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A Condensation-based Low Communication Linear Systems Solver Utilizing Cramer's Rule

Gabriel cramer (1704-1752). A Condensation-based Low Communication Linear Systems Solver Utilizing Cramer's Rule. Ken Habgood, Itamar Arel Department of Electrical Engineering & Computer Science The University of Tennessee. Outline. Motivation & problem statement Algorithm review

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A Condensation-based Low Communication Linear Systems Solver Utilizing Cramer's Rule

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  1. Gabriel cramer (1704-1752) A Condensation-based Low Communication Linear Systems Solver Utilizing Cramer's Rule Ken Habgood, ItamarArelDepartment of Electrical Engineering & Computer ScienceThe University of Tennessee

  2. Outline • Motivation & problem statement • Algorithm review • Numerical accuracy & stability • Parallel Implementation • Communication Results Source: http://tridane.faculty.asu.edu

  3. Introduction • Mainstream approach: Gaussian Elimination • e.g. LU decomposition • Looking for a lower communication overhead, efficient parallel solver • Targeting an unpopular approach: Cramer’s Rule

  4. LU Communication Pattern Communication for distributed LU decomposition L00 U00 U01 U02 L10 A11 A12 L20 A21 A22 • Three sequential steps • Top left computes and sends • Row and column leads compute and send • Remaining processors factorize their blocks • One-to-one communication • Idle time while leads processing Source: http://www.caam.rice.edu/~timwar/MA471F03/

  5. Outline • Motivation & problem statement • Algorithm review • Numerical accuracy & stability • Parallel Implementation • Communication Results Source: http://tridane.faculty.asu.edu

  6. Proposed Algorithm Flow

  7. Matrix “Mirroring” • Mirroring example • Applying Chio’s condensation yields:

  8. Outline • Motivation & problem statement • Algorithm review • Numerical accuracy & stability • Parallel Implementation • Communication Results Source: http://tridane.faculty.asu.edu

  9. Accuracy and Numerical Stability • Backward error estimation • Theoretical estimate of rounding error • E matrix depends on two items • The largest element in A or b • The growth factor of the algorithm • Same growth factor as LU-decomposition with partial pivoting

  10. Forward Error Comparisons

  11. Forward Error - Residual

  12. MATLAB Matrix Gallery

  13. Outline • Motivation & problem statement • Algorithm review • Numerical accuracy & stability • Parallel Implementation • Communication Results Source: http://tridane.faculty.asu.edu

  14. Serial Performance Results support the theoretical ~2.5x complexity ratio

  15. Algorithm Processing Flow

  16. Overview of Parallel Implementation

  17. Parallel Implementation (cont’)

  18. Communication Complexity • Two phases of parallel communication • Parallel Chio’s • Gather Columns • Overall Bandwidth N: Original matrix size, P: number of processors, F: gather columns size

  19. Communication Overhead

  20. Where’s the Breakeven Point? • Point at which Communication “dead time” matches computational workload • Assuming dC = .05 and N = 1000, the breakeven processors point would be P~142

  21. Closing Thoughts … • Proposed O(N3) Cramer’s Rule method • Significantly lower communications overhead • Many more “broadcasts” than “unicasts” • Comm. function of problem size not processors • Next steps … • Optimize parallel implementation • Spare matrix version

  22. Thank you

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