微波电路积分方程中的预处理共轭梯度算法

摘要

In this thesis,the conjugate gradient (CG) technique is applied to dense multilevel block-Toeplitz matrix equations from the mixed potential integral equation (MPIE).To significantly reduce the memory requirement and computational cost,the conjugate gradient (CG) method combined with the fast Fourier transformation (FFT),which is often referred to as the CG-FFT method is adapted.When FFT technique is used,the vector-vector multiplication in spectrl domain can replace the Topelitz matrix-vector multiplication in spatial domain.Therefore,the computational complexity of O(N<'2>) is reduced to O(N log N) per iteration.However,the FFT technique can't reduce the iteration number of CG method,which is largely depends on the spectral properties of the integral operator or the matrices of discrete linear systems.The preconditioning is simply a means of transforming the original linear system into one which has the same solution,but which is likely to be easier to solve by reducing the condition number of the operator equations.The banded diagonal matrix,symmetriv successive overrelaxation (SSOR), block diagonal matrix,sparse approximate invert and wavelet based sparse approximate inverse preconditioning technique are applied to CG method in this thesis.Our numerical calculations show that the PCG-FFT algorithms with these preconditioners converge much faster that the conventional one to the MPIE for microwave circuits.Some typical microstrip discontinuities are analyzed and the good results demonstrate the validity of all these proposed algorithms.

目录

英文文摘

Chapter 1 Introduction

References

Chapter 2 Discretization of MPIE and FFT Technique

2.1 Introduction

2.2 FFT Technique applied in matrix-vector multiplication

2.3 Discretization of MPIE

References

Chapter 3 Banded Diagonal and SSOR Preconditioning

3.1 Introduction

3.2 Banded Diagonal and SSOR Preconditioning

3.2.1 Theory Analysis

3.2.2 Results and Discussions

3.3 Block Diagonal Preconditioned CGAlgorithm

3.3.1 Theory Analysis

3.3.2 Results and Discussions

3.4 Conclusions

References

Chapter 4 Sparse Approximate Invert Preconditioning

4.1 Introduction

4.2 Sparse Approximate Invert Preconditioning

4.2.1 Algorithm of Multifrontal Method

4.2.2 Results and Discussions

4.3 Wavelet Based Sparse Approximate Inverse Preconditioning

4.3.1 discrete wavelet transform

4.3.2 Results and Discussions

4.4 Conclusions

References

Conclusions

Acknowledgments

List of Publications Produced During the Study