Approximate Deconvolution Models of Turbulence : Analysis Phenomenology and Numerical Analysis This volume presents a mathematical development of a recent approach to the modeling and simulation of turbulent flows based on methods for the approximate solution of inverse problems. The resulting Approximate Deconvolution Models or ADMs have some advantages over more commonly used turbulence models as well as some disadvantages. Our goal in this book is to provide a clear and complete mathematical development of ADMs while pointing out the difficulties that remain. In order to do so we present the analytical theory of ADMs along with its connections motivations and complements in the phenomenology of and algorithms for ADMs 1 Introduction 2 Large Eddy Simulation 3 Approximate Deconvolution Operators and Models 4 Phenomenology of ADMs 5 Time Relaxation Truncates Scales 6 The Leray Deconvolution Regularization 7 NS alpha and NS omega Deconvolution Regularizations