This book is an outgrowth of lectures by both authors in the graduate course of the first author at the Courant Institute during spring 2008 and 2010 on the topic of filtering turbulent dynamical systems as well as lectures by the second author at the North Carolina State University in a graduate course in fall 2009. The material is based on the authors' joint research as well as collaborations with Marcus Grote and Boris Gershgorin; the authors thank these colleagues for their explicit and implicit contributions to this material. Chapter 1 presents a detailed overview and summary of the viewpoint and material in the book. This book is designed for applied mathematicians, scientists and engineers, ranging from first- and second-year graduate students to senior researchers interested in filtering large-dimensional complex nonlinear systems.

The first author acknowledges the generous support of DARPA through Ben Mann and ONR through Reza Malek-Madani which funded the research on these topics and helped make this book a reality.