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 GABRIEL CRAMER ( )

EECS Department / University of Tennessee EECS Department / University of Tennessee Outline 2  Motivation & problem statement  Algorithm review  Numerical accuracy & stability  Parallel Implementation  Communication Results Source:

EECS Department / University of Tennessee EECS Department / University of Tennessee 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 3

EECS Department / University of Tennessee EECS Department / University of Tennessee LU Communication Pattern Source:  Communication for distributed LU decomposition L00 U00 U01U02 L10A11A12 L20A21A22  Three sequential steps 1. Top left computes and sends 2. Row and column leads compute and send 3. Remaining processors factorize their blocks  One-to-one communication  Idle time while leads processing 4

EECS Department / University of Tennessee EECS Department / University of Tennessee Outline 5  Motivation & problem statement  Algorithm review  Numerical accuracy & stability  Parallel Implementation  Communication Results Source:

EECS Department / University of Tennessee EECS Department / University of Tennessee Proposed Algorithm Flow 6

EECS Department / University of Tennessee EECS Department / University of Tennessee Matrix “Mirroring”  Mirroring example  Applying Chio’s condensation yields: 7

EECS Department / University of Tennessee EECS Department / University of Tennessee Outline 8  Motivation & problem statement  Algorithm review  Numerical accuracy & stability  Parallel Implementation  Communication Results Source:

EECS Department / University of Tennessee EECS Department / University of Tennessee 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 9

EECS Department / University of Tennessee EECS Department / University of Tennessee Forward Error Comparisons Matrix Size κ(A) Max MatlabMax GSL Avg Matlab Avg GSL 1000 x E E E E x E E E E x E E E E x E E E E x E E E E x E E E E x E E E E x E E E E-11 10

EECS Department / University of Tennessee EECS Department / University of Tennessee Forward Error - Residual Matrix Sizeκ(A) Max Residual Avg Residual 1000 x E E x E E x E E x E E x E E x E E x E E x E E-09 11

EECS Department / University of Tennessee EECS Department / University of Tennessee MATLAB Matrix Gallery Special Matrix Avg MatlabResidual Matlab Residual clement — Tridiagonal matrix with zero diagonal entries1.40E E E+144 lehmer — Symmetric positive definite matrix2.49E E E-06 circul — Circulant matrix3.23E E E-09 chebspec — Chebyshev spectral differentiation matrix9.12E E+042.0E-01 lesp — Tridiagonal matrix with real, sensitive eigenvalues9.56E E E-10 minij — Symmetric positive definite matrix5.14E E E-06 orthog — Orthogonal and nearly orthogonal matrices1.03E E E-08 randjorth — Random J-orthogonal matrix1.55E E E-04 12

EECS Department / University of Tennessee EECS Department / University of Tennessee Outline 13  Motivation & problem statement  Algorithm review  Numerical accuracy & stability  Parallel Implementation  Communication Results Source:

EECS Department / University of Tennessee EECS Department / University of Tennessee Serial Performance Results support the theoretical ~2.5x complexity ratio 14

EECS Department / University of Tennessee EECS Department / University of Tennessee Algorithm Processing Flow 15

EECS Department / University of Tennessee EECS Department / University of Tennessee Overview of Parallel Implementation 16

EECS Department / University of Tennessee EECS Department / University of Tennessee Parallel Implementation (cont’) 17

EECS Department / University of Tennessee EECS Department / University of Tennessee  Two phases of parallel communication  Parallel Chio’s  Gather Columns  Overall Bandwidth Communication Complexity N: Original matrix size, P: number of processors, F: gather columns size 18

EECS Department / University of Tennessee EECS Department / University of Tennessee Communication Overhead 19

EECS Department / University of Tennessee EECS Department / University of Tennessee  Point at which Communication “dead time” matches computational workload Where’s the Breakeven Point?  Assuming d C =.05 and N = 1000, the breakeven processors point would be P ~142 20

EECS Department / University of Tennessee EECS Department / University of Tennessee Closing Thoughts …  Proposed O(N 3 ) 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 21

EECS Department / University of Tennessee EECS Department / University of Tennessee Thank you 22