Fast Algorithm for Matrix-Vector Multiply of Asymmetric Multilevel Block-Toeplitz Matrices

Transcription

1 " Fast Algorithm for Matrix-Vector Multiply of Asymmetric Multilevel Block-Toeplitz Matrices B. E. Barrowes, F. L. Teixeira, and J. A. Kong Research Laboratory of Electronics, MIT, Cambridge, MA ElectroScience Laboratory and Department of Electrical Engineering, The Ohio State University, Columbus, OH 43210

2 " Presentation Outline I. Introduction and Motivation II. Definitions and Formulation III. Recursive Algorithms IV. Examples Scattering from arbitrary object 3-D SMCG V. Conclusion

3 " Introduction and Motivation Multilevel block-toeplitz (MBT) matrices often arise in the solution of electromagnetic scattering problems due to the translational invariance and convolutional nature of the Green s function. Discrete Dipole Approximation (DDA) 2-D and 3-D scattering based on cubic meshes Other situations, e.g. autocorrelation of 2-D random process Conjugant Gradient (CG) solvers are based on matrix-vector multiplies complexity memory storage MBT Fast multiply reduces these requirements 1-D FFT complexity memory storage

4 " Problem Formulation For 1-D point interactions: ents Interaction matrix is Toeplitz

5 " 2 3 ( ( E > A B *) ) -,,, ). /01 " " " " Definitions Let be a square matrix in the matrix equation where continuing until the level with " " arbitrary '& 89 :-; <=?> PSfrag replacements Example of a are boxed with a solid line. In the upper left hand corner, is enclosed by a dash-dot box, while at the lower left hand corner in delineated by a dashed box. Thin solid lines delineate interaction between different rows of a 2x2x2 cubic mesh, while bold solid line delineate between planes. asymmetric MBT matrix. Unique entries C*D

6 " from Fast Multiply Recursive Algorithms Fast FFT assisted multiply requires 4 steps 1. Assign unique entries of to 2. Insert zeros into 3. FFT of and, multiplication in the Fourier domain 4. Reconstruct

7 " block s order is: ) ), 1. Assign unique entries of to Proceed recursively from lower left to upper right Final For above, Because onlly is stored, memory requirement is reduced from to

8 > A ) 89 :-; <=?> B 2. Insert zeros into Given ),,, ) and, proceed recursively 1. Insert 0 s for unique entries along LHS of Toeplitz blocks 2. Zero pad to length of For, PSfrag replacements

9 " Convolution Picture Sfrag replacements Then / ts x z A u

10 " 3. Fast Fourier Transforms Now convolution becomes a multiplication in Fourier domain 1. FFT of and, multiplication in the Fourier domain For CG type solvers, perform Therefore only 2 FFTs required per iteration only once

11 " from 4. Reconstruct Some terms (corresponding to inserted zeros) must be removed from For and above, For further detail and sample Matlab codes, see Barrowes,, to appear October 2001.

12 " (For 3-D, this creates a can be seen to depend only on, Example I Scattering from an Arbitrarily Shaped Object Small dielectric spheres placed on lattice point (similar to DDA) Scattering from 1 small particle Self-consistent solution for scattering from many small particles For otherwise similar spheres on lattice points, the interaction between particles and which is translationally invariant.

13 " Example I Results MBT Fast Full length of Method Method (FFT length) Time (s) Time (s) error

14 " Example I Results 10 3 N g N(cube) ε eff error Mie theory i e 06i i i i i N g N(cube) ε eff error Mie theory i e 06i i i i i Scattering Intensity 10 1 Scattering Intensity Scattering Angle θ Scattering Angle θ

15 " Example II 3-D SMCG for Spheroids S 2 Strong interaction cloud S 1 PSfrag replacements y S x z

16 " " '& ( Example II Results Expansion Number of expansion order ( ) terms ( ) sion order. of FFTs (both forward and inverse) for expansion Number of expansion terms and total number Exact Solution 6x6x6 mesh 8x8x8 mesh 10x10x10 mesh 12x12x12 mesh 14x14x14 mesh PSfrag replacements Expansion Order ( ) Kullback-Leibler distance Time/iteration(s)

17 " Conclusion FFT-based method to expedite matrix-vector multiplies involv- A new ing multilevel block-toeplitz (MBT) matrices was presented. This methods applies to MBT matrices with any This is a minimal memory method storing only the unique entries of This method is applicable to many electromagnetic problems based on cubic meshes This method was shown to apply to scattering from an object of arbitrary shape and shown useful in an extension of the SMCG method to 3-D. This work was supported by National Science Foundation (NSF) under grant numbers ECS and ECS , by the Office of Naval Research (ONR) under contract numbers N and N , and through a NSF Graduate Fellowship.