Computational Science and Engineering – serie

Many engineering and scientific problems in design, control, and parameter estimation can be formulated as optimization problems that are governed by partial differential equations (PDEs). The complexities of the PDEs - and the requirement for rapid solution - pose significant difficulties. A particularly challenging class of PDE-constrained optimization problems is characterized by the need for real-time solution, i.e., in time scales that are sufficiently rapid to support simulation-based decision making.Real-Time PDE-Constrained Optimization, the first book devoted to real-time optimization for systems governed by PDEs, focuses on new formulations, methods, and algorithms needed to facilitate real-time, PDE-constrained optimization. In addition to presenting state-of-the-art algorithms and formulations, the text illustrates these algorithms with a diverse set of applications that includes problems in the areas of aerodynamics, biology, fluid dynamics, medicine, chemical processes, homeland security, and structural dynamics. Despite difficulties, there is a pressing need to capitalize on continuing advances in computing power to develop optimization methods that will replace simple rule-based decision making with optimized decisions based on complex PDE simulations.

Methods for the numerical simulation of dynamic mathematical models have been the focus of intensive research for well over 60 years, and the demand for better and more efficient methods has grown as the range of applications has increased. Mathematical models involving evolutionary partial differential equations (PDEs) as well as ordinary differential equations (ODEs) arise in many diverse applications such as fluid flow, image processing and computer vision, physics based animation, mechanical systems, relativity, earth sciences, and mathematical finance.This textbook develops, analyzes, and applies numerical methods for evolutionary, or time-dependent, differential problems. Both partial and ordinary differential equations are discussed from a unified viewpoint. The author emphasizes finite difference and finite volume methods, specifically their principled derivation, stability, accuracy, efficient implementation, and practical performance in various fields of science and engineering. Smooth and nonsmooth solutions for hyperbolic PDEs, parabolic type PDEs, and initial value ODEs are treated, and a practical introduction to geometric integration methods is included as well.The author bridges theory and practice by developing algorithms, concepts, and analysis from basic principles while discussing efficiency and performance issues and demonstrating methods through examples and case studies from a variety of application areas.

This book provides a comprehensive introduction to the mathematical methodology of parameter continuation, the computational analysis of families of solutions to nonlinear mathematical equations. It develops a systematic formalism for constructing abstract representations of continuation problems and for implementing these in an existing computational platform.Recipes for Continuation:Lends equal importance to theoretical rigor, algorithm development, and software engineering. Demonstrates the use of fully developed toolbox templates for single- and multisegment boundary-value problems to the analysis of periodic orbits in smooth and hybrid dynamical systems, quasi-periodic invariant tori, and homoclinic and heteroclinic connecting orbits between equilibria and/or periodic orbits. Shows the use of vectorization for optimal computational efficiency, an object-oriented paradigm for the modular construction of continuation problems, and adaptive discretization algorithms for guaranteed bounds on estimated errors. Contains extensive and fully worked examples that illustrate the application of the MATLAB-based Computational Continuation Core (COCO) to problems from recent research literature that are relevant to dynamical system models from mechanics, electronics, biology, economics, and neuroscience.

This book provides an introduction to representative nonrelativistic quantum control problems and their theoretical analysis and solution via modern computational techniques. The quantum theory framework is based on the Schrödinger picture, and the optimization theory, which focuses on functional spaces, is based on the Lagrange formalism. The computational techniques represent recent developments that have resulted from combining modern numerical techniques for quantum evolutionary equations with sophisticated optimization schemes. Both finite and infinite-dimensional models are discussed, including the three-level Lambda system arising in quantum optics, multispin systems in NMR, a charged particle in a well potential, Bose–Einstein condensates, multiparticle spin systems, and multiparticle models in the time-dependent density functional framework.This self-contained book covers the formulation, analysis, and numerical solution of quantum control problems and bridges scientific computing, optimal control and exact controllability, optimization with differential models, and the sciences and engineering that require quantum control methods.

This book provides an overview of global sensitivity analysis methods and algorithms, including their theoretical basis and mathematical properties. The authors use a practical point of view and real case studies as well as numerous examples, and applications of the different approaches are illustrated throughout using R code to explain their usage and usefulness in practice. Basics and Trends in Sensitivity Analysis: Theory and Practice in R covers a lot of material, including theoretical aspects of Sobol’ indices as well as sampling-based formulas, spectral methods, and metamodel-based approaches for estimation purposes; screening techniques devoted to identifying influential and noninfluential inputs; variance-based measures when model inputs are statistically dependent (and several other approaches that go beyond variance-based sensitivity measures); and a case study in R related to a COVID-19 epidemic model where the full workflow of sensitivity analysis combining several techniques is presented.This book is intended for engineers, researchers, and undergraduate students who use complex numerical models and have an interest in sensitivity analysis techniques and is appropriate for anyone with a solid mathematical background in basic statistical and probability theories who develops and uses numerical models in all scientific and engineering domains.

Computational methods are an integral part of most scientific disciplines, and a rudimentary understanding of their potential and limitations is essential for any scientist or engineer. This textbook introduces computational science through a set of methods and algorithms with the aim of familiarizing the reader with the field's theoretical foundations and providing the practical skills to use and develop computational methods. Methods in Computational Scienceextends the classical syllabus with new material, including high performance computing, adjoint methods, machine learning, randomized algorithms, and quantum computing,is centered around a set of fundamental algorithms presented in the form of pseudocode,presents theoretical material alongside several examples and exercises, andprovides Python implementations of many key algorithms.Methods in Computational Science is a textbook for computer science and data science students at the advanced undergraduate and graduate level. It is appropriate for the following courses: Advanced Numerical Analysis, Special Topics on Numerical Analysis, Topics on Data Science, Topics on Numerical Optimization, and Topics on Approximation Theory. Because the text is self-contained, it can also be used to support continuous learning for practicing mathematicians, data scientists, computer scientists, and engineers in the field of computational science.

Over seventy years ago, Richard Bellman coined the term "the curse of dimensionality" to describe phenomena and computational challenges that arise in high dimensions. These challenges, in tandem with the ubiquity of high-dimensional functions in real-world applications, have led to a lengthy, focused research effort on high-dimensional approximation—that is, the development of methods for approximating functions of many variables accurately and efficiently from data. This book provides an in-depth treatment of one of the latest installments in this long and ongoing story: sparse polynomial approximation methods. These methods have emerged as useful tools for various high-dimensional approximation tasks arising in a range of applications in computational science and engineering. It begins with a comprehensive overview of best s-term polynomial approximation theory for holomorphic, high-dimensional functions, as well as a detailed survey of applications to parametric differential equations. It then describes methods for computing sparse polynomial approximations, focusing on least squares and compressed sensing techniques.Sparse Polynomial Approximation of High-Dimensional Functions presents the first comprehensive and unified treatment of polynomial approximation techniques that can mitigate the curse of dimensionality in high-dimensional approximation, including least squares and compressed sensing. It develops main concepts in a mathematically rigorous manner, with full proofs given wherever possible, and it contains many numerical examples, each accompanied by downloadable code. The authors provide an extensive bibliography of over 350 relevant references, with an additional annotated bibliography available on the book's companion website (www.sparse-hd-book.com).This text is aimed at graduate students, postdoctoral fellows, and researchers in mathematics, computer science, and engineering who are interested in high-dimensional polynomial approximation techniques.

This textbook describes the mathematical principles of the finite element method, a technique that turns a (linear) partial differential equation into a discrete linear system, often amenable to fast linear algebra. Reflecting the author's decade of experience in the field, Mathematical Foundations of Finite Elements and Iterative Solvers examines the crucial interplay between analysis, discretization, and computations in modern numerical analysis; furthermore, it recounts historical developments leading to current state-of-the-art techniques. While self-contained, this textbook provides a clear and in-depth discussion of several topics, including elliptic problems, continuous Galerkin methods, iterative solvers, advection-diffusion problems, and saddle point problems.Accessible to readers with a beginning background in functional analysis and linear algebra, this text can be used in graduate-level courses on advanced numerical analysis, data science, numerical optimization, and approximation theory. Professionals in numerical analysis and finite element methods will also find the book of interest.

Reduced order modeling is an important, growing field in computational science and engineering, and this is the first book to address the subject in relation to computational fluid dynamics. It focuses on complex parametrization of shapes for their optimization and includes recent developments in advanced topics such as turbulence, stability of flows, inverse problems, optimization, and flow control, as well as applications.This book will be of interest to researchers and graduate students in the field of reduced order modeling.

This book discusses the foundations of the mathematical theory of finite element methods. The focus is on the concept of discrete stability and the exact sequence conforming elements and covers both coercive and non-coercive problems. Following a historical path of development, the author covers the Ritz and the Galerkin methods to Mikhlin's theory, the Lax-Milgram theorem and Cea's lemma to the Babuska theorem and Brezzi's theory, and finishes with an introduction to the discontinuous Petrov-Galerkin method with optimal test functions.Intended for graduate students in computational science, engineering and mathematics programs, Mathematical Theory of Finite Elements is also appropriate for graduate mathematics and mathematically-oriented engineering students. The book is appropriate for courses in real analysis, functional analysis, energy (Sobolev) spaces, and Hilbert space methods for PDEs.

Measure theory and measure-theoretic probability are fascinating subjects. Proofs describing profound ways to reason lead to results that are frequently startling, beautiful, and useful. Measure theory and probability also play roles in the development of pure and applied mathematics, statistics, engineering, physics, and finance. Indeed, it is difficult to overstate their importance in the quantitative disciplines. This book traces an eclectic path through the fundamentals of the topic to make the material accessible to a broad range of students.A Ramble through Probability: How I Learned to Stop Worrying and Love Measure Theory brings together the key elements and applications in a unified presentation aimed at developing intuition; contains an extensive collection of examples that illustrate, explain, and apply the theories; and is supplemented with videos containing commentary and explanations of select proofs on an ancillary website.

This comprehensive book focuses on numerical methods for approximating solutions to partial differential equations. Intended as a broad survey of methods, the aim is to introduce readers to the central concepts of various families of discretizations and solution algorithm and lay the foundation needed to understand more advanced material, The book is divided into four parts: Part I covers basic background on PDEs and numerical methods,Part II introduces the three main classes of numerical methods for PDEs that are the book’s focus (finite-difference, finite-element, and finite-volume methods),Part III discusses linear solvers and finite-element and finite-volume methods at a more advanced level, andPart IV presents further high-level topics on discretizations and solvers.The authors include over one hundred well-established definitions, theorems, corollaries, and lemmas, and summaries of and reference to in-depth treatments of more advanced mathematics when needed.Audience This book is intended for advanced undergraduate/first-year graduate and advanced graduate students in applied math, as well as students in science and engineering disciplines. It is also appropriate for researchers in the field of scientific computing.Chapters are designed to be stand-alone, allowing distinct paths through the text, making it appropriate for both single-semester and multisemester courses. It is appropriate for courses on numerical methods for PDEs and numerical linear algebra.

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.To deepen the reader’s understanding, the authors provide many applications from diverse areas of applied sciences, such as resource allocation, line fitting, graph coloring, the traveling salesman problem, game theory, and network flows. The book also includes more than 180 exercises, most of them with partial answers and about 70 with complete solutions, as well as a continuous illustration of the theory through examples and exercises.A First Course in Linear Optimization is intended to be read cover to cover and requires only a first course in linear algebra as a prerequisite. Its 13 chapters can be used as lecture notes for a first course in linear optimization.AudienceThis book is for a first undergraduate course in linear optimization, such as linear programming, linear optimization, and operations research. It is appropriate for students in operations research, mathematics, economics, and industrial engineering, as well as those studying computer science and engineering disciplines.