Michael Presho

Mode decomposition methods for flows in high-contrast porous media. Global-local approach

We apply dynamic mode decomposition (DMD) and proper orthogonal decomposition (POD) methods to flows in highly-heterogeneous porous media to extract the dominant coherent structures and derive reduced-order models via Galerkin projection. Permeability fields with high contrast are considered to investigate the capability of these techniques to capture the main flow features and forecast the flow evolution within a certain accuracy. A DMD-based approach shows a better predictive capability due to its ability to accurately extract the information relevant to long-time dynamics, in particular, the slowly-decaying eigenmodes corresponding to largest eigenvalues. Our study enables a better understanding of the strengths and weaknesses of the applicability of these techniques for flows in high-contrast porous media. Furthermore, we discuss the robustness of DMD- and POD-based reduced-order models with respect to variations in initial conditions, permeability fields, and forcing terms.

Generalized Multiscale Finite Element Methods. Nonlinear Elliptic Equations

In this paper we use the Generalized Multiscale Finite Element Method (GMsFEM) framework, introduced in [20], in order to solve nonlinear elliptic equations with high-contrast coefficients. The proposed solution method involves linearizing the equation so that coarse-grid quantities of previous solution iterates can be regarded as auxiliary parameters within the problem formulation. With this convention, we systematically construct respective coarse solution spaces that lend themselves to either continuous Galerkin (CG) or discontinuous Galerkin (DG) global formulations. Here, we use Symmetric Interior Penalty Discontinuous Galerkin approach. Both methods yield a predictable error decline that depends on the respective coarse space dimension, and we illustrate the effectiveness of the CG and DG formulations by offering a variety of numerical examples.

Application of a conservative, generalized multiscale finite element method to flow models

In this paper, we propose a method for the construction of locally conservative flux fields from Generalized Multiscale Finite Element Method (GMsFEM) pressure solutions. The flux values are obtained from an element-based postprocessing procedure in which an independent set of 4x4 linear systems need to be solved. To test the performance of the method we consider two heterogeneous permeability coefficients and couple the resulting fluxes to a two-phase flow model. The increase in accuracy associated with the computation of the GMsFEM pressure solutions is inherited by the postprocessed flux fields and saturation solutions, and is closely correlated to the size of the reduced-order systems. In particular, the addition of more basis functions to the enriched coarse space yields solutions that more accurately capture the behavior of the fine scale model. A number of numerical examples are offered to validate the performance of the method.

Multilevel Markov Chain Monte Carlo Method for High-Contrast Single-Phase Flow Problems

In this paper we propose a general framework for the uncertainty quantification of quantities of interest for high-contrast single-phase flow problems. It is based on the generalized multiscale finite element method (GMsFEM) and multilevel Monte Carlo (MLMC) methods. The former provides a hierarchy of approximations of different resolution, whereas the latter gives an efficient way to estimate quantities of interest using samples on different levels. The number of basis functions in the online GMsFEM stage can be varied to determine the solution resolution and the computational cost, and to efficiently generate samples at different levels. In particular, it is cheap to generate samples on coarse grids but with low resolution, and it is expensive to generate samples on fine grids with high accuracy. By suitably choosing the number of samples at different levels, one can leverage the expensive computation in larger fine-grid spaces toward smaller coarse-grid spaces, while retaining the accuracy of the final Monte Carlo estimate. Further, we describe a multilevel Markov chain Monte Carlo method, which sequentially screens the proposal with different levels of approximations and reduces the number of evaluations required on fine grids, while combining the samples at different levels to arrive at an accurate estimate. The framework seamlessly integrates the multiscale features of the GMsFEM with the multilevel feature of the MLMC methods following the work in [26], and our numerical experiments illustrate its efficiency and accuracy in comparison with standard Monte Carlo estimates.

A Resourceful Splitting Technique with Applications to Deterministic and Stochastic Multiscale Finite Element Methods

In this paper we use a splitting technique to develop new multiscale basis functions for the multiscale finite element method (MsFEM). The multiscale basis functions are iteratively generated using a Greens kernel. The Greens kernel is based on the first differential operator of the splitting. The proposed MsFEM is applied to deterministic elliptic equations and stochastic elliptic equations, and we show that the proposed MsFEM can considerably reduce the dimension of the random parameter space for stochastic problems. By combining the method with sparse grid collocation methods, the need for a prohibitive number of deterministic solves is alleviated. We rigorously analyze the convergence of the proposed method for both the deterministic and stochastic elliptic equations. Computational complexity discussions are also offered to supplement the convergence analysis. A number of numerical results are presented to confirm the theoretical findings.

A Novel Method for Solving Multiscale Elliptic Problems with Randomly Perturbed Data

We propose a method for efficient solution of elliptic problems with multiscale features and randomly perturbed coefficients. We use the multiscale finite element method (MsFEM) as a starting point and derive an algorithm for solving a large number of multiscale problems in parallel. The method is intended to be used within a Monte Carlo framework where solutions corresponding to samples of the randomly perturbed data need to be computed. We show that the proposed method converges to the MsFEM solution in the limit for each individual sample of the data. We also show that the complexity of the method is proportional to one solve using MsFEM (where the fine scale is resolved) plus N (number of samples) solves of linear systems on the coarse scale, as opposed to solving N problems using MsFEM. A set of numerical examples is presented to illustrate the theoretical findings.

Local–global model reduction of parameter-dependent, single-phase flow models via balanced truncation

Abstract In this paper we propose a method for the accurate calculation of output quantities resulting from a parameter-dependent, single-phase flow model. In particular, given a small-dimensional set of inputs (as compared to the fine model), we treat the problem using a combined local–global model reduction technique. The local model reduction is achieved through the use of the Generalized Multiscale Finite Element Method (GMsFEM) where a set of independently calculated basis functions are used in order to construct a suitable coarse approximation space. The multiscale basis function computations are localized to specified coarse subdomains, and follow an offline–online procedure in which a set of eigenvalue problems are used to capture the underlying behavior of the system. Because the offline stage accounts for a one-time preprocessing step, the online coarse space may be cheaply constructed for a given input state. We then apply balanced truncation (BT) to the online coarse system in order to obtain a global reduced-order approximation of the output state. BT recasts the model equation into a systems framework where the input–output mapping may be approximated through the spectral construction of a reduced-order model, and requires the solution of a set of Lyapunov equations. As the Lyapunov equations represent an expensive computation, the efficiency of the proposed method depends on the size of the online coarse space. The combined approach is shown to be flexible with respect to the online space and reduced dimensions, and may be readily modified in order to ensure that the resulting output errors are comparable.

An adapted Petrov–Galerkin multi-scale finite element method for singularly perturbed reaction–diffusion problems

In this paper, we propose the use of an adapted Petrov–Galerkin (PG) multi-scale finite element method for solving the singularly perturbed problem. The multi-scale basis functions that form the function space are constructed from both homogeneous and nonhomogeneous localized problems, which provide more flexibility. These PG multi-scale basis functions are shown to capture the originally perturbed information for the reaction–diffusion model, and reduce the boundary layer errors on graded (non-uniform) coarse meshes. We present the numerical experiment in order to demonstrate that our method acquires stable and convergent results in the , and energy norms. Due to the independent construction of the multi-scale bases, and the demonstrated accuracy by removing the resonance effect, the adapted PG multi-scale method is shown to be a suitable method for solving the singular perturbation problem.