Credit Scoring and Its Applications is recognized as the bible of credit scoring. It contains a comprehensive review of the objectives, methods, and practical implementation of credit and behavioral scoring. The authors review principles of the statistical and operations research methods used in building scorecards, as well as the advantages and disadvantages of each approach. The book contains a description of practical problems encountered in building, using, and monitoring scorecards and examines some of the country-specific issues in bankruptcy, equal opportunities, and privacy legislation. It contains a discussion of economic theories of consumers' use of credit, and readers will gain an understanding of what lending institutions seek to achieve by using credit scoring and the changes in their objectives. New to the second edition are: lessons that can be learned for operations research model building from the global financial crisis; current applications of scoring; discussions on the Basel Accords and their requirements for scoring; new methods for scorecard building and new expanded sections on ways of measuring scorecard performance; and survival analysis for credit scoring.
 Other unique features include methods of monitoring scorecards and deciding when to update them, as well as different applications of scoring, including direct marketing, profit scoring, tax inspection, prisoner release, and payment of fines.
 Audience: This book is an indispensable reference to credit analysts, scorecard developers, and credit risk managers employed by lending companies such as banks, finance houses, mortgage companies, credit card companies, retailers, mail order firms, utility companies, and insurance companies. Graduate students in mathematical finance, industrial mathematics, and statistics and senior undergraduate students in mathematics, statistics, and quantitative business studies courses will find this a most useful textbook.
 Contents: Chapter 1: The History and Philosophy of Credit Scoring; Chapter 2: The Practice of Credit Scoring; Chapter 3: Common Methods for Building Scorecards; Chapter 4: Alternative Methods of Building Scorecards; Chapter 5: Survival Methods; Chapter 6: Managing the Data in Scorecard Development; Chapter 7: Behavioral Scoring; Chapter 8: Measuring Scorecard Performance; Chapter 9: Implementation and Areas of Application; Chapter 10: Economic Issues; Chapter 11: Capital Requirements and the Basel Accords; Chapter 12: Securitization and the Subprime Mortgage Crisis; Chapter 13: Variable Pricing and Risk-Based Pricing.

Credit Scoring and Its Applications (Mathematics In Industry)

Lyn Thomas

Jonathan Crook

David Edelman

SIAM-Society for Industrial & Applied Mathematics

Numerous applications, including computational optimization and fluid dynamics, give rise to block linear systems of equations said to have the quasi-definite structure. In practical situations, the size or density of those systems can preclude a factorization approach, leaving only iterative methods as the solution technique. Known iterative methods, however, are not specifically designed to take advantage of the quasi-definite structure.
 This book discusses the connection between quasi-definite systems and linear least-squares problems, the most common and best understood problems in applied mathematics, and explains how quasi-definite systems can be solved using tailored iterative methods for linear least squares (with half as much work!). To encourage researchers and students to use the software, it is provided in MATLAB, Python, and Julia.
 The authors provide a concise account of the most well-known methods for symmetric systems and least-squares problems, research-level advances in the solution of problems with specific illustrations in optimization and fluid dynamics, and a website that hosts software in three languages.
 Audience: This book is intended for researchers and advanced graduate students in computational optimization, computational fluid dynamics, computational linear algebra, data assimilation, and virtually any computational field in which saddle-point systems occur. The software should appeal to all practitioners, even those not technically inclined.
 Contents: List of Algorithms; List of Theorems; Preface; Chapter 1: Introduction; Chapter 2: Preliminaries; Chapter 3: Overview of Existing Direct and Iterative Methods; Chapter 4: Fundamental Processes; Chapter 5: Iterative Methods Based on Reduced Equations; Chapter 6: Full-Space Iterative Methods; Chapter 7: Software and Numerical Experiments; Chapter 8: Discussion and Open Questions; Bibliography; Index.

Iterative Solution of Symmetric Quasi-Definite Linear Systems

Dominique Orban

Mario Arioli

Many physical, chemical, biomedical, and technical processes can be described by partial differential equations or dynamical systems. In spite of increasing computational capacities, many problems are of such high complexity that they are solvable only with severe simplifications, and the design of efficient numerical schemes remains a central research challenge. This book presents a tutorial introduction to recent developments in mathematical methods for model reduction and approximation of complex systems.
 Model Reduction and Approximation: Theory and Algorithms contains three parts that cover (I) sampling-based methods, such as the reduced basis method and proper orthogonal decomposition, (II) approximation of high-dimensional problems by low-rank tensor techniques, and (III) system-theoretic methods, such as balanced truncation, interpolatory methods, and the Loewner framework. The book is tutorial in nature, giving an accessible introduction to state-of-the-art model reduction and approximation methods; it covers a wide range of methods drawn from typically distinct communities (sampling based, tensor based, system-theoretic).
 Audience: This book is intended for researchers interested in model reduction and approximation, particularly graduate students and young researchers.
 Contents: Part I: Sampling-Based Methods; Chapter 1: Proper Orthogonal Decomposition for Linear-Quadratic Optimal Control; Chapter 2: A Tutorial on Reduced Basis Methods; Chapter 3: The Theoretical Foundation of Reduced Basis Methods; Part II: Tensor-Based Methods; Chapter 4: Low-Rank Methods for High-Dimensional Approximation and Model Order Reduction; Chapter 5: Model Reduction for High-Dimensional Parametric Problems by Tensor Techniques; Part III: System-Theoretic Methods; Chapter 6: Model Order Reduction Based on Systems Balancing; Chapter 7: Model Reduction by Rational Interpolation; Chapter 8: The Loewner Framework for Model Reduction; Chapter 9: Comparison of Methods for Parametric Model Order Reduction.

Model Reduction and Approximation: Theory and Algorithms

Science & Math

Albert Cohen

Peter Benner

Mario Ohlberger

Karen E. Wilcox

Mathematics

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 Schrodinger 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.
 Audience: This book is intended for mathematicians working on ODE/PDE control and optimization problems and the numerical analysis of differential equations; physicists; chemists; and engineers who focus on quantum control problems. It is suitable for advanced courses on ODE/PDE quantum control problems and provides extensively elaborated problems that help the reader develop insight into the main ideas and techniques of quantum control problems. It is also suitable for advanced graduate students and scientists of mathematics, the natural sciences, and engineering.
 Contents: Chapter 1: Introduction; Chapter 2: Quantum mechanics and the Schrödinger equation; Chapter 3: Optimal control theory for quantum systems; Chapter 4: Controllability of quantum systems; Chapter 5: Discretization schemes; Chapter 6: Numerical optimization methods; Chapter 7: Application to quantum control problems; Appendix.

Formulation and Numerical Solution of Quantum Control Problems

Alfio Borzi

Gabriele Ciaramella

Martin Sprengel

Inverse problems are found in many applications, such as medical imaging, engineering, astronomy, and geophysics, among others. To solve an inverse problem is to recover an object from noisy, usually indirect observations. Solutions to inverse problems are subject to many potential sources of error introduced by approximate mathematical models, regularization methods, numerical approximations for efficient computations, noisy data, and limitations in the number of observations; thus it is important to include an assessment of the uncertainties as part of the solution. Such assessment is interdisciplinary by nature, as it requires, in addition to knowledge of the particular application, methods from applied mathematics, probability, and statistics.
 This book bridges applied mathematics and statistics by providing a basic introduction to probability and statistics for uncertainty quantification in the context of inverse problems, as well as an introduction to statistical regularization of inverse problems. The author covers basic statistical inference, introduces the framework of ill-posed inverse problems, and explains statistical questions that arise in their applications.
 An Introduction to Data Analysis and Uncertainty Quantification for Inverse Problems includes:many examples that explain techniques which are useful to address general problems arising in uncertainty quantification; Bayesian and non-Bayesian statistical methods and discussions of their complementary roles; and analysis of a real data set to illustrate the methodology covered throughout the book.
 Audience: This book is intended for senior undergraduates and beginning graduate students in mathematics, engineering and physical sciences. The material spans from undergraduate statistics and probability to data analysis for inverse problems and probability distributions on infinite-dimensional spaces. It is also intended for researchers working on inverse problems and uncertainty quantification in geophysics, astrophysics, physics, and engineering. Because the statistical and probability methods covered have applications beyond inverse problems, the book may also be of interest to those people working in data science or in other applications of uncertainty quantification.
 Contents: Chapter 1: An introduction to inverse problems; Chapter 2: A primer on statistical methods; Chapter 3: Applications to inverse problems I; Chapter 4: Applications to inverse problems II; Chapter 5: A nonlinear parameter estimation problem; Appendix A: Some results from analysis; Appendix B: Conditional probability and expectation.

An Introduction to Data Analysis and Uncertainty Quantification for Inverse Problems

Luis Tenorio

If mathematical modeling is the process of turning real phenomena into mathematical abstractions, then numerical computation is largely about the transformation from abstract mathematics to concrete reality. Many science and engineering disciplines have long benefited from the tremendous value of the correspondence between quantitative information and mathematical manipulation. -from the Preface Fundamentals of Numerical Computation is an advanced undergraduate-level introduction to the mathematics and use of algorithms for the fundamental problems of numerical computation: linear algebra, finding roots, approximating data and functions, and solving differential equations. The book is organized with simpler methods in the first half and more advanced methods in the second half, allowing use for either a single course or a sequence of two courses. The authors take readers from basic to advanced methods, illustrating them with over 200 self-contained MATLAB functions and examples designed for those with no prior MATLAB experience. Although the text provides many examples, exercises, and illustrations, the aim of the authors is not to provide a cookbook per se, but rather an exploration of the principles of cooking.
 Professors Driscoll and Braun have developed an online resource that includes well-tested materials related to every chapter. Among these materials are lecture-related slides and videos, ideas for student projects, laboratory exercises, computational examples and scripts, and all the functions presented in the book.
 The authors describe their goals and intentions for the book in specific terms: Our guiding point of view is that the student users of the book are far more likely to apply and remix fundamental computational algorithms than to reinvent them. We want to prepare them for more advanced courses in numerical analysis, but we place a higher priority on conveying foundational skills in scientific computation. Accordingly, we emphasize knowing how to cast a problem into a form that can be coded and solved, tradeoffs between different methods for a problem, and assessing the correctness and convergence of the results. We do not, however, make much of an issue of optimizing the speed of implementations, except when orders of magnitude are in play, nor do we discuss parallelism.
 Above all, we emphasize linear algebra. Linear algebra is the lingua franca of scientific computing a fact in no small measure established and affirmed by MATLAB over more than three decades."
 Audience: Fundamentals of Numerical Computation is intended for advanced undergraduates in math, applied math, engineering, or science disciplines, as well as for researchers and professionals looking for an introduction to a subject they missed or overlooked in their education.
 Contents: Prologue: Two Imperfect Ideas; Chapter 1: Numbers, Problems, and Algorithms; Chapter 2: Square Linear Systems; Chapter 3: Overdetermined Linear Systems; Chapter 4: Roots of Nonlinear Equations; Chapter 5: Piecewise Interpolation and Calculus; Chapter 6: Initial-Value Problems for ODEs; Chapter 7: Matrix Analysis; Chapter 8: Krylov Methods in Linear Algebra; Chapter 9: Global Function Approximation; Chapter 10: Boundary-Value Problems; Chapter 11: Diffusion Equations; Chapter 12: Advection Equations; Chapter 13: Two-Dimensional Problems; Appendix A: Review of Linear Algebra.

Fundamentals of Numerical Computation

Tobin A. Driscoll

RIchard J. Braun

Highly oscillatory phenomena range across numerous areas in science and engineering and their computation represents a difficult challenge. A case in point is integrals of rapidly oscillating functions in one or more variables. The quadrature of such integrals has been historically considered very demanding. Research in the past 15 years (in which the authors played a major role) resulted in a range of very effective and affordable algorithms for highly oscillatory quadrature. This is the only monograph bringing together the new body of ideas in this area in its entirety.
 The starting point is that approximations need to be analyzed using asymptotic methods rather than by more standard polynomial expansions. As often happens in computational mathematics, once a phenomenon is understood from a mathematical standpoint, effective algorithms follow. As reviewed in this monograph, we now have at our disposal a number of very effective quadrature methods for highly oscillatory integrals-Filon-type and Levin-type methods, methods based on steepest descent, and complex-valued Gaussian quadrature. Their understanding calls for a fairly varied mathematical toolbox-from classical numerical analysis, approximation theory, and theory of orthogonal polynomials all the way to asymptotic analysis-yet this understanding is the cornerstone of efficient algorithms.
 Audience: The text is intended for advanced undergraduate and graduate students, as well as applied mathematicians, scientists, and engineers who encounter highly oscillatory integrals as a critical difficulty in their computations.
 Contents : Chapter 1: Introduction; Chapter 2: Asymptotic theory of highly oscillatory integrals; Chapter 3: Filon and Levin methods; Chapter 4: Extended Filon method; Chapter 5: Numerical steepest descent; Chapter 6: Complex-valued Gaussian quadrature; Chapter 7: A highly oscillatory olympics; Chapter 8: Variations on the highly oscillatory theme; Appendix A: Orthogonal polynomials.

Computing Highly Oscillatory Integrals

Alfredo Deaño

Daan Huybrechs

Arieh Iserles

Applied Numerical Linear Algebra introduces students to numerical issues that arise in linear algebra and its applications. A wide range of techniques are touched on, including direct to iterative methods, orthogonal factorizations, least squares, eigenproblems, and nonlinear equations. Inside Applied Numerical Linear Algebra, readers will find: <ul> <li>Clear and detailed explanations on a wide range of topics from condition numbers to the singular value decomposition.</li> <li>Material on nonlinear systems as well as linear systems.</li> <li>Frequent illustrations using discretizations of boundary-value problems or demonstrating other concepts.</li> <li>Exercises with detailed solutions at the end of the book.</li> <li>Supplemental material available at https://bookstore.siam.org/cl87/bonus.</li> </ul> This textbook is appropriate for junior and senior undergraduate students and beginning graduate students in the following courses: Advanced Numerical Analysis, Special Topics on Numerical Analysis, Topics on Data Science, Topics on Numerical Optimization, and Topics on Approximation Theory.

Applied Numerical Linear Algebra

William W. Hager

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