Model Order Reduction: Theory, Research Aspects and Applications (eBook)

Wil Schilders received the MSc degree in pure and applied mathematics from Nijmegen University in 1978, and a PhD in numerical analysis from Trinity College Dublin in 1980. Since 1980, he has been with Philips Electronics, where he developed algorithms for semiconductor device simulation, electronic circuit simulation, and electromagnetics problems. He wrote two volumes on the numerical simulation of semiconductor devices, and published a special volume on Numerical Methods in Electromagnetics. Since 1999, he is part-time professor in numerical analysis for industry at Eindhoven University of Technology. He developed a novel method known as the Schilders factorization for the solution of indefinite linear systems. Since more than a decade, his interest is in model order reduction, and he is a frequent organizer of workshops and symposia on this topic. Currently, he is with NXP Semiconductors, heading the Mathematics group. Henk van der Vorst is a leading expert in numerical linear algebra, in particular in iterative methods for linear systems and eigenproblems. The techniques developed and used in these areas are of very high interest in model order reduction. Van der Vorst was the (co-) author of novel and highly important techniques, including incomplete decompositions, Bi-CGSTAB, and the Jacobi-Davidson method. The Bi-CGSTAB paper was the most cited paper in mathematics  of the 1990’s according to ISI in 2000. For the Jacobi-Davidson method he received, together with co-author Sleijpen  a SIAG-LA Award for the best paper in numerical linear algebra over a three year period. Van der Vorst is Editor in Chief of the SIAM Journal SIMAX and he is member of the Netherlands Royal Academy of Sciences. Joost Rommes received the M.Sc. degree in computational science, the M.Sc.degree in computer science, and the Ph.D. degree in mathematics from UtrechtUniversity, Utrecht, The Netherlands, in 2002, 2003, and 2007, respectively. During his PhD studies he worked on eigensolution methods with applications in model order reduction. Some of his developed methods are now used in software for circuit simulation and power system analysis. Joost Rommes currently works at NXP Semiconductors on model order reduction. In the electronics industry, an increase in complexity at transistor level leads to much large models that can not be simulated without accurate reduction techniques. Due to specific properties of the models, there is also need for different reduction techniques that can deal with these properties. This book provides a wide range of reduction techniques.

Preface 5
Contents 7
List of Contributors 9
Part I Basic Concepts 13
Introduction to Model Order Reduction 14
1 Introduction 14
2 Transfer Function, Stability and Passivity 23
3 A Short Account of Techniques for Model Order Reduction 29
References 42
Linear Systems, Eigenvalues, and Projection 44
1 Introduction 44
2 Linear Systems 48
3 Subspace Methods 49
References 55
Part II Theory 58
Structure-Preserving Model Order Reduction of RCL Circuit Equations 60
1 Introduction 60
2 Formulation of General RCL Circuits as Integro-DAEs 62
3 Structure-Preserving Model Order Reduction 65
4 Equivalent First-Order Form of Integro-DAEs 68
5 Krylov-Subspace Projection and PRIMA 72
6 The SPRIM Algorithm 74
7 Pade-Type Approximation Property of SPRIM 77
8 Numerical Examples 78
9 Concluding Remarks 81
Acknowledgement 82
References 82
A Unified Krylov Projection Framework for Structure- Preserving Model Reduction 86
1 Introduction 86
2 A unified Krylov Projection Structure-Preserving Model Order Reduction Framework 87
3 Structure of Krylov Subspace and Arnoldi Process 93
4 RCL and RCS Systems 94
References 103
Model Reduction via Proper Orthogonal Decomposition 106
1 Introduction 106
2 Proper Orthogonal Decomposition 107
3 POD in Radiative Heat Transfer 113
4 Conclusions and Future Perspectives 116
Acknowledgments 118
References 118
PMTBR: A Family of Approximate Principal- components- like Reduction Algorithms 122
1 Introduction 122
2 Basic Algorithm 124
3 Algorithmic Variants 128
4 Analysis and Comparisons 134
5 Experimental Results 136
6 Conclusions 141
References 142
A Survey on Model Reduction of Coupled Systems. 144
1 Introduction 144
2 Coupled Systems 145
3 Model Reduction Approaches for Coupled Systems 148
4 Numerical Examples 157
References 164
Space Mapping and Defect Correction 168
1 Introduction 168
2 Fine and Coarse Models in Optimization 169
3 Space-Mapping Optimization 171
4 Defect Correction and Space Mapping 175
5 Manifold Mapping, the Improved Space Mapping Algorithm 178
6 Examples 181
7 Conclusions 186
References 186
Modal Approximation and Computation of Dominant Poles 188
1 Introduction 188
2 Transfer Functions, Dominant Poles and Modal Equivalents 188
3 Computing Dominant Poles 190
4 Generalizations 197
5 Numerical Examples 199
6 Conclusions 203
Acknowledgement 203
References 203
Some Preconditioning Techniques for Saddle Point Problems 206
1 Introduction 206
2 Properties of Saddle Point Systems 207
3 Preconditioned Krylov Subspace Methods 208
4 Block Preconditioners 210
5 Augmented Lagrangian Formulations 212
6 Constraint Preconditioning 213
7 Other Techniques 216
8 Numerical Examples 217
9 Conclusions 219
References 220
Time Variant Balancing and Nonlinear Balanced Realizations 224
1 Introduction 224
2 Time Varying Linear Systems 225
3 Sliding Interval Balancing 230
4 Nonlinear Balancing 235
5 Global Balancing 250
6 Mayer-Lie Interpolation 253
7 Nonlinear Model Reduction 255
8 How Far Can You Go? 256
9 Conclusions 258
References 259
Singular Value Analysis and Balanced Realizations for Nonlinear Systems 262
1 Introduction 262
2 Singular Value Analysis of Nonlinear Operators 263
3 Balanced Realization for Linear Systems 266
4 Basics of Nonlinear Balanced Realizations 269
5 Balanced Realizations Based on Singular Value Analysis of Hankel Operators 273
6 Model Order Reduction 276
7 Numerical Example 278
8 Conclusion 282
References 282
Part III Research Aspects and Applications 284
Matrix Functions 286
1 Introduction 286
2 Matrix Functions 286
3 Computational Aspects 289
4 The Exponential Function 296
5 The Matrix Sign Function 304
References 311
Model Reduction of Interconnected Systems 316
1 Introduction 316
2 Interconnected Systems Balanced Truncation 319
3 Krylov Techniques for Interconnected Systems 323
4 Examples of Structured Model Reduction Problems 327
5 Concluding Remarks 330
Acknowledgment 331
References 331
Quadratic Inverse Eigenvalue Problem and Its Applications to Model Updating — An Overview 334
1 Introduction 334
2 Challenges 336
3 Quadratic Inverse Eigenvalue Problem 338
4 Spill-Over Phenomenon 347
5 Least Squares Update 349
6 Conclusions 350
References 350
Data-Driven Model Order Reduction Using Orthonormal Vector Fitting 352
1 Identi.cation Problem 353
2 Vector Fitting 355
3 Orthonormal Vector Fitting 358
4 Example 362
5 Conclusion 364
6 Acknowledgements 364
A Sanathanan-Koerner Iteration 364
B Real-Valued State Space 366
References 368
Model-Order Reduction of High-Speed Interconnects Using Integrated Congruence Transform 372
1 High-Speed Interconnects and Its Effects on Signal Propagation 372
2 Time-Domain Macromodeling of High-Speed Interconnects 375
3 Time-Domain Macromodeling Through MOR 383
4 Numerical Computations 404
5 Conclusion 410
References 410
Model Order Reduction for MEMS: Methodology and Computational Environment for Electro- Thermal Models 414
1 Introduction 414
2 Applications 415
3 Model Order Reduction: Method and Numerical Results 416
4 Computational Environment 418
5 Error Estimation 420
6 Coupling of the Reduced Models 421
7 Model Order Reduction as a Fast Solver 423
8 Advanced Development 425
9 Summary 427
References 427
Model Order Reduction of Large RC Circuits 432
1 Introduction 432
2 Gaussian Elimination Background 433
3 RC-in RC-out Reduction 439
4 Elimination for Synthesis 446
5 Computational Aspects 451
6 Conclusion 455
References 456
Reduced Order Models of On-Chip Passive Components and Interconnects, Workbench and Test Structures 458
1 Extraction of the EM-FW State Models for Passive Components 458
2 Finite States Representation by Finite Integrals Technique 461
3 State Representation of the Boundary Conditions 465
4 ROMWorkBench 468
5 All Levels Reduced Order Modelling 470
6 Test Structures 471
7 Conclusions 476
Acknowledgment 478
References 478
Index 480