Abstract
A Bayesian method of inference known as “Stein variational gradient descent” was recently implemented for data assimilation problems, under the heading of “mapping particle filter”. In this manuscript, the algorithm is applied to another type of geoscientific inversion problems, namely history matching of petroleum reservoirs. In order to combat the curse of dimensionality, the commonly used Gaussian kernel, which defines the solution space, is replaced by a p-kernel. In addition, the ensemble gradient approximation used in the mapping particle filter is rectified, and the data assimilation experiments are re-run with more relevant settings and comparisons. Our experimental results in data assimilation are rather disappointing. However, the results from the subsurface inverse problem show more promise, especially as regards the use of p-kernels.
Highlights
Bayesian inference in data assimilation (DA) has been researched for several decades and is frequently applied in the petroleum industry today
The ensemble gradient approximation used in the mapping particle filter is rectified, and the data assimilation experiments are re-run with more relevant settings and comparisons
The Stein variational gradient descent (SVGD) algorithm was extended to p-kernels, and we discussed derivative and derivative-free implementations
Summary
Bayesian inference in data assimilation (DA) has been researched for several decades and is frequently applied in the petroleum industry today. Due to the typically vast computational cost of the forward simulation problem, classic Bayesian Monte Carlo methods, such as Markov chain Monte Carlo (MCMC) and importance sampling, are not applicable. Many of the modern derivative-free methods are based on the ensemble Kalman filter (EnKF, Evensen 2004). Previous applications of SVGD includes Bayesian logistic regression (Liu and Wang 2016), training of neural nets (Feng et al 2017), sequential filtering of the Lorenz-63 and -96 models (Pulido and van Leeuwen 2019; Pulido et al 2019), inference on a simple linear and nonlinear partial differential equation (PDE) using a subspace Hessian projection (Chen et al 2019) and a Gauss–Newton formulation (Detommaso et al 2018).
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