1 / 29

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

wiley
Download Presentation

Parallel Computations in Quantum Lanczos Representation Methods

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related