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In this paper, we consider the numerical solution of the poroelasticityproblem with stochastic properties. We present a Two-stage Markov Chain MonteCarlo method for geomechanical subsidence. In this work, we study twotechniques of preconditioning: (MS) multiscale method for model order reductionand (ML) machine learning technique. The purpose of preconditioning is the fastsampling, where a new proposal is first testes by a cheap multiscale solver orusing fast prediction of the neural network and the full fine grid computationswill be conducted only if the proposal passes the first step. To construct areduced order model, we use the Generalized Multiscale Finite Element Methodand present construction of the multiscale basis functions for pressure anddisplacements in stochastic fields. In order to construct a machine learningbased preconditioning, we generate a dataset using a multiscale solver and useit to train neural networks. The Karhunen-Loeve expansion is used to representthe realization of the stochastic field. Numerical results are presented fortwo- and three-dimensional model examples.

Preconditioning Markov Chain Monte Carlo Method for Geomechanical Subsidence using multiscale method and machine learning technique