Incorporating Posterior-Informed Approximation Errors into a\n Hierarchical Framework to Facilitate Out-of-the-Box MCMC Sampling for\n Geothermal Inverse Problems and Uncertainty Quantification

2018/10/09 by Oliver J. Maclaren, Ruanui Nicholson, Maclaren, Oliver J. +7
Computer Science · Engineering · Mathematics · #Computation (stat.CO) #FOS: Computer and information sciences #FOS: Mathematics #Gaussian Processes and Bayesian Inference #Markov Chains and Monte Carlo Methods #Numerical Analysis (math.NA) #Reservoir Engineering and Simulation Methods

paper · pdf · doi:10.48550/arxiv.1810.04350

openalex publication_date 2018/10/09 · openalex created_date 2022/08/02 · openalex updated_date 2026/07/28

Abstract

We consider geothermal inverse problems and uncertainty quantification from a\nBayesian perspective. Our main goal is to make standard, `out-of-the-box'\nMarkov chain Monte Carlo (MCMC) sampling more feasible for complex simulation\nmodels by using suitable approximations. To do this, we first show how to pose\nboth the inverse and prediction problems in a hierarchical Bayesian framework.\nWe then show how to incorporate so-called posterior-informed model\napproximation error into this hierarchical framework, using a modified form of\nthe Bayesian approximation error (BAE) approach. This enables the use of a\n`coarse', approximate model in place of a finer, more expensive model, while\naccounting for the additional uncertainty and potential bias that this can\nintroduce. Our method requires only simple probability modelling, a relatively\nsmall number of fine model simulations, and only modifies the target posterior\n-- any standard MCMC sampling algorithm can be used to sample the new\nposterior. These corrections can also be used in methods that are not based on\nMCMC sampling. We show that our approach can achieve significant computational\nspeed-ups on two geothermal test problems. We also demonstrate the dangers of\nnaively using coarse, approximate models in place of finer models, without\naccounting for the induced approximation errors. The naive approach tends to\ngive overly confident and biased posteriors while incorporating BAE into our\nhierarchical framework corrects for this while maintaining computational\nefficiency and ease-of-use.\n

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