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Parallel Computations in Quantum Lanczos Representation Methods

Parallel Computations in Quantum Lanczos Representation Methods. Hong Zhang and Sean C. Smith Quantum & Molecular Dynamics Group Center for Computational Molecular Science The University of Queensland, Australia. Outline. Introduction 2. Methodologies 3. Results

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Parallel Computations in Quantum Lanczos Representation Methods

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  1. Parallel Computations in Quantum Lanczos Representation Methods Hong Zhang and Sean C. Smith Quantum & Molecular Dynamics Group Center for Computational Molecular ScienceThe University of Queensland, Australia

  2. Outline • Introduction • 2. Methodologies • 3. Results • 4. Conclusions & Future Work

  3. 1. Introduction Lanczos (1950) 1. Wyatt: Recursive residue generation method 2. Iung and Leforestier; Yu and Nyman; Carrington: Spectrally transformed Lanczos algorithm 3. Manthe, Seideman, and Miller; Karlsson: GMRES/QMR DVR-ABC 4. Guo; Carrington: Symmetry adapted Lanczos method 5. Zhang and Smith: Lanczos representation methods.

  4. 1.1 Lanczos representation methods 1. Lanczos representation filter diagonalisation in unimolecular dissociation and rovibrational spectroscopy: recent non-zero total angular momentum (J > 0) calculations using parallel computing implementation. 2. TI Lanczos subspace wavepacket method; real Lanczos artificial boundary inhomogeneity (LABI); real single Lanczos propagation ABI in bimolecular reactions.

  5. 3. Systems: HOCl and HO2 Importance in atmospheric chemistry and in combustion chemistry, e.g., balance of stratospheric ozone, etc.

  6. [Zhang & Smith, PhysChemComm, 6: 12, 2003]

  7. 2. Methodologies 1. Transform the primary representation (DVR or FBR) of the fundamental scattering equations (e.g., TI Schrödinger equation or TI wavepacket-Lippmann-Schwinger equation) into the tridiagonal Lanczos representation. 2. Solve the eigen-problem orlinear systemwithin the subspace to obtain either eigen-pairs or subspace wavefunctions. 3. Extract physical information, e.g., bound states, resonances, and scattering (S) matrix elements.

  8. Representation transformation [Kouri, Arnold & Hoffman, CPL, 203: 166, 1993; Zhang H, Smith SC, JCP, 116: 2354, 2002]

  9. Solving subspace eigen-problem or linear system

  10. Algorithm (a) Choose Mth element of (Ej) to be arbitrary (but non-zero) and calculate (b) For k=M-1, M-2, …, 2, update scalar k-1: (c) Determine constant (true= c) by normalization or by

  11. Comparison with other eigen-problem/linear system solvers • QMR/MINRES • QL/QR • Forward recursion Efficiency very important, as linear system solver must be repeated for many different filter energies throughout spectrum. [Yu & Smith, BBPC,101: 400, 1997; Yu & Smith, CPL, 283: 69, 1998; Zhang & Smith, JCP, 115: 5751, 2001; Zhang & Smith, PCCP, 3: 2282, 2001]

  12. Representation Parallel computing Final analysis Conceptually difficult Propagation Computationally difficult: cpu time and memory In propagation, the most time consuming part is the matrix-vector multiplication. We use Message-passing interface (MPI) to perform parallel computation.

  13. J > 0 DVR Hamiltonian

  14. Matrix-vector multiplication

  15. a) Master processor (ID = 0) Perform the main propagation; write ,  elements in bound and resonance calculations; all other related works except the matrix-vector multiplications. b) Working/slave processors (ID = 1, 2, …) Perform the matrix-vector multiplications for each  component. Processor assignment

  16. Communications According to the Coriolis coupling rules, only two nearest neighbouring  components need to communicate. Load balancing jmin is different for each  component, but jmax is the same, i.e, the DVR size for  angle is changeable. For the highest or the lowest  components, only one Coriolis coupling required.

  17. Timing • Due to the communications and loading balance issues, the model doesn’t scale ideally with (J+1) for even spectroscopic symmetry or J for odd spectroscopic symmetry. • However, one can achieve wall clock times (e.g., for even symmetry J = 6 HO2 case) that are within about a factor of 2 of J = 0 calculations. For non parallel computing, the wall clock times will approximately be a factor of 7 of J = 0 calculations.

  18. 3. Recent Results • HO2: J = 0-50 bound state energies and resonance energies and widths using both Lanczos and Chebyshev method from parallel computing. • HOCl/DOCl: J = 0-30 ro-vibrational spectroscopy.

  19. Table 1 Selected HO2 bound state energies for J = 30 (even symmetry) for comparison.All energy units are in eV (Zhang & Smith, JPCA, 110: 3246, 2006 ).

  20. Fig. 1 Plot of the quantum logarithmic rates versus resonance energies. Thin dotted line - QM results; red line - Troe et al. SACM/CT calculations; green line – quantum average. (Zhang & Smith, JCP, 123: 014308, 2005)

  21. Fig. 2 Same as previous figure, except J = 30 (unpublished latest results).

  22. Table 2 Selected low vibrational energies at J = 0 for HOCl for comparison.All energy units are in cm-1 (Zhang, Smith, Nanbu & Nakamura,JPCA, 2006, in press).

  23. Table 3 Calculated HOCl ro-vibrational state energies in cm-1 with spectroscopic assignments for J = 20 (Zhang, Smith, Nanbu & Nakamura,JPCA, 2006, in press).

  24. Table 4 Comparison of experiments and calculations for selected HOCl far infrared transitions in cm-1 (Zhang, Smith, Nanbu & Nakamura, JPCA, 2006, in press).

  25. Table 5 Comparison of experiments and calculations for selected HO2 vibrational state energies in cm-1 from three latest PESs (Zhang & Smith, unpublished latest resuts).

  26. Table 6 Comparison of experiments and calculations for selected DOCl vibrational state energies in cm-1 from the latest ab initio PES and LHFD calculations (Zhang & Smith, unpublished latest resuts).

  27. Table 7 Comparison of experiments and calculations for selected DOCl ro-vibrational energies in cm-1 from the latest ab initio PES (Zhang & Smith, latest results; Hu et al., J Mol Spec, 209, 105).

  28. Development of Lanczos representation methods; Design of a parallel computing model; Combination of both has made rigorous quantum calculations possible for challenging J > 0 applications. Further comparative studies between QD/ST; Develop quantum statistical theories; Mixed QD/MD simulations in larger systems. 4. Conclusions and Future Work

  29. Prof J. Troe (The University of Göttingen) Prof S. Nanbu (Kyushu University) Prof H. Nakamura (IMS) Dr Marlies Hankel CCMS/MD members Australian Research Council Supercomputer time from APAC & UQ Acknowledgements

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