Approach Toward Linear Time QMC

1. The Materials Computation Center, University of Illinois David Ceperley and Eric deSturler (PIs), NSF DMR-03-25939 •  www.mcc.uiuc.edu Approach Toward Linear Time QMC a,cDavid Ceperley, aBryan Clark, b,dEric de Sturler,a,cJeongnim Kim, b,eChris Siefert University of Illinois at Urbana-Champaign, Departments of aPhysics, and bComputer Science, and c National Center for Supercomputer Applications, and dVirginia Tech, Department of Mathematics, and eSandia National Labs This work is supported by the Materials Computation Center (UIUC) NSF DMR 03-25939.

2. Geothermal Materials • Collaboration with geophysicists and geologists (at Carnegie Institute) toward calculating properties of geothermal materials • FeO, MgSo4 • Equation of state (eos) integral to understanding Earth’s interior. • Errors in eos magnified when making conclusion about Earth. • Use Quantum Monte Carlo. • Even QMC systematic errors need to be decrease. We’re currently actively working on this.

3. Collaboration • External NSF Funding • Geophysicists, mathematicians, physicists, etc.

4. QMC Errors • Pseudopotential error • Finite Size Effects • Fixed Node Error • Time step error, population bias, etc. Our contribution: Restrict Finite Size Effects by doing larger systems. Larger systems have many added benefits beyond simply reducing finite size effects. Can practically do about 1000 electrons. Finite systems have artificial effects associated with them. Would like to do much larger systems to quantify and remove these effects

5. Goal • Practice: • 1-2 orders of magnitude more electrons • Theory: • QMC in time O(n) Notation: Measure time for all n particles at once. We are actively collaborating with mathemeticians (Eric deSturler (VIT) and Chris Siefert (Sandia)) in attempting to accomplish this goal.

6. 2. Evaluate ratio 3. Accept if Hard Step! Ratio of Determinants especially hard QMC Steps 1. Move a particle

7. Current determinant calculation • Moving all n particles • Calculate determinant directly • Time: O(n3) • Moving each particle one at a time: • Note: Only 1 column/row changes • Use Shermann-Morisson • Update inverse and hence determinant • Time: O(n3) Our goal: Do better! Calculate ratio of determinants in time O(n) or O(n2)

8. Possible techniques • Sparse inverse updates • Truncated Matrix Method • Iterative Methods for Single Particle Updates + Speed up matrix-multiplication • Sparse Sampling of Bai and Golub

9. Sparse matrices Sparse Inverse • 500 particles • Green: M • Black: M-1 Off diagonal |i-j|

10. Truncated Matrices • Moving a single particle: • Select a domain of affected particles • Define Mnew(old) s.t it contains matrix elements of affected particles with the new (old) particles (and itself) • Evaluate Det[Mnew]/Det[Mold] • Cost: O(n) Physical intuition: All important information is in the large elements near the moved particle

11. Interpolation • Physical intuition support “sparse” matrices • “Interpolation” picture suggests wider applicability. • Small matrix gets many zeros of determinant correct.

12. Determinant Error System Parameters: • He-He Interaction • 5000 particles • 0.01635 ptcl/A3 • Interatomic spacing: a=2.54A • Fermi energy: 7.8 K 0 Determinant Error as a function of the number of particles included (out of 5000) -2 -4 -6 40 80 20 0 Number of Particles

13. Truncated Matrices Variational DM Model 500 Particles 25 Particle Cutoff Average is correct within errors BUT long tails.. 0.999 1.0 1.001

14. Extension: Removing error • Identify bad configurations and work harder when they happen. • Sample the error away • Bound error and cutoff when you know what you will do anyway.

15. Conclusion Method Cost Exact Practical Dense O(n) Inverse update Truncation O(n) ? O(n2) Iterative ? ? ? O(n) Sparse Sample Eventual Incorporation into QMCPack and PIMC++ Note: See poster on PIMC++ (written by Ken Esler and myself)