Accelerated Multilevel Monte Carlo With Kernel‐Based Smoothing and Latinized Stratification

Abstract

Heterogeneity and a paucity of measurements of key material properties undermine the veracity of quantitative predictions of subsurface flow and transport. For such model forecasts to be useful as a management tool, they must be accompanied by computationally expensive uncertainty quantification, which yields confidence intervals, probability of exceedance, and so forth. We design and implement novel multilevel Monte Carlo (MLMC) algorithms that accelerate estimation of the cumulative distribution functions (CDFs) of quantities of interest, for example, water breakthrough time or oil production rate. Compared to standard non-smoothed MLMC, the new estimators achieve a significant variance reduction at each discretization level by smoothing the indicator function with a Gaussian kernel or replacing standard Monte Carlo (MC) with the recently developed hierarchical Latinized stratified sampling (HLSS). After validating the kernel-smoothed MLMC and HLSS-enhanced MLMC methods on a single-phase flow test bed, we demonstrate that they are orders of magnitude faster than standard MC for estimating the CDF of breakthrough times in multiphase flow problems.

Transfer Learning on Multi-Fidelity Data

“…At the optimal mix of 573 LFS and 100 HFS, two of the five experiments, the CNN trained on 12 hours of these multi-fidelity data has lower RMSE than the mean RMSE of the CNN trained on 79 hours of the HFS data. This LFS/HFS ratio lies near the range, 1.5 − 5.5, suggested for multilevel Monte Carlo (Taverniers et al, 2020). For the datageneration budget of 12 hours, a mix dominated by the LFS data results in a CNN whose RMSE on test data exceeds 1.0, which indicates that the network's last Convolution Transpose 2 layer is not meaningfully trained.…”

“…The relative permeability for the th phase, k r , varies with the phase saturation, k r = k r (S ), in accordance with the Brooks-Corey constitutive model (Corey, 1954). Following Taverniers et al (2020) and many others, we neglect the capillary forces, i.e., assume pressure within the two phases to be equal, P 1 = P 2 ≡ P (x, t); that is a common assumption in applications to reservoir engineering and carbon sequestration. The computational domain D is a 150 m × 150 m square (Fig.…”

“…All the model parameters, except for the intrinsic permeability k(x), are assumed to be constant and known with certainty. The uncertain permeability k(x) is modeled as a second-order stationary random field, such that Y p = 31 terms to represent Y (x) (Taverniers et al, 2020). A representative realization of the resulting permeability field is shown in Fig.…”

“…Lower-fidelity solutions of (1), obtained on coarser meshes on which appropriately homogenized inputs θ hom are defined, are cheaper to compute but less accurate. We train a CNN on a mixture of these multi-fidelity data, using the theoretical results for multilevel Monte Carlo (MLMC) (Giles, 2008;Heinrich, 1998Heinrich, , 2001Taverniers et al, 2020) to guide our choice of the numbers of solutions u(x, t) of each kind. Similar to MLMC (Müller et al, 2013;Peherstorfer, 2019), the varying fidelity (aka "levels") of predictions of u can be achieved not only by solving (1) on different meshes, but also by replacing (1) with its cheaper-to-compute counterparts.…”

Nonparametric, data-based kernel interpolation for particle-tracking simulations and kernel density estimation

“…In many applications, discrete samples of a continuous, and potentially complex, random process are generated as output, even though a continuous solution is desired. Some examples are given by particle-tracking of passive solute transport (e.g., Fernàndez-Garcia & Sanchez-Vila, 2011;Pedretti & Fernàndez-Garcia, 2013;Siirila-Woodburn et al, 2015;Carrel et al, 2018), reactive particle transport (e.g., Ding et al, 2012Ding et al, , 2017Schmidt et al, 2017;Sole-Mari et al, 2017;Sole-Mari et al, 2019;Sole-Mari & Fernàndez-Garcia, 2018;Benson et al, 2019;Perez et al, 2019;Engdahl et al, 2017Engdahl et al, , 2019, and Monte Carlo simulation (e.g., Taverniers et al, 2020). A long history of statistical estimation has sought to best fit some continuous density function to a sequence of random samples, including maximum likelihood estimation (Brockwell & Davis, 2016) and kernel density estimation (Silverman, 1986).…”