Model Reduction and Approximation

Abstract

Many physical, chemical, biomedical, and technical processes can be described by partial differential equations or dynamical systems. In spite of increasing computational capacities, many problems are of such high complexity that they are solvable only with severe simplifications, and the design of efficient numerical schemes remains a central research challenge. This book presents a tutorial introduction to recent developments in mathematical methods for model reduction and approximation of complex systems. Model Reduction and Approximation: Theory and Algorithms contains three parts that cover (I) sampling-based methods, such as the reduced basis method and proper orthogonal decomposition, (II) approximation of high-dimensional problems by low-rank tensor techniques, and (III) system-theoretic methods, such as balanced truncation, interpolatory methods, and the Loewner framework; is tutorial in nature, giving an accessible introduction to state-of-the-art model reduction and approximation methods; and covers a wide range of methods drawn from typically distinct communities (sampling based, tensor based, system-theoretic). Preface -- part I. Sampling-based methods -- 1. Pod for linear-quadratic optimal control -- 2. A tutorial on RB-methods -- 3. The theoretical foundation of reduced basis methods -- part II. Tensor-based methods -- 4. Low-rank methods for high-dimensional approximation -- 5. Model reduction for high-dimensional parametric problems by tensor techniques -- part III. System-theoretic methods -- 6. Model order reduction based on systems building -- 7. Interpolatory model reduction -- 8. The loewner framework for model reduction -- 9. Comparison of methods for PMOR.