FEAST v4.0 with Applications - Sparse Days · FEAST v4.0 with Applications Eric Polizzi Department...
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FEAST v4.0 with Applications
Eric PolizziDepartment of Electrical and Computer Engineering
Department of Mathematics and StatisticsUniversity of Massachusetts, Amherst
Sparse Days, CERFACS, Toulouse, 2019
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Real-SpaceDiscretization
Real-SpaceDiscretization
Real-TimePropagation
Real-TimePropagation
Ground-State Calculations DFT/Kohn-Sham/All-electrons
Ground-State Calculations DFT/Kohn-Sham/All-electrons
Excited-State CalculationsTime-dependent DFT (TDDFT)
ALDA/AGGA
Excited-State CalculationsTime-dependent DFT (TDDFT)
ALDA/AGGA
From Molecules to Nanostructures
FEAST for First-Principle Calculations
NESSIE
www.nessie-code.org
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FEAST Solver LibraryDesign a robust, parallel and unified framework for solving the “interior” eigenvalue problems
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FEAST Solver LibraryDesign a robust, parallel and unified framework for solving the “interior” eigenvalue problems
Release dates v1.0 (2009): Hermitian problem v2.0 (2012): SMP+MPI+RCI interfaces v2.1 (2013): Adoption by Intel-MKL v3.0 (2015): Support for non-Hermitan v4.0 (fall 2019): Residual inverse iter.
-PFEAST (3 MPI levels)-IFEAST (FEAST w/o factorization)-mixed precision-non-linear (polynomial)
www.feast-solver.org
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Subspace iteration with RR
FEAST Algorithm- AX=BXΛ (Hermitian, Generalized)Hermitian, Generalized))
Standard iteration (power method)
Goal:
linear CV rate:
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Subspace iteration with RR
FEAST Algorithm- AX=BXΛ (Hermitian, Generalized)Hermitian, Generalized))
Standard iteration (power method)
Goal:
linear CV rate:
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Subspace iteration with RR
FEAST Algorithm- AX=BXΛ (Hermitian, Generalized)Hermitian, Generalized))
Standard iteration (power method)
Goal:
linear CV rate:
Shift-invert iteration
* 1 linear system solve by iteration* fast CV near the shift* slow CV elsewhere
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Subspace iteration with RR
FEAST Algorithm
Optimal filter for the M interior eigenpairs is given by thespectral projector
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Subspace iteration with RR
FEAST Algorithm
Optimal filter for the M interior eigenpairs is given by thespectral projector
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FEAST Algorithm: Numerical Quadrature Rational function filter Solving independent linear systems
(multiple shifts in complex plane)
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FEAST Algorithm: Numerical Quadrature Rational function filter Solving independent linear systems
(multiple shifts in complex plane)
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FEAST Algorithm: Numerical Quadrature Rational function filter Solving independent linear systems
(multiple shifts in complex plane)
Polizzi, Phys. Rev. B. (2009)Tang, Polizzi, SIAM SIMAX (2014) Guettel, Polizzi, Tang, Viaud, SIAM SISC (2015)
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FEAST Algorithm at a glance
M0
● Select search interval● Select quadrature rule● Select subspace size M
0
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FEAST Algorithm at a glance
M0
● Select search interval● Select quadrature rule● Select subspace size M
0
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FEAST non-Hermitian algorithmKestyn, Polizzi, Tang, SIAM, SISC (2015)
Right projector
Left projector
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FEAST non-Hermitian algorithmKestyn, Polizzi, Tang, SIAM, SISC (2015)
Right projector
Left projector
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PFEAST (James Kestyn, PhD thesis 2018)3 MPI communicators L1
Distribution of the spectrum (slicing)
L3(Row) Distributed direct solvers: Black-
box (cluster pardiso,mumps) and DD custom solvers
L2Ideal scalability - requires matrix copies
L1 and L3 can be used to reduce memory and increase performancesExample with 2L1 and 2L3:
New parallel FEAST interfaceslocal/global distributions
PFEASTKestyn, Kalantzis, Polizzi, Saad, supercomputing (2016)
FEAST-DD: Kalantzis, Kestyn, Polizzi, Saad, NLAA (2018)
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IFEAST- w/o factorization- (Brendan Gavin, PhD thesis 2018)
FEAST using inexact iterative solves
Example: Parsec Si2 (B=I)
#inner iterations is constant! CV rate is still linear
α=0.5
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IFEAST- w/o factorization- (Brendan Gavin, PhD thesis 2018)
FEAST using inexact iterative solves
Example: Parsec Si2 (B=I)
Generalization of previous work on inner-outer iterations with single real shift-invert. Robbé, Sadkane, Spence, SIMAX, 31(1), p.92, (2009)
#inner iterations is constant! CV rate is still linear
α=0.5
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IFEAST- w/o factorization- (Brendan Gavin, PhD thesis 2018)
FEAST using inexact iterative solves
Example: Parsec Si2 (B=I)
Generalization of previous work on inner-outer iterations with single real shift-invert. Robbé, Sadkane, Spence, SIMAX, 31(1), p.92, (2009)Formally equivalent to block restarted Krylov ideally suited for interior problem-Krylov eigenvalue strategy using FEAST with inexact system solves, Gavin, Polizzi: NLAA, (2018)
#inner iterations is constant! CV rate is still linear
α=0.5
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IFEAST- w/o factorization- (Brendan Gavin, PhD thesis 2018)
FEAST using inexact iterative solves
Example: Parsec Si2 (B=I)
Generalization of previous work on inner-outer iterations with single real shift-invert. Robbé, Sadkane, Spence, SIMAX, 31(1), p.92, (2009)Formally equivalent to block restarted Krylov ideally suited for interior problem-Krylov eigenvalue strategy using FEAST with inexact system solves, Gavin, Polizzi: NLAA, (2018)Equivalence to Polynomial filtering
#inner iterations is constant! CV rate is still linear
α=0.5
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IFEAST- w/o factorization- (Brendan Gavin, PhD thesis 2018)Example: Parsec standard Ga41As41H72, n=268K, m=10 lowest,m
0=20,nc=3
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IFEAST- w/o factorization- (Brendan Gavin, PhD thesis 2018)Example: Parsec standard Ga41As41H72, n=268K, m=10 lowest,m
0=20,nc=3
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IFEAST- w/o factorization- (Brendan Gavin, PhD thesis 2018)Example: Parsec standard Ga41As41H72, n=268K, m=10 lowest,m
0=20,nc=3
FEAST/PARDISO: ~few hours
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IFEAST- w/o factorization- (Brendan Gavin, PhD thesis 2018)Example: Parsec standard Ga41As41H72, n=268K, m=10 lowest,m
0=20,nc=3
FEAST/PARDISO: ~few hoursIFEAST/BiCGstab: ~few minutes ~100K mat-vec (1 rhs), 20 feast iterations
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IFEAST- w/o factorization- (Brendan Gavin, PhD thesis 2018)Example: Parsec standard Ga41As41H72, n=268K, m=10 lowest,m
0=20,nc=3
FEAST/PARDISO: ~few hoursIFEAST/BiCGstab: ~few minutes ~100K mat-vec (1 rhs), 20 feast iterationsARPACK: ~12K mat-vec (1rhs), ~2K restarts,
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IFEAST- w/o factorization- (Brendan Gavin, PhD thesis 2018)Example: Parsec standard Ga41As41H72, n=268K, m=10 lowest,m
0=20,nc=3
FEAST/PARDISO: ~few hoursIFEAST/BiCGstab: ~few minutes ~100K mat-vec (1 rhs), 20 feast iterationsARPACK: ~12K mat-vec (1rhs), ~2K restarts,A lot more mat-vec than standard Krylov (Arnoldi) but can be performed in parallel=> IFEAST fully in parallel, 1.7K mat-vec (1rhs)
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IFEAST- w/o factorization- (Brendan Gavin, PhD thesis 2018)Example: Parsec standard Ga41As41H72, n=268K, m=10 lowest,m
0=20,nc=3
FEAST/PARDISO: ~few hoursIFEAST/BiCGstab: ~few minutes ~100K mat-vec (1 rhs), 20 feast iterationsARPACK: ~12K mat-vec (1rhs), ~2K restarts,A lot more mat-vec than standard Krylov (Arnoldi) but can be performed in parallel=> IFEAST fully in parallel, 1.7K mat-vec (1rhs)
Difficulties: inverse free generalized problems (B≠I) and preconditioners
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IFEAST- w/o factorization- (Brendan Gavin, PhD thesis 2018)Example: Parsec standard Ga41As41H72, n=268K, m=10 lowest,m
0=20,nc=3
FEAST/PARDISO: ~few hoursIFEAST/BiCGstab: ~few minutes ~100K mat-vec (1 rhs), 20 feast iterationsARPACK: ~12K mat-vec (1rhs), ~2K restarts,A lot more mat-vec than standard Krylov (Arnoldi) but can be performed in parallel=> IFEAST fully in parallel, 1.7K mat-vec (1rhs)
Difficulties: inverse free generalized problems (B≠I) and preconditioners
Solution: Generalized IFEAST (based on Residual Inverse Iterations)
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Residual Inverse Iterations
• Generalization of previous work: *Golub G., Ye Q. Inexact Inverse Iteration for Generalized Eigenvalue Problems, BIT p671 (2000)
*See also (in the context of non-linear problems): A. Neumaier, Residual inverse iteration for the nonlinear eigenvalue problem, SIAM J. Numer. Anal. 22 (5) (1985)
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Residual Inverse Iterations
• Generalization of previous work: *Golub G., Ye Q. Inexact Inverse Iteration for Generalized Eigenvalue Problems, BIT p671 (2000)
*See also (in the context of non-linear problems): A. Neumaier, Residual inverse iteration for the nonlinear eigenvalue problem, SIAM J. Numer. Anal. 22 (5) (1985)
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Residual Inverse Iterations
• Generalization of previous work: *Golub G., Ye Q. Inexact Inverse Iteration for Generalized Eigenvalue Problems, BIT p671 (2000)
*See also (in the context of non-linear problems): A. Neumaier, Residual inverse iteration for the nonlinear eigenvalue problem, SIAM J. Numer. Anal. 22 (5) (1985)
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Residual Inverse Iterations
• Generalization of previous work: *Golub G., Ye Q. Inexact Inverse Iteration for Generalized Eigenvalue Problems, BIT p671 (2000)
*See also (in the context of non-linear problems): A. Neumaier, Residual inverse iteration for the nonlinear eigenvalue problem, SIAM J. Numer. Anal. 22 (5) (1985)
Three main consequences
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Residual Inverse Iterations
• Generalization of previous work: *Golub G., Ye Q. Inexact Inverse Iteration for Generalized Eigenvalue Problems, BIT p671 (2000)
*See also (in the context of non-linear problems): A. Neumaier, Residual inverse iteration for the nonlinear eigenvalue problem, SIAM J. Numer. Anal. 22 (5) (1985)
Three main consequences•IFEAST applicable to generalized systems and preconditioners
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Residual Inverse Iterations
• Generalization of previous work: *Golub G., Ye Q. Inexact Inverse Iteration for Generalized Eigenvalue Problems, BIT p671 (2000)
*See also (in the context of non-linear problems): A. Neumaier, Residual inverse iteration for the nonlinear eigenvalue problem, SIAM J. Numer. Anal. 22 (5) (1985)
Three main consequences•IFEAST applicable to generalized systems and preconditioners•Mixed-precision arithmetic (single precision direct/iterative solvers)
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Residual Inverse Iterations
• Generalization of previous work: *Golub G., Ye Q. Inexact Inverse Iteration for Generalized Eigenvalue Problems, BIT p671 (2000)
*See also (in the context of non-linear problems): A. Neumaier, Residual inverse iteration for the nonlinear eigenvalue problem, SIAM J. Numer. Anal. 22 (5) (1985)
Three main consequences•IFEAST applicable to generalized systems and preconditioners•Mixed-precision arithmetic (single precision direct/iterative solvers)•Applicable to non-linear eigenvalue problem
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Residual Inverse Iterations: Applications (Generalized+mixed)
Example: C6H6 (P2-FEM generalized), n=49K, m=6 lowest, m0=20 nc=5
Solverprecision
FEAST(pardiso)
● IFEAST● (bicgstab 30 iter.
max, jacobi prec.)double 7.94s (3 iter.) 51s (10 iter.)
single 5.18s (3 iter.) 33s (10 iter.)
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Residual Inverse Iterations: Application to non-linear problem
← reduced non-linear problem(reduced companion problem for polynomial eigenvalue)
Example: Butterfly problem
FEAST for nonlinear eigenvalue problems, Gavin, Miedlar, Polizzi, JCS (2018)
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FEAST non-linear (FEAST and Beyn)
\
n=50K, m=250
nc=8
m0=300
FEAST
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FEAST non-linear (FEAST and Beyn)
\
n=50K, m=250
nc=8
m0=300
FEAST
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FEAST non-linear (FEAST and Beyn)
\
n=50K, m=250
nc=8
m0=300
FEAST
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FEAST non-linear (FEAST and Beyn)
\
n=50K, m=250
nc=8
m0=300
FEAST
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FEAST non-linear (FEAST and Beyn)
\
n=50K, m=250
nc=8
m0=300
FEAST
FEAST proposed approach: solve the projected non-linear reduced system
(i) using companion problem for reduced system, or(ii) using Beyn’s method (beyond v4.0)
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FEAST non-linear (FEAST using Beyn)
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Conclusion
FEAST v4.0New implementation using Residual Inverse IterationsPFEAST (MPI-MPI-MPI)IFEAST (w/o factorization+basic preconditioners)All linear system solves using single precisionsNon-linear problems (polynomial)
New Direction (beyond 4.0): Hybrid solvers, svd, quaternions
Students: James Kestyn, Brendan Gavin, Braegan Spring, Julien BrenneckCollaborators: Y. Saad, A. Miedlar, P. TangFunding: NSF #1510010, #1739423, #1813480, Intel