Browse Books

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.

Excerpt

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.

Excerpt

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.

Excerpt

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.

Excerpt

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.

Excerpt

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.

Excerpt

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.

Excerpt

This new book focuses on computational methods developed in the last ten years that make use of splines to approximate functions and data and to solve boundary-value problems. The first half of the book works with bivariate spaces of splines defined on H-triangulations, T-meshes, and curved triangulations. Trivariate tensor-product splines and splines on tetrahedral partitions are also discussed. The second half of the book makes use of these spaces to solve boundary-value problems, with a special emphasis on elliptic PDEs defined on curved domains.

Excerpt

This book is intended for applied mathematicians, scientists, and engineers who use or are interested in learning about IMEX schemes. Readers should have some background in numerical methods for ODE systems and basic finite difference and finite volume discretization of evolutionary PDEs, along with a basic understanding of the relevant mathematical models. The book is suitable for students who have had a basic course in numerical analysis and are familiar with partial differential equations.

Excerpt

This book project arose out of the collaboration on Hidden Symmetries and Fusion Energy¹ funded by the Simons Foundation, initially between 2018 and 2022 and then for a renewed period from 2022 to 2025. This collaboration was formed to foster interactions among experts in numerical optimization, dynamical systems, analysis of partial differential equations, and plasma physics in order to find stellarator configurations with hidden symmetries. Given the diverse backgrounds of the participants, establishing a common language was the first challenge to tackle. The original idea was to gather a collection of definitions of fundamental concepts relevant to stellarator design, in a form similar to a dictionary. However, due to the complexity of both the phenomena at play and the various models describing them, the need for a different format quickly became clear. The final result is closer to an introduction to mathematical modeling for stellarator design, and the end goal is twofold: making these topics accessible to a broader audience of scientists, and stimulating new contributions to the field of stellarator research.

The biologist J. B. S. Haldane, when asked what we can learn about the Creator by examining the world, is said to have replied “an inordinate fondness for beetles” [705]. Today's biologists could be forgiven for pointing to an inordinate fondness for networks. Networks are ubiquitous in the life sciences: examples include genetic regulatory networks, neural circuitry, ecological food webs, phylogenetic trees, and epidemic networks. Networks are also common in many other branches of science, including physics, chemistry, computer science, electrical and electronic engineering, psychology, and sociology. In recent years there has been an explosion of interest in network-based modeling, and the research literature, including both theory and applications, now extends to many thousands of papers. The questions addressed and the techniques involved are extremely diverse, reflecting the broad range of backgrounds and interests of researchers.

This second edition of Matrix Analysis and Applied Linear Algebra differs substantially from the first edition in that this edition has been completely rewritten to include reformulations, extensions, and pedagogical enhancements. The goal in preparing this edition was to create an easily readable and flexible textbook that is adaptable for a single semester course or a more complete two-semester course. The following features are some of the characteristics of this edition.