Model reduction and approximation : theory and algorithms

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. List of Contributors 5 Contents 7 List of Figures 11 List of Tables 15 List of Algorithms 17 Preface 19 I Sampling-Based Methods 21 1 Proper Orthogonal Decomposition for Linear-Quadratic Optimal Control 23 1.1 Introduction 23 1.2 The POD method 25 1.3 Reduced-order modeling for evolution problems 43 1.4 The linear-quadratic optimal control problem 54 1.5 Numerical experiments 68 Bibliography 78 2 Reduced Basis Methods for Parametrized PDEs— A Tutorial Introduction for Stationary and Instationary Problems 85 2.1 Abstract 85 2.2 Introduction 85 2.3 Stationary problems 88 2.4 Instationary problems 127 2.5 Extensions and outlook 146 2.6 Exercises 148 Bibliography 151 3 The Theoretical Foundation of Reduced Basis Methods 157 3.1 Introduction 157 3.2 Elliptic PDEs 158 3.3 Parametric elliptic equations 160 3.4 Evaluating numerical methods 162 3.5 Comparing widths and entropies of  with those of  168 3.6 Widths of our two model classes 171 3.7 Numerical methods for parametric equations 175 3.8 Nonlinear methods in RBs 183 Bibliography 186 II Tensor-Based Methods 189 4 Low-Rank Methods for High-Dimensional Approximation and Model Order Reduction 191 4.1 Introduction 191 4.2 Tensor spaces 193 4.3 Low-rank approximation of order-two tensors 198 4.4 Low-rank approximation of higher-order tensors 203 4.5 Greedy algorithms for low-rank approximation 210 4.6 Low-rank approximation using samples 216 4.7 Tensor-structured parameter-dependent or stochastic equations 220 4.8 Low-rank approximation for equations in tensor format 230 Bibliography 240 5 Model Reduction for High-Dimensional Parametric Problems by Tensor Techniques 247 5.1 Introduction 247 5.2 The concept of tensor formats 248 5.3 Canonical format 249 5.4 Tucker format 251 5.5 SVD-based tensor formats 252 5.6 TT format 253 5.7 Optimization algorithms in TT format 260 5.8 Dynamical low-rank approximation 263 5.9 Black-box approximation of tensors 265 5.10 Quantized TT format 269 5.11 Numerical illustrations 270 Bibliography 273 III System-Theoretic Methods 279 6 Model Order Reduction Based on System Balancing 281 6.1 Introduction 281 6.2 BT for LTI systems 284 6.3 Balancing-related model reduction 286 6.4 BT for generalized systems 289 6.5 Numerical solution of linear matrix equations 295 6.6 Numerical examples 301 6.7 Conclusions and outlook 306 Bibliography 307 7 Model Reduction by Rational Interpolation 317 7.1 Introduction 317 7.2 Model reduction via projection 318 7.3 Model reduction by interpolation 321 7.4 Interpolatory projections for 2 optimal approximation 329 7.5 Model reduction with generalized coprime realizations 336 7.6 Realization-independent optimal 2 approximation 340 7.7 Interpolatory model reduction of parametric systems 342 7.8 Conclusions 347 Bibliography 348 8 A Tutorial Introduction to the Loewner Framework for Model Reduction 355 8.1 Introduction 355 8.2 The Loewner framework for linear systems 361 8.3 Reduced-order modeling from data 388 8.4 Summary 393 Bibliography 394 9 Comparison of Methods for Parametric Model Order Reduction of Time-Dependent Problems 397 9.1 Introduction 397 9.2 Methods for PMOR 399 9.3 Performance measures 404 9.4 Expectations 405 9.5 Benchmarks 406 9.6 Numerical results 411 9.7 Conclusions 424 Bibliography 424 Index 429 Model,Reduction,and,Approximation,9781611974812