Based on Extended Krylov Subspace for Transient Simulation of - - PowerPoint PPT Presentation

▶
based on extended krylov
SMART_READER_LITE
LIVE PREVIEW

Based on Extended Krylov Subspace for Transient Simulation of - - PowerPoint PPT Presentation

Dec 22, 2022 269 likes •542 views

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

slide-1
SLIDE 1

Efficient Matrix Exponential Method Based on Extended Krylov Subspace for Transient Simulation of Large-Scale Linear Circuits

Wenhui Zhao University of Hong Kong

1

slide-2
SLIDE 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

2

slide-3
SLIDE 3

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

slide-4
SLIDE 4
  • 1. Introduction

 1.1 Circuit Simulation

 Circuit simulation is to use mathematical models to

predict the behavior of an electronic circuit.

4

Ordinary differential equations (ODEs)

slide-5
SLIDE 5
  • 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−1𝐻 and 𝑐 𝑢 = 𝐷−1𝐶𝑣 𝑢 .

5

slide-6
SLIDE 6

For simplicity , we will use 𝐵 to represent the 𝐵 in the following part.

6

𝑦 𝑢 + ℎ = 𝑓𝐵ℎ𝑦 𝑢 + 𝑓𝐵 ℎ−𝜐 𝑐 𝑢 + 𝜐 𝑒𝜐

ℎ

𝑦 𝑢 + ℎ = 𝑓𝐵ℎ𝑦 𝑢 + 𝑓𝐵ℎ − 𝐽 𝐵−1𝑐 𝑢 + 𝑓𝐵ℎ − 𝐵ℎ + 𝐽 𝐵−2 𝑐 𝑢 + ℎ − 𝑐(𝑢) ℎ 𝑦 𝑢 + ℎ = [𝐽𝑜 0]𝑓𝐵

ℎ 𝑦(𝑢)

𝑓2 𝐵 = 𝐵 𝑋 𝐾 , 𝐾 = 0 1 0 , 𝑓2 = 0 1 , 𝑋 = 𝑐 𝑢 + ℎ − 𝑐(𝑢) ℎ 𝑐(𝑢) piece-wise linear (PWL) input transform

𝒇𝑩𝒊

Krylov subspace

slide-7
SLIDE 7
  • 1. Introduction

 1.2 Matrix Exponential Method (MEXP)

 Main computation is

𝑓𝐵ℎ𝑤 ≈ 𝛾𝑊

𝑛𝑓𝑈 𝑛ℎ𝑓1,

𝛾 = 𝑤 2

 Krylov subspace: 𝐿𝑛 = 𝑡𝑞𝑏𝑜{𝑤, 𝐵𝑤, 𝐵2𝑤, … 𝐵𝑛−1𝑤}  Arnoldi process: 𝐵𝑊

𝑛 = 𝑊 𝑛+1𝑈

𝑛

𝑊

𝑛 : orthonormal basis of 𝐿𝑛 𝐵, 𝑤

𝑈 𝑛 : contains the orthonormalization coefficients

 Error estimate: 𝑓𝑠𝑠 = 𝛾𝑢𝑛+1,𝑛 𝑓𝑛

𝑈 𝑓𝑈

𝑛ℎ𝑓1

 𝑢𝑛+1,𝑛 is the bottom right element of 𝑈

𝑛

7

slide-8
SLIDE 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

8

slide-9
SLIDE 9

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

slide-10
SLIDE 10
  • 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

slide-11
SLIDE 11
  • 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:

𝐿𝑚,𝑛 = 𝑡𝑞𝑏𝑜{𝐵−𝑚+1𝑤, … 𝐵−1𝑤, 𝑤, 𝐵𝑤, … 𝐵𝑛−1𝑤} 𝐿𝑛,𝑛 = 𝑡𝑞𝑏𝑜{𝑤, 𝐵−1𝑤, 𝐵𝑤, … 𝐵−𝑛+1𝑤, 𝐵𝑛−1𝑤}

11

slide-12
SLIDE 12
  • 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

  • nly augments in positive direction

𝐿𝑚,𝑛 = 𝑡𝑞𝑏𝑜{𝐵−𝑚+1𝑤, … 𝐵−1𝑤, 𝑤, 𝐵𝑤, … 𝐵𝑛−1𝑤}

 Equal number of negative and positive dimension may

lead to waste of runtime 𝐿𝑛,𝑛 = 𝑡𝑞𝑏𝑜{𝑤, 𝐵−1𝑤, 𝐵𝑤, … 𝐵−𝑛+1𝑤, 𝐵𝑛−1𝑤}

12

slide-13
SLIDE 13
  • 2. MEXP based on Extended Krylov Subspace

 2.2 Generalized Extended Krylov Subspace

 Generalized extended Krylov subspace with

unequal number of positive/negative dimensions: 𝐿𝑛,𝑙𝑛 = 𝑡𝑞𝑏𝑜{𝑤, 𝐵1𝑤, 𝐵2𝑤 … 𝐵𝑙𝑤, 𝐵−1𝑤, 𝐵𝑙+1𝑤, … 𝐵2𝑙𝑤, 𝐵−2𝑤, … , 𝐵𝑙𝑛−1𝑤, 𝐵−𝑛+1𝑤}

 Arnoldi-type process: 𝐵𝑊

𝑛 = 𝑊 𝑛+2𝑈

𝑛

 𝑈

𝑛 is a block Heisenberg matrix

 Posterior error estimate:

𝑓𝑠𝑠 = 𝛾𝜐𝑛+1,𝑛 𝑓𝑛

𝑈 𝑓𝑈

𝑛ℎ𝑓1

 𝜐𝑛+1,𝑛is the 2-by-2 bottom right block of 𝑈

𝑛

13

slide-14
SLIDE 14
  • 2. MEXP based on Extended Krylov Subspace

 How to compute 𝑈

𝑛 effectively and economically?

 From the construction of the generalized extended Krylov

subspace, we can get the following recursive relations:

14

slide-15
SLIDE 15
  • 2. MEXP based on Extended Krylov Subspace

 Can we compute 𝑈

𝑛 without extra matrix-vector products of 𝑊

𝑛+2 𝑈

𝐵𝑊

𝑛?

15

slide-16
SLIDE 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

16

slide-17
SLIDE 17

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

slide-18
SLIDE 18
  • 3. Numerical Results

 3.1 Improvement led by extended Krylov subspace  Example: RC ladder

 Stiff circuit; Matrix order: 1000;  Compute 𝑓𝐵ℎ𝑤 by four Krylov subspaces  Krylov subspace with different negative-positive ratios

k=0, 1, 2, 5 (dimension: 24)

18

slide-19
SLIDE 19
  • 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

slide-20
SLIDE 20
  • 3. Numerical Results

20

slide-21
SLIDE 21
  • 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

slide-22
SLIDE 22
  • 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

slide-23
SLIDE 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

23

slide-24
SLIDE 24

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

slide-25
SLIDE 25
  • 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

slide-26
SLIDE 26

Q & A

Thank you!

26