Modeling and Inverse Problems in the Presence of Uncertainty collects recent research-including the authors' own substantial projects-on uncertainty propagation and quantification. It covers two sources of uncertainty: where uncertainty is present primarily due to measurement errors and where uncertainty is present due to the modeling formulation itself. After a useful review of relevant probability and statistical concepts, the book summarizes mathematical and statistical aspects of inverse problem methodology, including ordinary, weighted, and generalized least-squares formulations. It then discusses asymptotic theories, bootstrapping, and issues related to the evaluation of correctness of assumed form of statistical models. The authors go on to present methods for evaluating and comparing the validity of appropriateness of a collection of models for describing a given data set, including statistically based model selection and comparison techniques. They also explore recent results on the estimation of probability distributions when they are embedded in complex mathematical models and only aggregate (not individual) data are available. In addition, they briefly discuss the optimal design of experiments in support of inverse problems for given models. The book concludes with a focus on uncertainty in model formulation itself, covering the general relationship of differential equations driven by white noise and the ones driven by colored noise in terms of their resulting probability density functions. It also deals with questions related to the appropriateness of discrete versus continuum models in transitions from small to large numbers of individuals. With many examples throughout addressing problems in physics, biology, and other areas, this book is intended for applied mathematicians interested in deterministic and/or stochastic models and their interactions. It is also s

A Modern Framework Based on Time-Tested MaterialA Functional Analysis Framework for Modeling, Estimation and Control in Science and Engineering presents functional analysis as a tool for understanding and treating distributed parameter systems. Drawing on his extensive research and teaching from the past 20 years, the author explains how functional analysis can be the basis of modern partial differential equation (PDE) and delay differential equation (DDE) techniques. Recent Examples of Functional Analysis in Biology, Electromagnetics, Materials, and MechanicsThrough numerous application examples, the book illustrates the role that functional analysis-a classical subject-continues to play in the rigorous formulation of modern applied areas. The text covers common examples, such as thermal diffusion, transport in tissue, and beam vibration, as well as less traditional ones, including HIV models, uncertainty in noncooperative games, structured population models, electromagnetics in materials, delay systems, and PDEs in control and inverse problems. For some applications, computational aspects are discussed since many problems necessitate a numerical approach.

Modeling and Inverse Problems in the Presence of Uncertainty collects recent research-including the authors' own substantial projects-on uncertainty propagation and quantification. It covers two sources of uncertainty: where uncertainty is present primarily due to measurement errors and where uncertainty is present due to the modeling formulation itself. After a useful review of relevant probability and statistical concepts, the book summarizes mathematical and statistical aspects of inverse problem methodology, including ordinary, weighted, and generalized least-squares formulations. It then discusses asymptotic theories, bootstrapping, and issues related to the evaluation of correctness of assumed form of statistical models. The authors go on to present methods for evaluating and comparing the validity of appropriateness of a collection of models for describing a given data set, including statistically based model selection and comparison techniques. They also explore recent results on the estimation of probability distributions when they are embedded in complex mathematical models and only aggregate (not individual) data are available. In addition, they briefly discuss the optimal design of experiments in support of inverse problems for given models. The book concludes with a focus on uncertainty in model formulation itself, covering the general relationship of differential equations driven by white noise and the ones driven by colored noise in terms of their resulting probability density functions. It also deals with questions related to the appropriateness of discrete versus continuum models in transitions from small to large numbers of individuals. With many examples throughout addressing problems in physics, biology, and other areas, this book is intended for applied mathematicians interested in deterministic and/or stochastic models and their interactions. It is also s

A Modern Framework Based on Time-Tested MaterialA Functional Analysis Framework for Modeling, Estimation and Control in Science and Engineering presents functional analysis as a tool for understanding and treating distributed parameter systems. Drawing on his extensive research and teaching from the past 20 years, the author explains how functional analysis can be the basis of modern partial differential equation (PDE) and delay differential equation (DDE) techniques. Recent Examples of Functional Analysis in Biology, Electromagnetics, Materials, and MechanicsThrough numerous application examples, the book illustrates the role that functional analysis-a classical subject-continues to play in the rigorous formulation of modern applied areas. The text covers common examples, such as thermal diffusion, transport in tissue, and beam vibration, as well as less traditional ones, including HIV models, uncertainty in noncooperative games, structured population models, electromagnetics in materials, delay systems, and PDEs in control and inverse problems. For some applications, computational aspects are discussed since many problems necessitate a numerical approach.

The research detailed in this monograph was originally motivated by our interest in control problems involving partial and delay differential equations. Our attempts to apply control theory techniques to such prob lems in several areas of science convinced us that in the need for better and more detailed models of distributed/ continuum processes in biology and mechanics lay a rich, interesting, and challenging class of fundamen tal questions. These questions, which involve science and mathematics, are typical of those arising in inverse or parameter estimation problems. Our efforts on inverse problems for distributed parameter systems, which are infinite dimensional in the most common realizations, began about seven years ago at a time when rapid advances in computing capabilities and availability held promise for significant progress in the development of a practically useful as well as theoretically sound methodology for such problems. Much of the research reported in our presentation was not begun when we outlined the plans for this monograph some years ago. By publishing this monograph now, when only a part of the originally intended topics are covered (see Chapter VII in this respect), we hope to stimulate the research and interest of others in an area of scientific en deavor which has exceeded even our optimistic expectations with respect to excitement, opportunity, and stimulation. The computer revolution alluded to above and the development of new codes allow one to solve rather routinely certain estimation problems that would have been out of the question ten years ago."

These notes are based on (i) a series of lectures that I gave at the 14th Biennial Seminar of the Canadian Mathematical Congress held at the University of Western Ontario August 12-24, 1973 and (li) some of my lectures in a modeling course that I have cotaught in the Division of Bio-Medical Sciences at Brown during the past several years. An earlier version of these notes appeared in the Center for Dynamical Systems Lectures Notes series (CDS LN 73-1, November 1973). I have in this revised and extended version of those earlier notes incorporated a number of changes based both on classroom experience and on my research efforts with several colleagues during the intervening period. The narrow viewpoint of the present notes (use of optimization and control theory in biomedical problems) reflects more the scope of the CMC lectures given in August, 1973 than the scope of my own interests. Indeed, my real interests have included the modeling process itself as well as the contributions made by investiga tors who employ the techniques and ideas of control theory, systems analysis, dif ferential equations, and stochastic processes. Some of these contributions have quite naturally involved application of optimal control theory. But in my opinion many of the interesting efforts being made in modeling in the biomedical sciences encompass much more than the use of control theory."

Through several case study problems from industrial and scientific research laboratory applications, Mathematical and Experimental Modeling of Physical and Biological Processes provides students with a fundamental understanding of how mathematics is applied to problems in science and engineering. For each case study problem, the authors discuss why a model is needed and what goals can be achieved with the model. Exploring what mathematics can reveal about applications, the book focuses on the design of appropriate experiments to validate the development of mathematical models. It guides students through the modeling process, from empirical observations and formalization of properties to model analysis and interpretation of results. The authors also describe the hardware and software tools used to design the experiments so faculty/students can duplicate them. Integrating real-world applications into the traditional mathematics curriculum, this textbook deals with the formulation and analysis of mathematical models in science and engineering. It gives students an appreciation of the use of mathematics and encourages them to further study the applied topics. Real experimental data for projects can be downloaded from CRC Press Online.

Inverse problems arise in a number of important practical applications, ranging from biomedical imaging to seismic prospecting. This book provides the reader with a basic understanding of both the underlying mathematics and the computational methods used to solve inverse problems. It also addresses specialized topics like image reconstruction, parameter identification, total variation methods, nonnegativity constraints, and regularization parameter selection methods. Because inverse problems typically involve the estimation of certain quantities based on indirect measurements, the estimation process is often ill-posed. Regularization methods, which have been developed to deal with this ill-posedness, are carefully explained in the early chapters of Computational Methods for Inverse Problems. The book also integrates mathematical and statistical theory with applications and practical computational methods, including topics like maximum likelihood estimation and Bayesian estimation. Several web-based resources are available to make this monograph interactive, including a collection of MATLAB m-files used to generate many of the examples and figures. These resources enable readers to conduct their own computational experiments in order to gain insight. They also provide templates for the implementation of regularization methods and numerical solution techniques for other inverse problems. Moreover, they include some realistic test problems to be used to further develop and test various numerical methods.

Interface problems arise when there are two different materials, such as water and oil, or the same material at different states, such as water and ice. If partial or ordinary differential equations are used to model these applications, the parameters in the governing equations are typically discontinuous across the interface separating the two materials or states, and the source terms are often singular to re?ect source/sink distributions along codimensional interfaces. Because of these irregularities, the solutions to the differential equations are typically nonsmooth or even discontinuous. As a result, many standard numerical methods based on the assumption of smoothness of solutions do not work or work poorly for interface problems. The Immersed Interface Method provides an introduction to the immersed interface method (IIM), a powerful numerical method for solving interface problems and problems defined on irregular domains for which analytic solutions are rarely available. This book gives a complete description of the IIM, discusses recent progress in the area, and describes numerical methods for a number of classic interface problems. It also contains many numerical examples that can be used as benchmark problems for numerical methods designed for interface problems on irregular domains. The IIM is a sharp interface method that has been coupled with evolution schemes such as the level set and front tracking methods and has been used in both finite difference and finite element formulations to solve several moving interface and free boundary problems. In particular, the authors discuss the IIM's applications to Stefan problems and unstable crystal growth, incompressible Stokes and Navier-Stokes flows with moving interfaces, an inverse problem identifying unknown shapes in a region, a nonlinear interface problem of magnetorheological ?uids containing iron particles, and other problems. The book also contains several applications of free boundary and moving interface problems, including examples from physics, computational fluid mechanics, mathematical biology, material science, and other fields. The IIM, which is based on uniform or adaptive Cartesian/polar/spherical grids or triangulations, is simple enough to be implemented by researchers and graduate students with a reasonable background in differential equations and numerical analysis yet powerful enough to solve complicated problems with high-order accuracy. Since interfaces or irregular boundaries are one dimension lower than solution domains, the extra costs in dealing with interfaces or irregular boundaries are generally insigni?cant, and many software packages based on uniform Cartesian/polar/spherical grids, such as the FFT and fast Poisson solvers, can be applied easily with the IIM. The most recent IIM computer codes and packages are available online.

Gender-Structured Population Modeling gives a unified presentation of and mathematical framework for modeling population growth by couple formation. It provides an overview of both past and present modeling results. The authors focus on pair formation (marriage) and two-sex models with different forms of the marriage function - the basis of couple formation - and discuss which of these forms might make a better choice for a particular population (the United States). It also provides results on model analysis, gives an up-to-date review of mathematical demography, discusses numerical methods, and puts deterministic modeling of human populations into historical perspective. The authors describe several models and derive the theoretical results that demonstrate the validity of these models. The numerical methods for approximating the solutions of the differential models - the equivalent of creating discrete simulators - are delineated. Simulation results are compared with actual demographic data to show some of the difficulties concerning the availability of data and to show that mathematical demography provides reasonable qualitative and quantitative estimates. The models in this book can be applied to different sets of data.

Smart Material Systems: Model Development describes several novel applications currently under investigation that exploit the unique actuator and sensor capabilities of smart material compounds. In addition to present and projected applications, this book provides comprehensive coverage of both linear and nonlinear modeling techniques necessary to characterize materials in a manner that facilitates transducer design and control development. The author focuses on ferroelectric, magnetic, and shape memory compounds and also addresses applications exploiting amorphous and ionic polymers, magnetorheological compounds, and fiber optic sensors. Three classes of nonlinear models are discussed, all of which provide unified characterization frameworks for the broad class of combined compounds. An extensive discussion of structural models based on linear and nonlinear constitutive models is included, as is a chapter on numerical techniques for approximating solutions to the structural systems. characterization, including significant discussion of the physics underlying nonlinear behavior, the development of nonlinear constitutive relations, the development of system models based on these constitutive relations, and the formulation of numerical techniques for approximating solutions to the resulting system models. To aid the understanding of both mathematical and physical terminology, an extensive glossary of terms is provided in an appendix. Furthermore, MATLAB software for representative models is provided at an accompanying website. By providing a unified treatment of both linear and nonlinear characterization frameworks, Smart Material Systems: Model Development encompasses both low to moderate drive levels, which constitute the primary focus of most present texts, and the high drive regimes dictated by present and future applications. This will significantly enhance the design of transducers and control systems which exploit the unique actuator and sensor capabilities provided by smart material compounds. students and researchers in applied mathematics, materials science, mechanical, aerospace, electrical and civil engineering, physics, chemistry, and related fields involving applications that utilize advanced materials. Individuals ranging from students entering the field to experts within constituent disciplines will find Smart Material Systems: Model Development useful.

This book is composed of eleven chapters, written by experts in their respective fields, on topics ranging from control of the Navier-Stokes equations to nondestructive evaluation - all of which are modeled by distributed parameters systems. Written by the plenary speakers for the Conference on Future Directions in Distributed Parameter Systems (October 2000), the volume addresses the state of the art, open questions, and important research directions in applications modeled by partial differential equations and delay systems, with the ultimate goal of passing along these perspectives to researchers in the field. This book compiles a broad range of material, much of which is available only in research journals. Its goal is to disseminate to the research community the perspectives and challenges posed for a variety of current research applications. Topics include electromagnetic theory for dielectric and conductive materials, flow control, cardiovascular and respiratory models, homogenization and systems theory, optimal and geometric control, reduced-order models for large-scale systems, smart materials, and nondestructive evaluation and structural health monitoring for systems, including nuclear power plants. A unifying theme throughout the book focuses on the necessity of considering model development, numerical approximation, parameter estimation through inverse problem techniques, and control design in concert to achieve the stringent design criteria dictated by present applications. Research Directions in Distributed Parameter Systems is intended for graduate students, academic and industrial research scientists, and program managers from funding agencies supporting research focused on distributed parameter systems.

Globalization and the possibility of bioterrorist acts have highlighted the pressing need for the development of theoretical and practical mathematical frameworks that may be useful in our systemic efforts to anticipate, prevent, and respond to acts of destabilization. Bioterrorism: Mathematical Modeling Applications in Homeland Security collects the detailed contributions of selected groups of experts from the fields of biostatistics, control theory, epidemiology, and mathematical biology who have engaged in the development of frameworks, models, and mathematical methods needed to address some of the pressing challenges posed by acts of terror. The ten chapters of this volume touch on a large range of issues in the subfields of biosurveillance, agroterrorism, bioterror response logistics, deliberate release of biological agents, impact assessment, and the spread of fanatic behaviors.

This book is devoted to giving a modern view of iterative methods for solving linear and nonlinear equations, which are the basis for many, if not most, of the models of phenomena in science and engineering; their efficient numerical solution is critical to progress in these areas. The text provides motivating examples mainly from boundary value problems with partial differential equations, and many of the chapters contain links to MATLAB code, which is provided per anonymous ftp by the author. This is the first book to be published on nonlinear equations since the mid-1980s. Although it stresses recent developments in this area, considerable material on linear equations has been incorporated. It focuses on a small number of methods and treats them in depth. The author provides a complete analysis of the conjugate gradient and generalized minimum residual iterations as well as recent advances including Newton-Krylov methods, incorporation of inexactness and noise into the analysis, new proofs and implementations of Broyden's method, and globalization of inexact Newton methods. Examples, methods, and algorithmic choices are based on applications to infinite dimensional problems such as partial differential equations and integral equations. The analysis and proof techniques are constructed with the infinite dimensional setting in mind and the computational examples and exercises are based on the MATLAB environment.

Much recent research has concentrated on the efficient solution of large sparse or structured linear systems using iterative methods. A language loaded with acronyms for a thousand different algorithms has developed, and it is often difficult even for specialists to identify the basic principles involved. Here is a book that focuses on the analysis of iterative methods. The author includes the most useful algorithms from a practical point of view and discusses the mathematical principles behind their derivation and analysis. Several questions are emphasized throughout: Does the method converge? If so, how fast? Is it optimal, among a certain class? If not, can it be shown to be near-optimal? The answers are presented clearly, when they are known, and remaining important open questions are laid out for further study. Greenbaum includes important material on the effect of rounding errors on iterative methods that has not appeared in other books on this subject. Additional important topics include a discussion of the open problem of finding a provably near-optimal short recurrence for non-Hermitian linear systems; the relation of matrix properties such as the field of values and the pseudospectrum to the convergence rate of iterative methods; comparison theorems for preconditioners and discussion of optimal preconditioners of specified forms; introductory material on the analysis of incomplete Cholesky, multigrid, and domain decomposition preconditioners, using the diffusion equation and the neutron transport equation as example problems. A small set of recommended algorithms and implementations is included.

This book presents a carefully selected group of methods for unconstrained and bound constrained optimization problems and analyzes them in depth both theoretically and algorithmically. It focuses on clarity in algorithmic description and analysis rather than generality, and while it provides pointers to the literature for the most general theoretical results and robust software, the author thinks it is more important that readers have a complete understanding of special cases that convey essential ideas. A companion to Kelley's book, Iterative Methods for Linear and Nonlinear Equations (SIAM, 1995), this book contains many exercises and examples and can be used as a text, a tutorial for self-study, or a reference. Iterative Methods for Optimization does more than cover traditional gradient-based optimization: it is the first book to treat sampling methods, including the Hooke-Jeeves, implicit filtering, MDS, and Nelder-Mead schemes in a unified way, and also the first book to make connections between sampling methods and the traditional gradient-methods. Each of the main algorithms in the text is described in pseudocode, and a collection of MATLAB codes is available. Thus, readers can experiment with the algorithms in an easy way as well as implement them in other languages.

A practical handbook for understanding and using fast adaptive composite grid (FAC) methods for discretization and solution of partial differential equations (PDEs). Contains fundamental concepts. These so called FAC are characterized by their use of a composite grid, which is nominally the union of various uniform grids. FAC is capable of producing a composite grid with tailored resolution, and a corresponding solution with commensurate accuracy, at a cost proportional to the number of composite grid points. Moreover, special asynchronous versions of the fast adaptive composite grid methods (AFAC) studied here have seemingly optimal complexity in a parallel computing environment. Most of the methods treated in this book were discovered only within the last decade, and in many cases their development is still in its infancy. While this book is therefore not meant to be comprehensive, it does provide a theoretical and practical guide to multilevel adaptive methods and relevant discretization techniques. It also contains new material, which is included to fill in certain gaps and to expose new avenues of research. Also, because adaptive refinement seems to demand a lot of attention to philosophical issues, personal perspectives are often brought freely into the discussion.

Developments in optimization theory, including emphasis on large problems and on interior-point methods for linear programming, have begun to appear in production software. Here is a reference tool that includes discussions of these areas and names software packages that incorporate the results of theoretical research. After an introduction to the major problem areas in optimization and an outline of the algorithms used to solve them, a data sheet is presented for each of the 75 software packages and libraries in the authors' survey. These include information on the capabilities of the packages, how to obtain them, and addresses for further information. Standard optimization paradigms are addressed - linear, quadratic, and nonlinear programming; network optimization; unconstrained and bound-constrained optimization; least-squares problems; nonlinear equations; and integer programming. The most practical algorithms for the major fields of numerical optimization are outlined, and the software packages in which they are implemented are described. This format will aid current and potential users of optimization software in classifying the optimization problem to be solved, determining appropriate algorithms, and obtaining the software that implements those algorithms. Readers need only a basic knowledge of vector calculus and linear algebra to understand this book.

This volume addresses recent developments in mathematical modeling in three areas of optical science: diffractive optics, photonic band gap structures, and waveguides. Particular emphasis is on the formulation of mathematical models and the design and analysis of new computational approaches. The book contains cutting-edge discourses on emerging technology in optics that provides significant challenges and opportunities for applied mathematicians, researchers, and engineers.

Each of the three topics is presented through a series of survey papers to provide a broad overview focusing on the mathematical models. Chapters present model problems, physical principles, mathematical and computational approaches, and engineering applications corresponding to each of the three areas. Although some of the subject matter is classical, the topics presented are new and represent the latest developments in their respective fields.

Neural networks and fuzzy systems are model free control design approaches that represent an advantage over classical control when dealing with complicated nonlinear actuator dynamics. This book brings neural networks and fuzzy logic together with dynamical control systems. Each chapter presents powerful control approaches for the design of intelligent controllers to compensate for actuator nonlinearities such as time delay, friction, deadzone, and backlash that can be found in all industrial motion systems, plus a thorough development, rigorous stability proofs, and simulation examples for each design. In the final chapter, the authors develop a framework to implement intelligent control schemes on actual systems. Rigorous stability proofs are further verified by computer simulations, and appendices contain the computer code needed to build intelligent controllers for real-time applications. Neural networks capture the parallel processing and learning capabilities of biological nervous systems, and fuzzy logic captures the decision-making capabilities of human linguistics and cognitive systems.

Here is a book for anyone who would like to become better acquainted with the modern tools of numerical analysis for several significant computational problems arising in finance. The authors review some important aspects of finance modeling involving partial differential equations and focus on numerical algorithms for the fast and accurate pricing of financial derivatives and for the calibration of parameters. Option pricing has become a technical topic that requires sophisticated numerical methods for robust and fast numerical solutions. This book explores the best numerical algorithms and discusses them in depth, from their mathematical analysis up to their implementation in C with efficient numerical libraries. Much of this information is not available elsewhere. In particular, this is one of the few books that gives detailed coverage of the following topics:* Mathematical results and efficient algorithms for pricing American options.* Modern algorithms with adaptive mesh refinement for European and American options. Regularity and error estimates are derived and give strong support to the mesh adaptivity, an essential tool for speeding up the numerical implementations.* Calibration of volatility with European and American options. * The use of automatic differentiation of computer codes for computing gree.

The research detailed in this monograph was originally motivated by our interest in control problems involving partial and delay differential equations. Our attempts to apply control theory techniques to such prob­ lems in several areas of science convinced us that in the need for better and more detailed models of distributed/ continuum processes in biology and mechanics lay a rich, interesting, and challenging class of fundamen­ tal questions. These questions, which involve science and mathematics, are typical of those arising in inverse or parameter estimation problems. Our efforts on inverse problems for distributed parameter systems, which are infinite dimensional in the most common realizations, began about seven years ago at a time when rapid advances in computing capabilities and availability held promise for significant progress in the development of a practically useful as well as theoretically sound methodology for such problems. Much of the research reported in our presentation was not begun when we outlined the plans for this monograph some years ago. By publishing this monograph now, when only a part of the originally intended topics are covered (see Chapter VII in this respect), we hope to stimulate the research and interest of others in an area of scientific en­ deavor which has exceeded even our optimistic expectations with respect to excitement, opportunity, and stimulation. The computer revolution alluded to above and the development of new codes allow one to solve rather routinely certain estimation problems that would have been out of the question ten years ago.