1. CSE245: Computer-Aided Circuit Simulation and Verification
Multipole method based on the thesis of Leslie Greengrad, 1987
Chung-Kuan Cheng

2. Outlines Strategy A 2 Dimensional Example Source grouping
Destination grouping
A 3 Dimensional Example
Remark

3. Strategy: Definition Dipole:
a pair of equal and opposite electric charges or magnetic poles of opposite sign separated especially by a small distance
Multipole:
having or involving several or many poles

4. Strategy: The applications
System of huge variables
Simulation of physical systems
EM wave, stellar clusters, protein folding, acoustics, turbulence
Learning theory
Kernel methods, support vector machines
Graphics and vision
Light scattering

5. Strategy Derivation of Potential vs. Field
Special case of matrix vector multiplication
Si= sumj f(zi, zj) mj
Multipole: two reference systems based on the local regions of obj i and obj j.
Complexity O(n2) => O(n logn)

6. Strategy of Multipole Method

7. A 2 Dimensional Example

8. Multipole: Source Grouping

9. Separation of the region

10. Source Grouping

11. Definition of Regions

12. Destination Grouping

13. Multipole Algorithm

14. Regions of Algorithm

15. Boundary Condition

16. A 3 Dimensional Example

17. 3 Dimension Space

18. 3 Dimension: Polar Coordinate System

19. Legendre Polynomials

20. Legendre Polynomials

21. Source Grouping

22. Destination Grouping

23. Destination Grouping

24. Remark Format of the matrix vector multiplication Dimensions
2D Case: Complex space
3D Case: Polar coordinate system
Data structure: hierarchy of multiple level regions
Error estimation
Boundary conditions
Sensitivity derivation