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Stanimire Tomov 1 Andrew Canning 2 , Jack Dongarra 1 , Osni Marques 2 Christof Vömel 2 and Lin-Wang Wang 2 Innovative Computing Laboratory 1 Lawrence Berkeley National Laboratory 2 University of Tennessee Computational Research Division.

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Efficient Eigensolvers for Large-scale Electronic Nanostructure Calculations

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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

Efficient Eigensolvers for Large-scale Electronic Nanostructure Calculations


Supported by:

U.S. DOE, Office of Science

SC05, Seattle


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

problem formulation
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
solution approach
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. Knyazev’s LOBPCG method
      • Supports blocking
      • Supports preconditioning
      • Developed and integrated in NanoPSE (S.Tomov and J.Langou)
solution approach1
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
  • 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
bulk band bb preconditioner
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?
bb preconditioner1
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 (r+A) = 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)
bb preconditioner continued
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 (R–QBB 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 w>0 is a dumping parameter
bb preconditioner continue
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)
numerical results
Numerical results

64 atoms of Cd48-Se34

512 atoms of Cd48-Se34

  • Tests on a bulk problem
  • 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
numerical results1
Numerical results

64 atoms of Cd48-Se34

512 atoms of Cd48-Se34

  • Tests with “perturbed” potential (simulate a quantum dot)
  • Factor of 2 speedup
  • Increasing with increasing problem size
numerical results2
Numerical results
  • Tests with “perturbed” potential (simulate a quantum dot)
  • Localized wave-functions with density charge confinement simulating a quantum dot
numerical results3
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
numerical results4
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
numerical results5
Numerical results
  • Comparison of diagonal (in red) vsBB 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
  • 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