Modeling And Inverse Problems In The Presence Of Uncertainty (chapman & Hall/crc Monographs And Research Notes In Mathematics)

**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 suitable for scientists in biology, medicine, engineering, and physics working on basic modeling and inverse problems, uncertainty in modeling, propagation of uncertainty, and statistical modeling. "Writing a research monograph on a 'hot topic' such as 'uncertainty propagation' is a somewhat daunting undertaking. Nontheless, we decided to collect our own views, supported by our own research efforts over the past 12-15 years on a number of aspects of this topic and summarize these for the possible enlightenment it might provide (for us, our students and others). The research results discussed below are thus necessarily filled with a preponderance of references to our own research reports and papers. In numerous of the references below (given at the conclusion of each chapter), we refer to CRSC-TRXX-YY. These refer to early Technical Report versions of manuscripts which can be found on the Center for Research in Scientific Computation website at North Carolina State University where XX refers to the year, e.g., XX = 03 is 2003, XX = 99 is 1999, while the YY refers to the number of the report in that year. These can be found at and downloaded from http://www.ncsu.edu/crsc/reports.html where they are listed by year. Our presentation here has an intended audience from the community of investigators in applied mathematics interested in deterministic and/or stochastic models and their interactions as well as scientists in biology, medicine, engineering and physics interested in basic modeling and inverse problems, uncertainty in modeling, propagation of uncertainty, and statistical modeling. We owe great thanks to our former and current students, postdocs and colleagues for their patience in enduring lectures, questions, feedback, and some proofreading. Special thanks are due (in no particular order) to Zack Kenz, Keri Rehm, Dustin Kaparun, Jared Catenacci, Katie Link, Kris Rinnevatore, Kevin Flores"-- Provided by publisher Front Cover 1 Contents 6 Preface 14 Chapter 1: Introduction 16 Chapter 2: Probability and Statistics Overview 18 Chapter 3: Mathematical and Statistical Aspects of Inverse Problems 68 Chapter 4: Model Selection Criteria 132 Chapter 5: Estimation of Probability Measures Using Aggregate Population Data 172 Chapter 6: Optimal Design 210 Chapter 7: Propagation of Uncertainty in a Continuous Time Dynamical System 224 Chapter 8: A Stochastic System and Its Corresponding Deterministic System 324 Frequently Used Notations and Abbreviations 398