Fast Parallel Iterative Matrix Diagonalization

Windsor Hsu, Hong Yun Wang, Bernd Pfrommer

Abstract:

We find that with our data layout, the 3D Fast Fourier Transform (FFT) involved in matrix-vector multiplies is the most difficult part to parallelize, and limits the speedup for small problem sizes. For larger problem sizes, the speedup becomes better because operations requiring little communication dominate the FFT.

• Introduction
  • The Physical Problem Under Consideration
  • Formalism
• The Specific Form of the Matrix
• Three Algorithms for finding some lowest eigenvalues and eigenvectors of a Hermitian matrix
  • A Conjugate Gradient Algorithm to Solve the Minimization Problem
  • A Conjugate Gradient Minimization Scheme Without Matrix Inversion
  • A New Conjugate Gradient Algorithm For Minimization With Constraints
  • Comparison of the Algorithms
• Implementation
  • Data Layout
  • Changes to the Existing Code
• Visualization
• Timing of the Diagonalization Routine and Scaling Behavior
  • Speedup
  • Time Breakdown for the Diagonalization Routine
  • Scaling Behavior
• The LPO Routine
  • Inverse 3D-FFT Design Decisions
  • Canned 1D-FFT Routine
  • Overlapping Communication with Computation
  • The Blocking Factor
  • Timing Model for the Inverse 3D-FFT Routine
  • Performance of PVM
  • Performance of the Inverse 1D-FFT Routine
  • Accuracy of the Timing Model
• Conclusion
• References

Administration