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Efficient Matrix Exponential Method Based on Extended Krylov Subspace for Transient Simulation of Large-Scale Linear Circuits Wenhui Zhao University of Hong Kong 1

Outline  Introduction  Circuit Simulation  Matrix Exponential Method(MEXP)  MEXP based on Extended Krylov Subspace  Problem of Stiff Circuit  Generalized Extended Krylov Subspace  Numerical Results  Conclusion 2

Outline  Introduction  Circuit Simulation  Matrix Exponential Method(MEXP)  MEXP based on Extended Krylov Subspace  Problem of Stiff Circuit  Generalized Extended Krylov Subspace  Numerical Results  Conclusion 3

1. Introduction  1.1 Circuit Simulation  Circuit simulation is to use mathematical models to predict the behavior of an electronic circuit. Ordinary differential equations (ODEs) 4

1. Introduction  1.2 Matrix Exponential Method (MEXP) The numerical system to be solved in transient circuit analysis is a set of  differential algebraic equations (DAE) 𝐷𝑦 𝑢 = 𝐻𝑦 𝑢 + 𝐶𝑣 𝑢 𝐷, 𝐻 and 𝐶: susceptance, conductance and input matrix, respectively 𝑣(𝑢) : collects the voltage and current sources The essence of MEXP lies in transforming the above equation to an  ODE 𝑦 𝑢 = 𝐵𝑦 𝑢 + 𝑐 𝑢 where 𝐵 = 𝐷 −1 𝐻 and 𝑐 𝑢 = 𝐷 −1 𝐶𝑣 𝑢 . 5

𝑦 𝑢 + ℎ = 𝑓 𝐵ℎ 𝑦 𝑢 + 𝑓 𝐵 ℎ−𝜐 𝑐 𝑢 + 𝜐 𝑒𝜐 ℎ 0 piece-wise linear (PWL) input 𝑦 𝑢 + ℎ = 𝑓 𝐵ℎ 𝑦 𝑢 + 𝑓 𝐵ℎ − 𝐽 𝐵 −1 𝑐 𝑢 𝐵 −2 𝑐 𝑢 + ℎ − 𝑐(𝑢) + 𝑓 𝐵ℎ − 𝐵ℎ + 𝐽 ℎ transform ℎ 𝑦(𝑢) 𝑦 𝑢 + ℎ = [𝐽 𝑜 0]𝑓 𝐵 𝑓 2 1 , 𝑋 = 𝑐 𝑢 + ℎ − 𝑐(𝑢) = 𝐵 𝑋 , 𝐾 = 0 1 0 , 𝑓 2 = 0 𝐵 𝑐(𝑢) 0 𝐾 0 ℎ 𝒇 𝑩𝒊 Krylov subspace in the following part. For simplicity , we will use 𝐵 to represent the 𝐵 6

1. Introduction  1.2 Matrix Exponential Method (MEXP)  Main computation is 𝑛 ℎ 𝑓 1 , 𝑓 𝐵ℎ 𝑤 ≈ 𝛾𝑊 𝑛 𝑓 𝑈 𝛾 = 𝑤 2  Krylov subspace: 𝐿 𝑛 = 𝑡𝑞𝑏𝑜{𝑤, 𝐵𝑤, 𝐵 2 𝑤, … 𝐵 𝑛−1 𝑤} 𝑛  Arnoldi process: 𝐵𝑊 𝑛 = 𝑊 𝑛+1 𝑈 𝑊 𝑛 : orthonormal basis of 𝐿 𝑛 𝐵, 𝑤  𝑛 : contains the orthonormalization coefficients 𝑈  𝑈 𝑓 𝑈 𝑛 ℎ 𝑓 1  Error estimate: 𝑓𝑠𝑠 = 𝛾𝑢 𝑛+1,𝑛 𝑓 𝑛 𝑛  𝑢 𝑛+1,𝑛 is the bottom right element of 𝑈 7

Outline  Introduction  Circuit Simulation  Matrix Exponential Method(MEXP)  MEXP based on Extended Krylov Subspace  Problem of Stiff Circuit  Generalized Extended Krylov Subspace  Numerical Results  Conclusion 8

Outline  Introduction  Circuit Simulation  Matrix Exponential Method(MEXP)  MEXP based on Extended Krylov Subspace  Problem of Stiff Circuit  Generalized Extended Krylov Subspace  Numerical Results  Conclusion 9

2. MEXP based on Extended Krylov Subspace  2.1 Problem for Stiff Circuits  Stiff circuits:  Time constants differ by a large magnitude  Real parts of eigenvalues are well-separated  Shortcomings of Krylov subspace:  Tend to capture the dominant eigenvalues first  Tend to undersample of small magnitude eigenvalues 10

2. MEXP based on Extended Krylov Subspace  2.1 Problem for Stiff Circuits  Traditional extended Krylov subspace:  Merits: Capture the small magnitude eigenvalues because of the basis vectors from negative power of the matrix  Demerits: Computation of negative dimensions are more expensive than the computation of positive dimensions  Existing extended Krylov subspace: 𝐿 𝑚,𝑛 = 𝑡𝑞𝑏𝑜{𝐵 −𝑚+1 𝑤, … 𝐵 −1 𝑤, 𝑤, 𝐵𝑤, … 𝐵 𝑛−1 𝑤} 𝐿 𝑛,𝑛 = 𝑡𝑞𝑏𝑜{𝑤, 𝐵 −1 𝑤, 𝐵𝑤, … 𝐵 −𝑛+1 𝑤, 𝐵 𝑛−1 𝑤} 11

2. MEXP based on Extended Krylov Subspace  2.1 Problem of the Stiff Circuit  Shortcoming of existing extended Krylov subspace:  Negative dimension 𝑚 need to be prespecified, subspace only augments in positive direction 𝐿 𝑚,𝑛 = 𝑡𝑞𝑏𝑜{𝐵 −𝑚+1 𝑤, … 𝐵 −1 𝑤, 𝑤, 𝐵𝑤, … 𝐵 𝑛−1 𝑤}  Equal number of negative and positive dimension may lead to waste of runtime 𝐿 𝑛,𝑛 = 𝑡𝑞𝑏𝑜{𝑤, 𝐵 −1 𝑤, 𝐵𝑤, … 𝐵 −𝑛+1 𝑤, 𝐵 𝑛−1 𝑤} 12

2. MEXP based on Extended Krylov Subspace  2.2 Generalized Extended Krylov Subspace  Generalized extended Krylov subspace with unequal number of positive/negative dimensions: 𝐿 𝑛,𝑙𝑛 = 𝑡𝑞𝑏𝑜{𝑤, 𝐵 1 𝑤, 𝐵 2 𝑤 … 𝐵 𝑙 𝑤, 𝐵 −1 𝑤, 𝐵 𝑙+1 𝑤, … 𝐵 2𝑙 𝑤, 𝐵 −2 𝑤, … , 𝐵 𝑙𝑛−1 𝑤, 𝐵 −𝑛+1 𝑤} 𝑛  Arnoldi-type process: 𝐵𝑊 𝑛 = 𝑊 𝑛+2 𝑈 𝑛 is a block Heisenberg matrix  𝑈  Posterior error estimate: 𝑈 𝑓 𝑈 𝑛 ℎ 𝑓 1 𝑓𝑠𝑠 = 𝛾𝜐 𝑛+1,𝑛 𝑓 𝑛 𝑛  𝜐 𝑛+1,𝑛 is the 2-by-2 bottom right block of 𝑈 13

2. MEXP based on Extended Krylov Subspace 𝑛 effectively and economically?  How to compute 𝑈  From the construction of the generalized extended Krylov subspace, we can get the following recursive relations: 14

2. MEXP based on Extended Krylov Subspace 𝑛 without extra matrix-vector  Can we compute 𝑈 𝑈 products of 𝑊 𝐵𝑊 𝑛 ? 𝑛+2 15

Outline  Introduction  Circuit Simulation  Matrix Exponential Method(MEXP)  MEXP based on Extended Krylov Subspace  Problem of Stiff Circuit  Generalized Extended Krylov Subspace  Numerical Results  Conclusion 16

Outline  Introduction  Circuit Simulation  Matrix Exponential Method(MEXP)  MEXP based on Extended Krylov Subspace  Problem of Stiff Circuit  Generalized Extended Krylov Subspace  Numerical Results  Conclusion 17

3. Numerical Results  3.1 Improvement led by extended Krylov subspace  Example: RC ladder  Stiff circuit; Matrix order: 1000;  Compute 𝑓 𝐵ℎ 𝑤 by four Krylov subspaces  Krylov subspace with different negative-positive ratios k=0, 1, 2, 5 (dimension: 24) 18

3. Numerical Results  3.1 Improvement led by extended Krylov subspace  Extended Krylov subspace enjoys higher accuracy but increases runtime as a trade off 19

3. Numerical Results 20

3. Numerical Results  3.2 Performance of MEXP based on different Krylov subspace with real circuit examples  Example: three linear circuit examples  Run 100 time step with a constant step size  Allow the subspace dimension to vary dynamically to satisfy a tolerance of 10 −6 21

3. Numerical Results  Standard Krylov subspace requires a much larger order of the subspace than extended Krylov subspace  The best breakdown of positive and negative dimensions in extended Krylov subspace is generally problem dependent 22

Outline  Introduction  Circuit Simulation  Matrix Exponential Method(MEXP)  MEXP based on Extended Krylov Subspace  Problem of Stiff Circuit  Generalized Extended Krylov Subspace  Numerical Results  Conclusion 23

Outline  Introduction  Circuit Simulation  Matrix Exponential Method(MEXP)  MEXP based on Extended Krylov Subspace  Problem of Stiff Circuit  Generalized Extended Krylov Subspace  Numerical Results  Conclusion 24

4. Conclusion We have investigated the use of extended Krylov subspace to enhance  the accuracy of numerical approximation of MEXP-vector product, which in turn benefits the MEXP-based transient circuit simulation. We generalize the extended Krylov subspace to allow unequal  positive/negative dimensions to maximize the overall performance in circuit simulation. Numerical results have confirmed the efficiency of the proposed method.  25

Q & A Thank you! 26