Applying kriging proxies for Markov chain Monte Carlo in reservoir simulation
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ORIGINAL PAPER
Applying kriging proxies for Markov chain Monte Carlo in reservoir simulation Ilya Fursov 1 & Mike Christie 1 & Gabriel Lord 1
# The Author(s) 2020
Abstract One way to quantify the uncertainty in Bayesian inverse problems arising in the engineering domain is to generate samples from the posterior distribution using Markov chain Monte Carlo (MCMC) algorithms. The basic MCMC methods tend to explore the parameter space slowly, which makes them inefficient for practical problems. On the other hand, enhanced MCMC approaches, like Hamiltonian Monte Carlo (HMC), require the gradients from the physical problem simulator, which are often not available. In this case, a feasible option is to use the gradient approximations provided by the surrogate (proxy) models built on the simulator output. In this paper, we consider proxy-aided HMC employing the Gaussian process (kriging) emulator. We overview in detail the different aspects of kriging proxies, the underlying principles of the HMC sampler and its interaction with the proxy model. The proxy-aided HMC algorithm is thoroughly tested in different settings, and applied to three case studies—one toy problem, and two synthetic reservoir simulation models. We address the question of how the sampler performance is affected by the increase of the problem dimension, the use of the gradients in proxy training, the use of proxy-for-the-data and the different approaches to the design points selection. It turns out that applying the proxy model with HMC sampler may be beneficial for relatively small physical models, with around 20 unknown parameters. Such a sampler is shown to outperform both the basic Random Walk Metropolis algorithm, and the HMC algorithm fed by the exact simulator gradients. Keywords Markov chain Monte Carlo . Uncertainty quantification . Gaussian process . Proxy model . Reservoir simulation Mathematics subject classification (2010) 65C40, 60G15, 68U20
1 Introduction: HMC samplers and proxy models One of the common ways to tackle an inverse problem is finding a single model which is calibrated (history matched) to reproduce the observation data. A more advanced approach involves quantification of uncertainties (UQ), e.g. by generating a range of models consistent both with the prior information, and with the observed data, or, strictly speaking, sampling from the target posterior distribution. This is usually done by employing a Markov chain Monte Carlo (MCMC) sampler. Sampling from the posterior target and UQ is more challenging and computationally demanding than the mere
* Ilya Fursov [email protected] 1
Heriot-Watt University, Edinburgh, UK
model calibration, but may be more appropriate, since having the information on the uncertainties of the simulation model allows a higher quality decision making. An MCMC sampler generates a Markov chain—i.e. a sequence of points/models {x i} where a new model x i + 1 is found based only on the current model x i, and does not depend on the preceding models x i − 1, x i − 2, … given x i. A properly desig
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