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, andc National Center for Supercomputer Applications, and

dVirginia Tech, Department of Mathematics, andeSandia National Labs

This work is supported by the Materials Computation Center (UIUC) NSF DMR 03-25939.

The Materials Computation Center, University of IllinoisDavid Ceperley and Eric deSturler (PIs), NSF DMR-03-25939 •  www.mcc.uiuc.edu

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

€

Ψ(R)

€

Ψ (R')

€

Ψ (R')

Ψ (R)

2. Evaluate ratio 3. Accept if

€

Ψ (R')

Ψ (R)> r ∈ 0,1{ }

Hard Step!

Ratio of Determinants especially hard

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)

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

1 .4 .2 .07 .001

1

1

.4 .3 1

.01 1

.002 1€

log1

NM ij∑

•500 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[Mnew]/Det[Mold] 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)

0 20 40 80-6

Number of Particles

0

-2

-4

System Parameters:He-He Interaction5000 particles0.01635 ptcl/A3

Interatomic spacing: a=2.54AFermi energy: 7.8 K

Truncated Matrices

Variational DM Model

500 Particles

25 Particle Cutoff

Average is correct within errors BUT long tails..

1.00.999 1.001

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

Method Cost Exact Practical

Inverse update

Truncation

Iterative

Sparse Sample

O(n)

O(n)

O(n2)

O(n)

Dense

?

?

?

?

Eventual Incorporation into QMCPack and PIMC++

Note: See poster on PIMC++ (written by Ken Esler and myself)