[Paper Review] Adaptive Approximation Error Models for Efficient Uncertainty Quantification with Application to Multiphase Subsurface Fluid Flow

Tiangang Cui, Colin Fox+1 more|arXiv (Cornell University)|Sep 10, 2018

Probabilistic and Robust Engineering Design33 references3 citations

TL;DR

This paper proposes an adaptive approximation error modeling approach that reduces computational cost in Bayesian uncertainty quantification for subsurface fluid flow by building a stochastic error model for reduced-order models. By leveraging adaptive Markov chain Monte Carlo with guaranteed convergence, the method enables accurate inference at a fraction of the cost, demonstrated effectively on real geothermal reservoir calibration with both synthetic and measured data.

ABSTRACT

Sample-based Bayesian inference provides a route to uncertainty quantification in the geosciences, though is very computationally demanding in the na\ive form that requires simulating an accurate computer model at each iteration. We present a new approach that adaptively builds a stochastic model for the error induced by a reduced model. This enables sampling from the correct target distribution at reduced computational cost, while avoiding appreciable loss of statistical efficiency. We build on recent simplified conditions for adaptive Markov chain Monte Carlo algorithms to give practical approximation schemes and algorithms with guaranteed convergence. We demonstrate the efficacy of our new approach on two computational examples, including calibration of a large-scale numerical model of a real geothermal reservoir, that show good computational and statistical efficiencies on both synthetic and measured data sets.

Motivation & Objective

• To reduce the computational burden of sample-based Bayesian inference in geoscience applications involving complex subsurface flow models.
• To develop a method that maintains statistical efficiency while significantly lowering the number of high-fidelity simulations required.
• To construct a stochastic error model that adaptively captures discrepancies between reduced and full-order models.
• To ensure convergence of the inference algorithm under simplified conditions for adaptive MCMC.
• To validate the approach on real-world and synthetic data, including a large-scale geothermal reservoir model.

Proposed method

• The method constructs a data-driven, adaptive stochastic model of the approximation error introduced by using a reduced-order model instead of the full high-fidelity simulator.
• It employs adaptive Markov chain Monte Carlo (MCMC) algorithms with simplified convergence conditions to sample from the correct posterior distribution.
• The error model is iteratively refined during the MCMC process using information from previous simulations to improve accuracy and efficiency.
• The approach avoids the need for repeated high-fidelity simulations at every MCMC step by learning and compensating for model reduction errors.
• The framework is designed to be general and applicable to multiphase subsurface flow problems with complex physics.
• Convergence is guaranteed under mild conditions, ensuring reliable inference despite approximation.

Experimental results

Research questions

• RQ1 Can adaptive error modeling reduce computational cost in Bayesian uncertainty quantification without sacrificing statistical accuracy?
• RQ2 How can approximation errors from reduced-order models be systematically modeled and corrected during MCMC sampling?
• RQ3 What conditions ensure convergence of adaptive MCMC when using error models in place of full simulations?
• RQ4 How does the method perform on real-world subsurface flow problems with measured data?
• RQ5 Can the approach maintain high statistical efficiency while drastically reducing the number of high-fidelity simulations?

Key findings

• The proposed method achieves accurate Bayesian inference with significantly reduced computational cost compared to naive sampling with full models.
• The adaptive error model enables sampling from the correct posterior distribution by compensating for discrepancies introduced by model reduction.
• The approach demonstrates good computational and statistical efficiency on both synthetic and real measured data sets.
• The method was successfully applied to a large-scale geothermal reservoir model, showing practical viability for real-world geoscience problems.
• Guaranteed convergence is achieved under simplified conditions, ensuring robustness of the inference process.
• The framework maintains statistical efficiency by minimizing the number of required high-fidelity simulations while preserving posterior accuracy.