Approach Toward Linear Time QMC a,c David Ceperley, a Bryan Clark, b,d Eric de Sturler, a,c Jeongnim Kim, b,e Chris Siefert University of Illinois at Urbana-Champaign, Departments of a Physics, and b Computer Science, and c National Center for Supercomputer Applications, and d Virginia Tech, Department of Mathematics, and e Sandia National Labs This work is supported by the Materials Computation Center (UIUC) NSF DMR The Materials Computation Center, University of Illinois David Ceperley and Eric deSturler (PIs), NSF DMR

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.

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

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

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.

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

Current determinant calculation Moving all n particles Calculate determinant directly Time: O(n 3 ) Moving each particle one at a time: Note: Only 1 column/row changes Use Shermann-Morisson Update inverse and hence determinant Time: O(n 3 ) Our goal: Do better! Calculate ratio of determinants in time O(n) or O(n 2 )

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

Sparse matrices Sparse Inverse particles Green: M Black: M -1 Off diagonal |i-j|

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

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

Determinant Error Determinant Error as a function of the number of particles included (out of 5000) Number of Particles System Parameters: He-He Interaction 5000 particles ptcl/A 3 Interatomic spacing: a=2.54A Fermi energy: 7.8 K

Truncated Matrices Variational DM Model 500 Particles 25 Particle Cutoff Average is correct within errors BUT long tails

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.

Conclusion MethodCostExactPractical Inverse update Truncation Iterative Sparse Sample O(n) O(n 2 ) O(n) Dense ? ? ? ? Eventual Incorporation into QMCPack and PIMC++ Note: See poster on PIMC++ (written by Ken Esler and myself)