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This book is based on a series of 10 lectures given at an NSF-CBMS conference entitled Computational Methods in Optimal Control. The conference took place between July 23 and July 27, 2018, at Jackson State University. The organizers were Tor A. Kwembe, Jun Liu, Yonghua Yan, and Zhenbu Zhang. Talks by the principal speaker, William W. Hager, were complemented by talks given by Vladislav Bukshtynov, Matthias Heinkenschloss, Michael Hinze, Suzanne Lenhart, Dmitriy Leykekhman, and Anil V. Rao. The complementary talks focused on applications arising in partial differential equations and in mathematical biology, and recent software developments. The conference sponsors included the National Science Foundation through DMS award 1743826, the Conference Board of the Mathematical Sciences (CBMS), the Lockheed Martin Corporation, and Nissan. Research results presented in this book were funded by a series of grants from the National Science Foundation, the Office of Naval Research, and the Defense Advanced Research Projects Agency, which are gratefully acknowledged. The software package 𝔾ℝ𝕆𝕌ℙ-𝕀𝕀, introduced during the conference, was used to solve several control problems presented in the book. See https://gpops2.com/ for license arrangements. The optimization package SuiteOPT and its Polyhedral Active Set Algorithm (PASA), which are available from the following website, were used for other applications:

License arrangements are discussed within the software. The MATLAB codes used for either plots or numerical experiments in the book can be downloaded from the following website:

This comprehensive textbook focuses on numerical methods for approximating solutions to partial differential equations (PDEs). The authors present a broad survey of these methods, introducing readers to the central concepts of various families of discretizations and solution algorithms and laying the foundation needed to understand more advanced material. The authors include over 100 well-established definitions, theorems, corollaries, and lemmas and summaries of and references to in-depth treatments of more advanced mathematics when needed.

This self-contained textbook provides the foundations of linear optimization, covering topics in both continuous and discrete linear optimization. It gradually builds the connection between theory, algorithms, and applications so that readers gain a theoretical and algorithmic foundation, familiarity with a variety of applications, and the ability to apply the theory and algorithms to actual problems.

Measure theory and probability are fascinating subjects, with proofs that describe profound ways to reason, leading to results that are frequently startling, beautiful, and useful. Measure theory and probability also played colorful roles in the development of pure and applied mathematics, statistics, engineering, physics, and finance. Indeed, it would be very difficult to overstate the central importance of measure theory and probability in the quantitative disciplines. This book is intended to provide a widely accessible introduction to these subjects.

Uncertainty quantification serves a central role for simulation-based analysis of physical, engineering, and biological applications using mechanistic models. From a broad perspective, the field of uncertainty quantification can be described as the synthesis of mathematical, statistical, and computational theory and methods to quantify uncertainties associated with mechanistic models and their parameters, simulation codes, observed data, and predicted responses for applications whose complexity can preclude sole reliance on sampling-based methods. Hence the field is inherently interdisciplinary and can require the synthesis of theory inherent to considered applications.

Today the conjugate gradient (CG) algorithm is almost always the iterative method of choice for solving linear systems with symmetric positive definite matrices. It was developed independently by M. R. Hestenes in the United States and E. L. Stiefel in Switzerland at the beginning of the 1950s. After a visit of Stiefel in 1951 to the Institute of Numerical Analysis (INA), located on the campus of the University of California at Los Angeles where Hestenes was working, they wrote a joint paper [50] which was published in the December 1952 issue of the Journal of the National Bureau of Standards. For the early history of CG, we refer the reader to [8] and the references therein as well as to [38, 79, 80].

The story of machine learning is one of rigorous success. It is frequently employed by scientists and practitioners around the globe in various areas of application ranging from economics to chemistry, from medicine to engineering, from gaming to astronomy, and from speech processing to computer vision. While the remarkable success of machine learning methods speaks for itself, they are often applied in an ad hoc manner without much care for their mathematical foundation or for their algorithmic intricacies. Therefore, we decided to write this book. Our goal is to provide the necessary background on commonly used machine learning algorithms and to highlight important implementational and numerical details. The book is based on a well-received practical lab course, which we established within the mathematics studies course at the University of Bonn, Germany, in 2017. The course has been taught and successively enhanced each year since then.

This textbook covers various machine learning methods applied to asset and liability management, as well as asset pricing. We shortened the title to Machine Learning for Asset Management and Pricing for practical reasons, but also more fundamental ones. First, we do not give much space to liabilities in this book. It would not render justice to the field of asset and liability management (ALM) to include liabilities in the title. It is, however, important for a student to realize that the comprehensive problem of ALM can be handled (at least in theory) using the same theories and methods as asset management or liability management.

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More than 25 years have passed since the first edition of this book was published in 1996. Least squares and least-norm problems have become more significant with every passing decade, and applications have grown in size, complexity, and variety. More advanced techniques for data acquisition give larger amounts of data to be treated. What counts as a large matrix has gone from dimension 1000 to 10⁶. Hence, iterative methods play an increasingly crucial role for the solution of least squares problems. On top of these changes, methods must be adapted to new generations of multiprocessing hardware.

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This book introduces elements of set-valued analysis, convex analysis, and nonsmooth analysis — which are relatively modern branches of mathematical analysis — and highlights their relevance for and applications to the analysis of dynamical systems, especially those that arise or are of interest in control theory and control engineering.

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It goes without saying that partial differential equation models have been hugely successful in modeling phenomena in a practically endless number of natural and social sciences, engineering, and other settings. However, despite this success, solutions of partial differential equation models have, in some important settings, failed to agree with observations. On the other hand, the nonlocal, integral equation models considered in this book have been shown to more faithfully agree with observations in many settings in which partial differential equations fail to do so. The fundamental difference between partial differential equations and nonlocal models is that for the former, a point interacts only with points within an infinitesimal neighborhood needed to define derivatives, whereas for nonlocal models, points separated by a finite distance interact with each other.

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This book is intended as an introduction, at the advanced undergraduate or beginning graduate level, to the mathematics behind practical computational methods for approximating solutions to common problems arising in calculus, differential equations, and linear algebra. The text has developed out of courses I have taught on these topics over the past 10+ years at the University of Kentucky and Portland State University. These courses have always included a mixture of students majoring in mathematics, computer science, and a variety of disciplines within engineering, and this book is written with such an audience in mind. A heavier emphasis has been put on the theory supporting the numerical methods under consideration than is typical in introductory texts (at the advanced undergraduate level) on these topics, but this has not been done at the expense of actual computations. Theory, implementation, and experimentation are all essential to a proper understanding of the subject, and I have tried to strike a good balance between them in the main text and exercises. Supplementary material, including example code and PDF slides containing many of the figures and tables in the text, can be found at https://bookstore.siam.org/ot198/bonus.

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This is a textbook—as opposed to a research monograph—on optimal transport. It is written for advanced undergraduate students, beginning graduate students, and researchers in applied mathematics as well as in neighboring fields like physics, engineering, computer science, data science, and machine learning who want to understand cornerstone concepts, results, and methods such as Wasserstein distances, Kantorovich duality, Brenier's theorem, or the Sinkhorn algorithm, at other than a superficial level. The book builds up the mathematical theory rigorously and from scratch, aided by intuitive arguments, informal discussion, and carefully selected applications (to economics, many-particle physics, partial differential equations, data science, and machine learning). I have tried to maximize clarity and transparency of key ideas and results, rather than generality of setting or minimality of assumptions. References to some extensions are given in the Notes at the end of the book.

This book is a revised, updated, and augmented version of the out-of-print classic book Finite Element Methods for Navier-Stokes Equations by Girault and Raviart published by Springer in 1986 [217]. The incompressible Navier-Stokes equations model the flow of incompressible Newtonian fluids and are used in many practical applications, including computational fluid dynamics. In addition to the basic theoretical analysis, this book presents a fairly exhaustive treatment of the up-to-date finite element discretizations of incompressible Navier-Stokes equations, and a variety of numerical algorithms used in their computer implementation, complemented with some numerical experiments. It covers the cases of standard and nonstandard boundary conditions and their numerical discretizations via the finite element methods. Both conforming and nonconforming finite elements are examined in detail, as well as their stability or instability. The topic of time-dependent Navier-Stokes equations, which was missing from [217], is now presented in two chapters. In the same spirit as [217], we have tried as much as possible to make this book self-contained, and therefore we have either proved or recalled all the theoretical results required. This book can be used as a textbook for advanced graduate students.

Control design for linear time-invariant (LTI) systems has been a topic of interest for decades. However, new results and developments are continuously being contributed to the open literature as new opportunities and challenges arise from real-world problems. While Proportional Integral Derivative (PID) controllers are the industry standard, designing such controllers still keeps researchers and academicians occupied. One major drawback of PID controllers is that the “derivative” part—critical for the anticipation capabilities of the controller—is quite sensitive to noise in measurements, and therefore additional filtering implementations are needed to securely utilize PID controllers. This issue is one of the challenges faced in the real world: filter design is not a straightforward task; it is laborious and will also limit the bandwidth of the closed-loop system.

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This book grew out of multiple stimulating conversations with Nathan Kutz while I was a postdoc at the University of Washington. It began with joking about taking our favorite dynamical systems textbook, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields by J. Guckenheimer and P. Holmes, and rewriting it chapter by chapter with modern datadriven techniques. Our idea was to highlight how so much of the dynamical systems theory we were taught can now be explicitly implemented using the widely available computational techniques that typically fall under the umbrella of data analysis. Ideas as simple as optimally fitting data to models could lead the reader right back to Guckenheimer and Holmes's textbook since one is now in a position to apply all the pencil-and-paper techniques that have been developed over more than a century of dynamical systems theory. Similarly, finding changes of variable that recast complex dynamics in the form of the phenomenological models that have been examined in detail by not only Guckenheimer and Holmes but also nearly anyone who has taught or published in dynamical systems is now accessible using neural networks. The goal has always been to showcase a suite of computational methods that can be combined with analysis to better understand the increasingly complicated models and datasets that describe our complex world.

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This book is a continuation of my earlier book Spline Functions: Computational Methods published in 2015 by SIAM; see [[Schum15]]. I have written it because a number of important topics were not treated in that book. These include several new types of splines with important computational applications, such as H-splines, splines on T-meshes, and splines on curved triangulations. Moreover, there has been a lot of progress on the use of splines for solving boundary-value problems (BVPs). In [[Schum15]] I have just one chapter on the Ritz-Galerkin method for polygonal domains. This book includes eight chapters on this topic, with a special emphasis on methods for solving BVPs on domains with curve boundaries.

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This book focuses on IMEX methods, with particular emphasis on their application to systems of PDEs. IMEX methods have proven to be highly effective for solving a wide range of evolutionary problems in various contexts, including the kinetic theory of rarefied gases, linear and nonlinear waves, viscoelasticity, multiphase flows, radiation hydrodynamics, traffic flows, and shallow water, among others.