Modeling and inverse problems in the presence of uncertainty 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. 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