Preface	6
	Contents	9
	1 Model Order Reduction of Integrated Circuitsin Electrical Networks	12
	1.1 Introduction	12
	1.2 Basic Models	14
	1.2.1 Coupling	15
	1.3 Simulation of the Full System	18
	1.3.1 Standard Galerkin Finite Element Approach	18
	1.3.2 Mixed Finite Element Approach	19
	1.4 Model Order Reduction Using POD	24
	1.4.1 Numerical Investigation	27
	1.4.2 Numerical Investigation, Position of the Semiconductor in the Network	28
	1.4.3 MOR for the Nonlinearity with DEIM	30
	1.4.4 Numerical Implementation and Results with DEIM	31
	1.5 Residual-Based Sampling	34
	1.5.1 Numerical Investigation for Residual Based Sampling	37
	1.6 PABTEC Combined with POD MOR	38
	1.6.1 Decoupling	39
	1.6.2 Model Reduction Approach	41
	1.6.3 Numerical Experiments	42
	References	45
	2 Element-Based Model Reduction in Circuit Simulation	49
	2.1 Introduction	49
	2.2 Circuit Equations	50
	2.2.1 Graph-Theoretic Concepts	51
	2.2.2 Modified Nodal Analysis and Modified Loop Analysis	51
	2.2.3 Linear RLC Circuits	55
	2.2.3.1 Passivity	56
	2.2.3.2 Stability	57
	2.2.3.3 Reciprocity	57
	2.3 Model Reduction of Linear Circuits	57
	2.3.1 Balanced Truncation for RLC Circuits	58
	2.3.2 Balanced Truncation for RC Circuits	62
	2.3.2.1 RCI Circuits	62
	2.3.2.2 RCV Circuits	64
	2.3.2.3 RCIV Circuits	66
	2.3.3 Numerical Aspects	69
	2.4 Model Reduction of Nonlinear Circuits	71
	2.5 Solving Matrix Equations	76
	2.5.1 ADI Method for Projected Lyapunov Equations	77
	2.5.2 Newton's Method for Projected Riccati Equations	78
	2.6 MATLAB Toolbox PABTEC	81
	2.7 Numerical Examples	85
	References	92
	3 Reduced Representation of Power Grid Models	96
	3.1 Introduction	96
	3.2 System Description	98
	3.2.1 Basic Definitions	98
	3.2.2 Benchmark Systems	103
	3.2.2.1 A Test Circuit Example	103
	3.2.2.2 Linear Subdomain for Non-linear Electro-Quasistatic Field Simulations	103
	3.3 Terminal Reduction Approaches	105
	3.3.1 (E)SVDMOR	105
	3.3.2 TermMerg	110
	3.3.2.1 The k-Means Clustering Algorithm	111
	3.3.2.2 The Reduction Step	112
	3.3.3 SparseRC	112
	3.3.3.1 MOR via Graph Partitioning and EMMP	113
	3.3.4 MOR for Many Terminals via Interpolation	115
	3.3.4.1 Tangential Interpolation and the Loewner Concept	115
	3.4 ESVDMOR in Detail	117
	3.4.1 Stability, Passivity, Reciprocity	117
	3.4.1.1 Stability	118
	3.4.1.2 Passivity	119
	3.4.1.3 Reciprocity	121
	3.4.2 Error Analysis	123
	3.4.2.1 Particular Error Bounds	124
	3.4.2.2 Total ESVDMOR Error Bound	126
	3.4.3 Implementation Details	129
	3.4.3.1 The Implicitly Restarted Arnoldi Method	130
	3.4.3.2 The Jacobi-Davidson SVD Method	131
	3.4.3.3 Efficiency Issues	135
	3.4.3.4 Truncated SVD of the Input Response Moment Matrix MI	136
	3.4.3.5 Truncated SVD of the Output Response Moment Matrix MO	138
	3.5 Summary and Outlook	140
	References	141
	4 Coupling of Numeric/Symbolic Reduction Methods for Generating Parametrized Models of NanoelectronicSystems	144
	4.1 Introduction	144
	4.1.1 Symbolic Modeling of Analog Circuits	146
	4.2 Hierarchical Modelling and Model Reduction	146
	4.2.1 Workflow for Subsystem Reductions	147
	4.2.2 Subsystem Sensitivities	149
	4.2.3 Subsystem Ranking	151
	4.2.4 Algorithm for Hierarchical Model Reduction	153
	4.3 Implementations	153
	4.4 Applications	155
	4.4.1 Differential Amplifier	155
	4.4.2 Reduction of the Transmission Line L1 by Using an Adapted PABTEC Algorithm	158
	4.4.3 Operational Amplifier	159
	4.5 Conclusions	164
	References	164
	5 Low-Rank Cholesky Factor Krylov Subspace Methodsfor Generalized Projected Lyapunov Equations	166
	5.1 Introduction	166
	5.2 Balanced Truncation	167
	5.2.1 Introduction to Balanced Truncation	167
	5.2.2 Numerical Methods for Projected, Generalized Lyapunov Equations	169
	5.3 Low-Rank Cholesky Factor Krylov Subspace Methods	170
	5.3.1 Low-Rank Krylov Subspace Methods	171
	5.3.2 Low-Rank Cholesky Factor Preconditioning	172
	5.3.3 Low-Rank Pseudo Arithmetic	173
	5.3.4 Approximate LRCF-ADI Preconditioning	177
	5.3.5 Selected Low-Rank Krylov Subspace Methods	178
	5.3.6 Reduced Lyapunov Equation	180
	5.4 Numerical Results	182
	5.4.1 Model Problems	183
	5.4.2 Different Krylov Subspace Methods and Their Efficiency with Respect to the Selection of Shifts	185
	5.4.3 Truncated QR? Decomposition	188
	5.4.4 Evolution of the Rank Representations in the Low-Rank CG Method	192
	5.4.5 Numerical Solution Based on Reduced LyapunovEquations	194
	5.4.6 Incomplete LU Versus LU	194
	5.4.7 Parallel Approach	197
	5.5 Conclusions	200
	References	200
	Index	203