Springer, London, New York, 2011, 355 pp. - ISBN 978-3-642-14940-5 A mathematical model is an intellectual device that works. The models have various purposes and, according to these purposes they differ by the level of simplification. There are many possible ways to construct the mathematical model we need. We can start from a detailed model - a hypothesis which reflects all our knowledge of the first principles and then go down the stair of simplification. We can go the opposite way and use a metaphor or an analogy and start from a very simple model, then add more details and develop a better and more complicated intellectual tool for our needs. In 1980, R. Peierls suggested that one might distinguish seven different types of models and demonstrated various types of confusion that can result if the nature of the model is misunderstood. He proposed the term model making. Model reduction is one of the main operations in model making. Following Peierls we can state that the main difference between models is "the degree of simplification or exaggeration they involve". The technology of model reduction should answer the modern challenges of the struggle with complexity.ContentsAveraging of Fast-Slow Systems – (Marshall Slemrod). The Use of Global Sensitivity Methods for the Analysis, Evaluation and Improvement of Complex Modelling Systems – (Alison S. Tomlin and Tilo Ziehn). Optimisation and Linear Control of Large Scale Nonlinear Systems: A Review and a Suite of Model Reduction-Based Techniques – (Constantinos Theodoropoulos). Universal Algorithms, Mathematics of Semirings and Parallel Computations – (Grigory L. Litvinov, Victor P. Maslov, Anatoly Ya. Rodionov, and Andrei N. Sobolevski). Scaling Invariant Interpolation for Singularly Perturbed Vector Fields (SPVF) – (Viatcheslav Bykov, Vladimir Gol'dshtein, and Ulrich Maas). Think Globally, Move Locally: Coarse Graining of Effective Free Energy Surfaces – (Payel Das, Thomas A. Frewen, loannis G. Kevrekidis, and Cecilia Clementi). Extracting Functional Dependence from Sparse Data Using Dimensionality Reduction: Application to Potential Energy Surface Construction – (Sergei Manzhos, Koichi Yamashita, and Tucker Carrington). A Multilevel Algorithm to Compute Steady States of Lattice Boltzmann Models – (Giovanni Samaey, Christophe Vandekerckhove, and Wim Vanroose). Time Step Expansions and the Invariant Manifold Approach to Lattice Boltzmann Models – (David J. Packwood, Jeremy Levesley, and Alexander N. Gorban). The Shock Wave Problem Revisited: The Navier-Stokes Equations and Brenner's Two Velocity Hydrodynamics – (Francisco J. Uribe). Adaptive Simplification of Complex Systems: A Review of the Relaxation-Redistribution Approach – (Eliodoro Chiavazzo and Ilya Karlin). Geometric Criteria for Model Reduction in Chemical Kinetics via Optimization of Trajectories – (Dirk Lebiedz, Volkmar Reinhardt, Jochen Siehr, and Jonas Unger). Computing Realizations of Reaction Kinetic Networks with Given Properties – (Gabor Szederkenyi, Katalin M. Hangos, and David Csercsik). A Drift-Filtered Approach to Diffusion Estimation for Multiscale Processes – (Yves Frederix and Dirk Roose). Model Reduction of a Higher-Order KdV Equation for Shallow Water Waves – (Tassos Bounds, Ко van der Weele, Giorgos Kanellopoulos, and Kostis Andriopoulos). Coarse Collective Dynamics of Animal Groups – (Thomas A. Frewen, Iain D. Couzin, Allison Kolpas, Jeff Moehlis, Ronald Coifman, and Ioannis G. Kevrekidis). Self-Simplification in Darwin's Systems – (Alexander N. Gorban). Author Index Subject Index
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