Multiscale Model Reduction and Learning

Many engineering problems have multiscale features. These problems usually require some model reduction since the computational cost of a fine-scale solution is extremely expensive. Existing model reduction methods such as Generalized Multiscale Finite Element Method (GMsFEM) and Non-local multi-continuum approach (NLMC) have shown extensive success in solving multiscale problems especially on various flow simulation problems. However, there are still challenges in developing effective multiscale models for flow in more complicated heterogeneous media. The geometries of domain, coexistence of multiple continuum, and lack of observation data can all give rise to the difficulty of developing the reduced-order model. In this thesis, I will concentrate on the development of novel multiscale methods following the idea of the existing model reduction methods to address such problems. Moreover, deep learning techniques are combined to overcome certain difficulties met along model construction. These proposed models are targeted to tackle specific problems, where the performance is verified both numerically and analytically. For instance, flow simulation within a heterogeneous thin domain is one of such challenging problems. Though homogenization methods are proven to be successful when the media have clear scale separation, that’s not always the case for flow simulation within a capillary system. Using only one basis function in each coarse region can lead to large errors. We thus design a customized GMsFEM instead, which is able to automatically enrich the approximation space and significantly reduce the error. When simulating flow in a fractured vuggy reservoir, on the other hand, I develop a coarse solver under the framework of GMsFEM by combining it with multi-continuum model and Discrete Fracture Model (DFM). Instead of treating the media as a single continuum, I treat the multiscale formation hierarchically and consider it as a coupled system of matrix, fractures and vugs. This allows us to explicitly represent the mass transfers between continuum as well as model the local effects of the discrete fractures. We further investigate how deep learning can facilitate multiscale model construction for nonlinear flow dynamics. Utilizing a multi-layer neural network to approximate the reduced order model, the observed data can be easily incorporated to adjust the model. Deep learning techniques are also used to conduct model reduction. With a soft thresholding operator as an activation function, a novel neural network is proposed which can identify important multiscale features that are crucial in modeling the underlying flow. The forward input-output maps are thus learned in a reduced way. Extensive applications to engineering problems and numerical analysis are presented in supplement of the proposed approaches. It is shown that our proposed methods can significantly advance the computational efficiency and accuracy for multiscale flow simulation in various heterogeneous media

Multiscale simulations for upscaled multi-continuum flows

We consider in this paper a challenging problem of simulating fluid flows, in complex multiscale media possessing multi-continuum background. As an effort to handle this obstacle, model reduction is employed. In \cite{rh2}, homogenization was nicely applied, to find effective coefficients and homogenized equations (for fluid flow pressures) of a dual-continuum system, with new convection terms and negative interaction coefficients. However, some degree of multiscale still remains. This motivates us to propose the generalized multiscale finite element method (GMsFEM), which is coupled with the dual-continuum homogenized equations, toward speeding up the simulation, improving the accuracy as well as clearly representing the interactions between the dual continua. In our paper, globally, each continuum is viewed as a system and connected to the other throughout the domain. We take into consideration the flow transfers between the dual continua and within each continuum itself. Such multiscale flow dynamics are modeled by the GMsFEM, which systematically generates either uncoupled or coupled multiscale basis (to carry the local characteristics to the global ones), via establishing local snapshots and spectral decomposition in the snapshot space. As a result, we will work with a system of two equations coupled with some interaction terms, and each equation describes one of the dual continua on the fine grid. Convergence analysis of the proposed GMsFEM is accompanied with the numerical results, which support the favorable outcomes.Comment: 35 pages, 6 figures, 4 tables, submitted to Journal of Computational and Applied Mathematic

Hierarchical Upscaling and Model Reduction Techniques for Multiscale Dual-continuum Systems

Simulation in media with multiple interacting continua is often challenging due to distinct properties of the continua, multiple scales and high contrast. Thus, some type of model reduction is required. One of the approaches is a multi-continuum technique, where every process in each continuum is modeled separately and an interaction term is added. Direct numerical simulation in multiscale multi-continuum media is very expensive as it requires a large number of degrees of freedom to completely resolve the micro-scale variation. In this work, we present efficient upscaling and model reduction methods for multiscale dual-continuum systems. We first consider the numerical homogenization of a multiscale dual-continuum system where the interaction terms between the continua are scaled as O(1/ε²) where ε is the microscopic scale. Computing the effective coefficients of the homogenized equations can be expensive because one needs to solve local cell problems for a large number of macroscopic points. We develop a hierarchical approach for solving these cell problems at a dense network of macroscopic points with an essentially optimal computation cost. The method employs the fact that neighboring representative volume elements (RVEs) share similar features; and effective properties of the neighboring RVEs are close to each other. The hierarchical approach reduces computation cost by using different levels of resolution for cell problems at different macroscopic points. Solutions of the cell problems which are solved with a higher level of resolution are employed to correct the solutions at neighboring macroscopic points that are computed by approximation spaces with a lower level of resolution. We then consider the case where the interaction terms of the dual-continuum system are scaled as O(1/ε). We derive the homogenized problem that is a dual-continuum system which contains features that are not in the original two scale problem. In particular, the homogenized dual-continuum system contains extra convection terms and negative interaction coefficients while the interaction coefficient between the continua in the original two scale system obtains both positive and negative values. We prove rigorously the homogenization convergence and homogenization convergence rate. Homogenization of dual-continuum system of this type has not been considered before. We present the numerical examples for computing effective coefficients using hierarchical finite element methods. We assume the above mentioned homogenized equation still possess some degree of multiscale and high contrast features caused by channels in the media. This motivates us to develop the generalized multiscale finite element method (GMsFEM) for an upscaled multiscale dual-continuum equations with general convection and interaction terms. GMs- FEM systematically generates either uncoupled or coupled multiscale basis, via establishing local snapshots and spectral decomposition in the snapshot space. Then the global problem is solved in the constructed multiscale space with a reduced dimensional structure. Convergence analysis of the proposed GMsFEM is accompanied with the numerical results, which support the theoretical results

Model Reduction, Bayesian & Deep Learning Approaches for Flows in Fractured Porous Media

Numerical modelling of flow problems in fractured porous media has important applications in many engineering areas, such as unconventional reservoir simulation and nuclear waste disposal. Simulation of the flow problems in porous media is challenging as numerical discretization results in a very fine mesh for capturing the finest scales and high contrast of the physical properties. On the other hand, the effects of fractures are often modelled by multi-continuum models, resulting coupled systems of equations describing the interactive flow of different continua in heterogenous porous media. While multi-continuum models are widely adopted by different applications, for instance, naturally fractured porous media is modelled by dual porosity approach, shale gas production is modelled by the interactive flow of organic matter, inorganic matter and multiscale fractures in a heterogeneous media, and vuggy carbonate reservoir simulation is characterized by the complex interaction between matrix, fractures and vugs, numerical solutions on the fine grid are often prohibitively expensive in these complex multiscale problems. Extensive research effort had been devoted to developing efficient methods for solving multiscale problems at reduced expense, for example, numerical homogenization approaches and multiscale methods, including Multiscale Finite Element Methods, Variational Multiscale Methods, Heterogeneous Multiscale Methods. The common goal of these methods is to construct numerical solvers on the coarse grid, which is typically much coarser than the fine grid which captures all the heterogeneities in the medium properties. In numerical homogenization approaches, effective properties are computed and the global problem is formulated and solved on the coarse grid. However, these approaches are limited to the cases when the medium properties possess scale separation. In this dissertation, we discuss and analyze novel multiscale model reduction techniques with different model problems arising from flows in porous media and numerical discretization techniques, which can be used for obtaining accurate coarse-scale approximations, even in the case of absence of scale separation. On the other end, Bayesian approaches have been developed for forward and inverse problems to address the uncertainties associated with the solution and the variations of the field parameters, and neural networks approaches are proposed for prediction of flow problems. In the dissertation, we also present methodologies of combining model reduction approaches with Bayesian approaches and deep learning approaches for efficient solution sampling and prediction for flow problems in porous media

State Of the Art Report in the fields of numerical analysis and scientific computing. Final version as of 16/02/2020 deliverable D4.1 of the HORIZON 2020 project EURAD.: European Joint Programme on Radioactive Waste Management

Document information Project Acronym EURAD Project Title European Joint Programme on Radioactive Waste Management Project Type European Joint Programme (EJP) EC grant agreement No. 847593 Project starting / end date 1 st June 2019-30 May 2024 Work Package No. 4 Work Package Title Development and Improvement Of NUmerical methods and Tools for modelling coupled processes Work Package Acronym DONUT Deliverable No. 4.

Generalized Multiscale Finite Element Methods for Transport Problems with Heterogeneous Media

Many practical problems are modeled by partial differential equations with highly heterogeneous coefficients. Classical numerical methods for solving these problems typically require very fine computational meshes, and are therefore very expensive to use. In order to solve these problems efficiently, one needs some types of model reduction, which is typically based on upscaling techniques or multiscale methods. In upscaling methods, the heterogeneous coefficient is carefully replaced by an effective medium, so that the system can be solved on a much coarser grid. In multiscale methods, one attempts to represent the solution by some multiscale basis functions. These basis functions are constructed carefully and are usually based on some local cell problems. The purpose is to capture the fine scale properties of the true solution by using a few multiscale basis functions, with the aim of reducing computational costs. There are many existing approaches for multiscale methods, but few works are done about transport problem. And we know there are many applications of transport equation in real life. As such, some efficient model reduction methods are required to handle transport problems. In the dissertation, we consider H(curl)-elliptic problems, transport equations, and Boltzmann equations. We will consider a multiscale method called generalized multiscale finite element method (GMsFEM). We first construct an adaptive multiscale method for solving H(curl)-elliptic problems in highly heterogeneous media. This problem is motivated by electromagnetic wave propagation. And there are few existing works applying upscaling techniques on curl operator. In our method, we will first construct a suitable snapshot space, and a dimensional reduction procedure to identify important modes of the solution. We next develop and analyze an a posteriori error indicator, and the corresponding adaptive algorithm. In addition, we will construct a coupled offline-online adaptive algorithm, which provides an adaptive strategy to the selection of offline and online basis functions. Our theory shows that the convergence is robust with respect to the heterogeneities and contrast of the media. We present several numerical results to illustrate the performance of our method. We next consider solving transport equations. Most of existing multiscale approaches use spatial multiscale basis functions or upscaling, and there are very few works that design spacetime multiscale functions to solve the transport equation on a coarse grid. For the time dependent problems, the use of space-time multiscale basis functions offers several advantages as the spatial and temporal scales are intrinsically coupled. By using the GMsFEM idea with a space-time framework, one obtains a better dimensional reduction taking into account features of the solutions in both space and time. In addition, the time-stepping can be performed using much coarser time step sizes compared to the case when spatial multiscale basis are used. Our scheme is based on space-time snapshot spaces and model reduction using space-time spectral problems derived from the analysis. We give the analysis for the well-posedness and the spectral convergence of our method. We also present some numerical examples to demonstrate the performance of the method. In all examples, we observe a good accuracy with a few basis functions. We finally solve Boltzmann equations, which are used to describe the statistical behavior of a large number of particles driven by the same physics laws. Depending on the media and the particles to be modeled, the equation has slightly different forms. In this dissertation, we investigate a model Boltzmann equation with highly oscillatory media in the small Knudsen number regime, and study the numerical behavior of GMsFEM in the fluid regime when high oscillation in the media presents. The method is divided into the offline and online steps. In the offline step, basis functions are prepared from a snapshot space via a well-designed generalized eigenvalue problem (GEP), and these basis functions are then utilized to patch up for a solution through DG formulation in the online step to incorporate specific boundary and source information. We prove the wellposedness of the method on the Boltzmann equation, and show that the GEP formulation provides a set of optimal basis functions that achieve spectral convergence. Such convergence is independent of the oscillation in the media, or the smallness of the Knudsen number, making it one of the few methods that simultaneously achieve numerical homogenization and asymptotic preserving properties across all scales of oscillations and the Knudsen number