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