Efficient Eigensolvers for Large-scale Electronic Nanostructure Calculations

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• Efficient Eigensolvers for Large-scale Electronic
  Nanostructure Calculations
• ________________________________________________

Stanimire Tomov1 Andrew Canning2, Jack
Dongarra1, Osni Marques2 Christof Vömel2 and
Lin-Wang Wang2 Innovative Computing
Laboratory 1 Lawrence Berkeley National
Laboratory 2 University of Tennessee
Computational Research
Division
Supported by U.S. DOE, Office of Science
Alex Zunger Gabriel Bester Joonhee An Alberto Franceschetti Wesley JonesKim Kwiseon Peter Graf Jack Dongarra Julien Langou Stanimire Tomov Lin-Wang Wang Andrew Canning Osni MarquesChristof Vömel M. Claudia Troparevsky
SC05, Seattle 11/16/2005
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Outline

• Background
• Problem formulation
• Solution approach
• Iterative Conjugate Gradients (CG) type
  eigensolvers
• Preconditioning
• The Bulk-band (BB) preconditioner
• Numerical results
• Conclusions

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Background

• Quantum dots
• Tiny crystals ranging from a few hundred to few
  thousand atoms in size made by humans
• Electronic properties critically depend on shape
  and size
• Colors of light absorbed and emitted can be
  tuned by the quantum dot size
• Absorbed energy can lift an electron from its
  valence band to its conduction band (generate
  electrical current)
• Electron falling back from conduction to valence
  band lead to loss of energy, emitted as light
• The mathematical simulation leads to eigen-value
  problems
• Different electronic properties than their bulk
  material
• But still, bulk material properties may be
  useful we found ways to use them in designing
  preconditioners that would significantly
  accelerate quantum dots electronic structure
  calculations

Total electron charge density of a quantum dot
of gallium arsenide, containing just 465 atoms.
Quantum dots of the same material but different
sizes have different band gaps and emit
different colors
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Problem formulation

• Solve a single particle Schrödinger-type
  equation (E) (- 0.5 ? V ) ?i
  ?i ?i with periodic boundary conditions
• Many electronic nano-structure calculations lead
  to it
• Leads to a discrete eigenvalue problem
• H ?i Ei ?i ,
  where H is Hermitian
• Many additional requirements
• Find a few (4-10) interior eigenvalues closest to
  a given point Eref
• Repeated eigenvalues are allowed (degeneracy up
  to 4), etc.
• The problem size requires a parallel iterative
  solution approach

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Solution approach

• Phase 1 Iterative eigen-solvers
• Conjugate Gradients (CG) type with spectral
  transformation
• Based on their previous successful use in the
  field
• Folded spectrum solve for (H-Eref)2 to get
  interior eigen-states(L.W.Wang A. Zunger,
  1993)
• Developed library of 3 non-linear CG
  eigen-solvers
• The library includes the A. Knyazevs LOBPCG
  method
• Supports blocking
• Supports preconditioning
• Developed and integrated in NanoPSE (S.Tomov and
  J.Langou)

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Solution approach

• We use the Nanoscience Problem Solving
  Environment (NanoPSE) package
• Integrate various nano-codes (developed over 10
  years)
• Its design goal provide a software context for
  collaboration
• Features easy install runs on many platforms,
  etc.
• Collected and maintained by Wesley Jones (NREL)
• Results
• 43 improvement in speed and 49 in number of
  matrix-vector products
• On a InAs nanowire system of 70,000 atoms,
  eigen-system of size 2,265,827 (A. Canning and
  G. Bester)
• Results are good reference algorithm
  implementation were very efficient
• But limited by the effectiveness of the available
  preconditioner
• Phase 2 Preconditioning

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Preconditioning

• Preconditioning term coming from accelerating
  the convergence of iterative solvers for linear
  systems Ax b in particular,
  find operator/preconditioner
  T ?A-1 s.t. (TA) x Tb be
  easier to solve
• Preconditioning for eigenproblems
• Harder problem / not as straightforward
• Can be shown that efficient preconditioners for
  linear systems are efficient preconditioners for
  CG-type eigensolvers

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Bulk Band (BB) Preconditioner

• Basic idea
• Use the electronic properties of the bulk
  materials constituent for the nanostructure in
  designing a preconditioner
• What does it mean and how?

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BB preconditioner
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BB preconditioner

• Find electronic properties of the bulk materials
• Solve (E) on infinite crystal (bulk material)
• Because of the periodicity solve just on the
  primary cell (much smaller problem) Find
  solution in form (Bloch theorem) ?nk
  (r ) unk( r) eikr, unk (rA) unk( r)
• Denote span?nk as BB space
• Denote by HBB the Hamiltonian stemming from a
  bulk problem if ? ? BB space, HBB-1 ? is easy to
  compute
• Note that if H stems from a bulk problem HBB-1 is
  the exact preconditioner for H (H-1)

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BB preconditioner, continued

• Decompose the current residual R as R
  QBB R (R QBB R)where QBB is the L2
  projection in the BB space
• Use HBB-1 to precondition the QBB R component of
  R and a diagonal preconditioner D-1 for the
  (RQBB R) component, i.e. (1) T R ?
  HBB-1 QBB R D-1 (R QBB R)
• TR in (1) is just one example
• Preconditioners of form (1) are refered to in the
  literature as additive another variation is
  (2) T R ? HBB-1 QBB R w D-1
  R,where wgt0 is a dumping parameter

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BB preconditioner, continue

• (2) can be viewed as a multilevel (two-level)
  preconditioner correct the low frequency
  components of R with HBB-1 and smooth the high
  frequencies with D-1
• How to choose w in (2) also present in (1)?
• Avoid the problem of determining it by
  considering a multiplicative multilevel version
  of the BB preconditioner
  r1 D-1 R r2 r1 HBB-1 QBB (R
  H r1) T R ? r2 D-1 (R H r2)

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Numerical results

• Tests on a bulk problem

64 atoms of Cd48-Se34
512 atoms of Cd48-Se34

• The BB preconditioner should be most efficient
  for this case (speedup of factor 3, increasing
  with problem size increase)
• We start with arbitrary initial guess
• Here BB space dimension is ? 1.5 of solution
  space dimension

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Numerical results

• Tests with perturbed potential (simulate a
  quantum dot)

64 atoms of Cd48-Se34
512 atoms of Cd48-Se34

• Factor of 2 speedup
• Increasing with increasing problem size

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Numerical results

• Tests with perturbed potential (simulate a
  quantum dot)

• Localized wave-functions with density charge
  confinement simulating a quantum dot

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Numerical results

• Various perturbations with the BB multiplicative
  preconditioner

64 atoms of Cd48-Se34
512 atoms of Cd48-Se34

• Not that sensitive to perturbation increase

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Numerical results

• BB vs diagonal preconditioning on a bigger system
  (4096 atoms of Cd48-Se34) for various
  perturbations

BB multiplicative preconditioning
Diagonal preconditioning

• Speedup exceeding a factor of 3
• Goes to about factor of 7 for perturbation 4

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Numerical results

• Comparison of diagonal (in red) vs BB
  preconditoining (in green) using folded
  spectrum (H-Eref)2

64 atoms of Cd48-Se34
512 atoms of Cd48-Se34

• The speedup from the H case is multiplied by a
  factor of 2
• A speedup of factor 4 for small problems
  increasing with problem size increase

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Conclusions

• A new preconditioning technique was presented
• Numerical results show the efficiency of the BB
  preconditioning
• A factor of 4 speedup for small problems with
  folded spectrum (compared to diagonal
  preconditioning)
• Increased efficiency with problem size increase
• More testing has to be done
• On bigger problems
• With real quantum dots