | Roberts, G.O. and Tweedie, R.L. (1996) Exponential Convergence of Langevin Distributions and Their Discrete Approximations. Bernoulli, 2, 341-363. https://doi.org/10.2307/3318418 |
| [8] | Gavalas, G.R., Shah, P.C., Seinfeld, J.H., et al. (1976) Reservoir History Matching by Bayesian Estimation. Society of Petroleum Engineers Journal, 16, 337-350. https://doi.org/10.2118/5740-PA |
| [9] | Karhunen, K. (1947) Uber linearemethoden in der Wahrscheinlichkeitsrech-nung. Annales Academiae Scientiarum Fennicae, 37, 3-79. |
| [10] | Loeve, M. (1978) Probability Theory II. Spring-er-Verlag New York Inc., New York. |
| [11] | Sarma, P., Louis, J.D. and Aziz, K. (2008) Kernel Principal Component Analysis for Efficient, Differentiable Parameterization of Multipoint Geostatistics. Mathematical Geosciences, 40, 3-32. https://doi.org/10.1007/s11004-007-9131-7 |
| [12] | Jafarpour, B. and McLaughlin, D.B. (2009) Reservoir Characteri-zation with the Discrete Cosine Transform. SPE Journal, 14, 182-201. https://doi.org/10.2118/106453-PA |
| [13] | Sahni, I. and Roland, R.N. (2005) Multiresolution Wavelet Analysis for Improved Reservoir Description. SPE Reservoir Evaluation and Engineering, 8, 53-69. https://doi.org/10.2118/87820-PA |
| [14] | Rockova, V. and George, E.I. (2014) EMVS: The EM Approach to Bayesi-an Variable Selection. Journal of the American Statistical Association, 109, 828-846. https://doi.org/10.1080/01621459.2013.869223 |
| [15] | Jafarpour, B., Goyal, V.K., McLaughlin, D.B. and Freeman, W.T. (2010) Compressed History Matching: Exploiting Transform-Domain Sparsity for Regularization of Nonlinear Dynamic Data Integration Problems. Mathematical Geosciences, 42, 1-27. https://doi.org/10.1007/s11004-009-9247-z |
| [16] | Wang, Y.T., Deng, W. and Lin, G. (2021) Bayesian Sparse Learn-ing with Preconditioned Stochastic Gradient MCMC and Its Applications. Journal of Computational Physics, 432, 110134. https://doi.org/10.1016/j.jcp.2021.110134 |
| [17] | Deng, W., Zhang, X., Liang, F.M. and Lin, G. (2019) An Adaptive Empirical Bayesian Method for Sparse Deep Learning. Advances in Neural Information Processing Systems 32. |
| [18] | Andrew, M.S., Voss, J. and Wilberg, P. (2004) Conditional Path Sampling of SDEs and the Langevin MCMC Method. Communications in Mathematical Sciences, 2, 685-697. https://doi.org/10.4310/CMS.2004.v2.n4.a7 |
| [19] | Martin, J., Lucas, C.W., Burstedde, C., et al. (2012) A Stochastic Newton MCMC Method for Large-Scale Statistical Inverse Problems with Application to Seismic Inversion. SIAM Journal on Scientific Computing, 34, 1460-1487.
https://doi.org/10.1137/110845598 |
