1 / 30

# Component-Averaged Domain Decomposition: CADD, KACZ, CARP

This talk outlines the concepts and applications of Component-Averaged Domain Decomposition (CADD), including the algebraic explanation, the Kaczmarz algorithm (KACZ), and the CARP method. It also discusses the parallel implementation and convergence properties of CADD.

## Component-Averaged Domain Decomposition: CADD, KACZ, CARP

An Image/Link below is provided (as is) to download presentation Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

### Presentation Transcript

1. CADD:Component-AveragedDomain Decomposition Dan Gordon Computer Science University of Haifa Rachel Gordon Aerospace Engg. Technion Component-Averaged Domain Decomposition

2. Outline of Talk • CADD – Algebraic explanation • CADD – DD explanation • The Kaczmarz algorithm (KACZ) • KACZ CARP (a CADD method) • Applications of CARP • CARP-CG: CG acceleration of CARP • Sample results Component-Averaged Domain Decomposition

3. CADD – Algebraic Explanation – 1 • The equations are divided into blocks (not necessarily disjoint) • Initial estimate: vector x=(x1,…,xn) • Suppose x1 is a variable (component of x) that appears in 3 blocks • x1 is “cloned” as y1 , z1 , t1 in the different blocks. • Perform one (or more) iterative operation(s) on each block (independently, in parallel) Component-Averaged Domain Decomposition

4. CADD – Algebraic Explanation – 2 • The internal iterations in each block produce 3 new values for the clones of x1: y1’ , z1’ , t1’ • The next iterative value of x1 is x1’ = (y1’ + z1’ + t1’)/3 • The next iterate is x’ = (x1’ , ... , xn’) • Repeat iterations as needed for convergence Component-Averaged Domain Decomposition

5. CADD – non-overlapping domains clone of x1 y 1 x x 1 0 external grid point of A domain A domain B Component-Averaged Domain Decomposition

6. CADD – overlapping domains domain B clone of x1 y x 1 x 0 1 external grid point of A A∩B domain A Component-Averaged Domain Decomposition

7. CADD – Parallel Implementation • Every processor is in charge of one domain • Mode: BLOCK-PARALLEL • All processors operate in parallel (each on its domain) • Processors exchange clone values • All values are updated • New values = average of clones Component-Averaged Domain Decomposition

8. CADD Vs. Standard DD Methods • Difference is in handling external grid points: • CADD: A clone of the external grid point is modified by the domain's equations and contributes to new value of grid point. • Standard DD: Value of external grid point is fixed (contributes to RHS) Component-Averaged Domain Decomposition

9. KACZ: The Kaczmarz algorithm • Iterative method, due to Kaczmarz (1937). Rediscovered for CT as ART • Basic idea: Consider the hyperplane defined by each equation • Start from an arbitrary initial point • Successively project current point onto the next hyperplane, in cyclic order Component-Averaged Domain Decomposition

10. KACZ: Geometric Description initial point eq. 1 eq. 2 eq. 3 Component-Averaged Domain Decomposition

11. KACZ with Relaxation Parameter • KACZ can be used with a relaxation parameter λ • λ=1: project exactly on the hyperplane • λ<1: project in front of hyperplane • λ>1: project beyond the hyperplane • Cyclicrelaxation: Eq. i is assigned a relaxation parameter λi Component-Averaged Domain Decomposition

12. Convergence Properties of KACZ • KACZ with relaxation (0< λ <2)converges for consistent systems: • Herman, Lent & Lutz, 1978 • Trummer, 1981 • For inconsistent systems, KACZ converges cyclically: • Tanabe, 1971 • Eggermont, Herman & Lent, 1981 (for cyclic relaxation parameters). Component-Averaged Domain Decomposition

13. Another view of KACZ • Given the system Ax = b • Consider the "normal equations" system AATy = b, x=ATy • Well-known: KACZ is simply SOR applied to the normal equations • The relaxation parameter of KACZ is the usual relax. par. of SOR Component-Averaged Domain Decomposition

14. CARP: Component-Averaged Row Projections • CARP is a CADD method with KACZ iterations in each domain: • The system Ax=b is divided into blocks B1,…,Bn (need not be disjoint) • Each processor is assigned one (or more) block • All the blocks are processed in parallel • Results from blocks are “merged” to form the next iterate • Merging is done by CADD averaging Component-Averaged Domain Decomposition

15. Overview of CARP domain A domain B averaging KACZ iterations KACZ iterations cloning KACZ in superspace (with cyclic relaxation) Component-Averaged Domain Decomposition

16. Convergence of CARP • Averaging Lemma: the component-averaging and cloning operations of CARP are equivalent to KACZ row-projections in a certain superspace (with λ= 1) • CARP is equivalent to KACZ in the superspace, with cyclic relaxation parameters – known to converge Component-Averaged Domain Decomposition

17. "Historical" Note • The term "component averaging" was first used by Censor, Gordon & Gordon (2001) for CAV and BICAV, used for image reconstruction in CT • These are Cimmino-type algorithms, with weights related to the sparsity of the system matrix • CADD, CARP use this concept in a different sense Component-Averaged Domain Decomposition

18. CARP Application: Solution of stiff linear systems from PDEs • Elliptic PDEs w/large convection term result in stiff linear systems (large off-diagonal elements) • CARP is very robust on these systems, as compared to leading solver/preconditioner combinations • Downside: Not always efficient Component-Averaged Domain Decomposition

19. CARP Application: Electron Tomography(joint work with J.-J. Fernández) • 3D reconstructions: Each processor is assigned a block of consecutive slices. Data is in overlapping blobs. • The blocks are processed in parallel. • The values of shared variables are transmitted between the processors which share them, averaged, and redestributed. Component-Averaged Domain Decomposition

20. CARP-CG: CG acceleration of CARP • CARP is KACZ in some superspace (with cyclic relaxation parameter) • Björck & Elfving (BIT 79): developed CGMN, which is a (sequential)CG-acceleration of KACZ (double sweep, fixed relax. parameter) • We extended this result to allow cyclic relaxation parameters • Result: CARP-CG Component-Averaged Domain Decomposition

21. CARP-CG: Properties • Same robustness as CARP • Very significant improvement in performance on stiff linear systems derived from elliptic PDEs • Very competitive runtime compared to leading solver/preconditioner combinations on systems derived from convection-dominated PDEs • Improved performance in ET Component-Averaged Domain Decomposition

22. CARP-CG on PDEs CG acceleration of projection methods was done before, but with block projections, requiring multi-coloring. CARP-CG avoids this. Tests were run on 9 convection-dominated PDEs, comparing CARP-CG with restarted GMRES, CGS and Bi-CGSTAB, with and without various preconditioners. Also tested: CGNR. Domain: unit cube – 80x80x80. Component-Averaged Domain Decomposition

23. 512,000 equations Component-Averaged Domain Decomposition

24. 512,000 equations Component-Averaged Domain Decomposition

25. 512,000 equations Component-Averaged Domain Decomposition

26. 512,000 equations Component-Averaged Domain Decomposition

27. 512,000 equations Component-Averaged Domain Decomposition

28. CARP-CG: ET results Component-Averaged Domain Decomposition

29. Future research on CADD • Different internal solvers: 1. Guaranteed convergence? 2. Symmetrizability? • Combine CADD with multigrid • CFD applications • Additional biomedical applications • CADD with preconditioners • Compare CADD with other DD methods, such as additive Schwarz Component-Averaged Domain Decomposition

30. CARP Publications • CARP: SIAM J. Sci. Comput. 27 (2005) 1092-1117. http://cs.haifa.ac.il/~gordon/carp.pdf • Electron tomography: J. Par. & Dist. Comput. (2007), in press. http://cs.haifa.ac.il/~gordon/eltom.pdf • CARP-CG: Submitted for publication. Component-Averaged Domain Decomposition

More Related