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Efficient Eigensolvers for Large-scale Electronic Nanostructure Calculations

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## Efficient Eigensolvers for Large-scale Electronic Nanostructure Calculations

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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

11/16/2005

Outline

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

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

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

- 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 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

- 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

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 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 …

- 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 …

- (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

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 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 results

- Tests with “perturbed” potential (simulate a quantum dot)

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

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 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 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

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

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