This chapter discusses the methods of solving PEPS or other two-dimensional tensor network states, including variational optimization and the annealing simulation. The variation optimization determines the local terms by minimizing the ground-state energy. The annealing simulation takes the full or simple update strategy to filter out the ground state through the imaginary time evolution. The nonlinear effect arises in evaluating the derivative of uniform PEPS and is avoided by utilizing automatic differentiation. Both variational optimization and the annealing simulation involve a contraction of double-layer tensor network states. This contraction is the primary technical barrier in the study of PEPS. A nested tensor network approach is introduced to combat this difficulty.

Coarse-graining renormalization aims to reformulate a tensor network model with a coarse-grained one at a larger scale. It has attracted particular attention in recent years because it opens a new avenue to unveil the entanglement structure of a tensor network model under the scaling transformation. This chapter reviews and compares the tensor renormalization group (TRG) and other coarse-graining methods developed in the past two decades. The methods can be divided into two groups according to whether or not the renormalization effect of the environment tensors is incorporated in the optimization of local tensors. The local optimization methods include TRG, HOTRG (a variant of TRG based on the higher-order singular value decomposition), tensor network renormalization (TNR), and loop-TNR. The global optimization methods include the second renormalized TRG and HOTRG, referred to as SRG and HOSRG, respectively. Among all these coarse-graining methods, HOTRG and HOSRG are the only two that can be readily extended and efficiently applied to three-dimensional classical or two-dimensional quantum lattice models.

The purpose of this chapter is to introduce the diagrammatic representation of tensors and tensor network states and discuss some basic properties or formulas of matrices or tensors used in various renormalization group methods. It includes, for example, the singular value decomposition and the polar decomposition of matrices, the higher-order singular value decomposition of tensors, and the automatic differentiation and its backpropagation scheme. Several Trotter-Suzuki decomposition formulas of non-commutate operators are also introduced.

Chapter 3 starts out with a physics motivation, as well as a mathematical statement of the problem that will be tackled in later sections. After a brief discussion of analytical differentiation, the bulk of the chapter is devoted to increasingly better finite-difference approximations, like the forward difference and the central difference. These are explicitly derived using Taylor expansions, and also applied to second derivatives and to points on a grid. A section introduces the useful tool of Richardson extrapolation, which reappears in later chapters. The chapter also includes an original discussion of automatic differentiation, which is built up from the concept of dual numbers. The chapter is rounded out by a physics project, which studies the kinetic energy in single-particle quantum mechanics, and a problem set. The physics project involves different wave functions and provides the groundwork for the project in the integrals chapter.

First derivatives (gradients) are needed for most of the algorithms described in the book. Here, we describe how these gradients can be computed efficiently for functions that have the form of arising in deep learning. The reverse mode of automatic differentiation, often called “back-propagation” in the machine learning community, is described for several problems with nested-composite and progressive structure that arises in neural network training. We provide another perspective on these techniques, based on a constrained optimization formulation and optimality conditions for this formulation.

The ad-core module in MRST offers an object-oriented framework for rapid prototyping of new reservoir simulators based on automatic differentiation (AD-OO). The framework simplifies the task of changing and extending existing simulation models in MRST or implementing brand new ones. The MRST textbook (Lie, Cambridge University Press, 2019, Chapter 12) presents a model hierarchy for the black-oil equations, discretized by a standard fully implicit method, and describes how to (automatically) select time steps and configure linear and nonlinear solvers. Herein, we present a further modularization of the AD-OO framework that aims to simplify the implementation of more complex flow models and other types of discretizations and solution strategies. To this end, we view the reservoir simulator as a graph of functional relationships and their dependencies and introduce the new concept of so-called state functions to define these functional relationships and compute discrete quantities required for the linearized governing equations. Using the graph perspective, it is relatively simple to not only visualize and understand the data flow of highly complex reservoir simulators, but also replace components of the graph and/or extend the graph with new branches as needed. The result is a versatile family of reservoir simulators that can easily be configured to run different types of multiphase, multicomponent models and at the same time support a number of different spatial and temporal discretizations. The state-function concept also has a built-in compute cache that helps you to systematically eliminate redundant function evaluations. The chapter explains the new concept in detail and exemplifies its use by showcasing implicit, explicit, and adaptive-implicit versions of the same physical processes. We also demonstrate the use of consistent and high-resolution schemes to improve simulation accuracy. Applications to complex flow physics (EOR models, compositional flow, fractured reservoirs) are discussed in other chapters.

Many different factors contribute to elapsed runtime of reservoir simulators. Once the cases become larger and more complex, the required wait time for results can become prohibitive. This chapter discusses three features recently introduced into the MRST AD-OO framework to make simulation of large cases more efficient. By analysing the sparsity pattern of the Jacobians for some of the most common operations involved in computing residual flow equations, we have developed different implementations of automatic differentiation that offer better memory usage and requires fewer floating point operations. Using these so-called AD backends ensures (much) faster assembly of linearized systems. Likewise, these systems can be solved much faster by utilizing external packages for linear algebra; herein, primarily represented by the AMGCL header-only C++ library for solving large sparse linear systems with algebraic multigrid (AMG) methods. Last, but not least, the new ``packed problem'' format simplifies the management of multiple simulation cases and enables automatic restart of simulations and an ability for early inspection of results from large batch simulations. Altogether, these features are essential if you are working with bigger simulation models and want timely results that persist across MATLAB sessions.

This chapter explains how you can discretize the basic equations for single-phase, compressible flow by use of the discrete differential and averaging operators introduced in Chapter 4. These operators enable you to implement the flow equations in a compact form similar to the continuous mathematical description. By using automatic differentiation, you can automatically linearize and assemble the corresponding linear system without having to explicitly derive and implement expressions for partial derivatives in the Jacobian matrix. The combination of discrete operators and automatic differentiation with a flexible grid structure, a highly vectorized and interactive scripting language, and a powerful graphical environment, is the main reason MRST has proven to be an efficient tool for developing new proof-of-concept codes. To demonstrate this, we first develop a compact solver for compressible flow, and then extend the basic single-phase model to include pressure-dependent viscosity, non-Newton fluid behavior, and temperature effects.

The Appendix gives you a quick introduction to MRST. We start by explaining how to download, install, and get started using the software. We also discuss the release policy, terms of use, and how the software has been organized into a small core that offers basic functionality and a large set of add-on modules that implement specific solvers, simulators, and tools. We briefly review all modules that are part of the 2018a release. The final two sections discuss rapid prototyping using MATLAB and MRST, present some powerful constructs you can use to speed up your MATLAB code, and explain the key ideas of automatic differentiation and how it has been implemented in MRST. Altogether, this Appendix should provide you with enough knowledge of MRST so that you can effectively use the software to further your understanding of the material presented in the main part of the book.