An optimized subspace iteration eigensolver applied to sequences of - - PowerPoint PPT Presentation

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Mitglied der Helmholtz-Gemeinschaft

An optimized subspace iteration eigensolver applied to sequences

• f dense eigenproblems in ab initio

simulations

PMAA14, Lugano, Switzerland. July 4th

• E. Di Napoli and M. Berljafa

SLIDE 2

Motivation: electronic structure computation

Full-potential Linearized Augmented Plane Waves (FLAPW) self-consistent field cycle

Initial guess for charge density

nstart(r)

Compute discretized Kohn-Sham equations Solve a set of eigenproblems

P(ℓ)

k1 ...P(ℓ) kN

Compute new charge density

n(ℓ)(r)

Converged?

|n(ℓ) −n(ℓ−1)| < η

OUTPUT Electronic structure,

... No Yes

1 every P(ℓ)

k

: A(ℓ)

k x = B(ℓ) k λx is a generalized eigenvalue problem;

2 A and B are DENSE and hermitian (B is positive definite); 3 required: lower 2÷10 % of eigenpairs; 4 momentum vector index: k = 1 : 10÷100; 5 iteration cycle index: ℓ = 1 : 20÷50.

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Outline

Sequences of correlated eigenproblems The algorithm: Chebyshev Filtered Sub-space Iteration method (ChFSI) ChFSI parallelization and numerical tests

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Outline

Sequences of correlated eigenproblems The algorithm: Chebyshev Filtered Sub-space Iteration method (ChFSI) ChFSI parallelization and numerical tests

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Sequences of eigenproblems

Definitions and solving strategies

Definition: Eigenproblem sequence

A sequence of eigenproblems is a finite and index-ordered set of problems {P}N . = P(1) ···P(ℓ) ···P(N) with same size = n such that the eigenpairs of P(ℓ) are used (directly or indirectly) to initialize P(ℓ+1).

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Sequences of eigenproblems

Definitions and solving strategies

Definition: Eigenproblem sequence

A sequence of eigenproblems is a finite and index-ordered set of problems {P}N . = P(1) ···P(ℓ) ···P(N) with same size = n such that the eigenpairs of P(ℓ) are used (directly or indirectly) to initialize P(ℓ+1).

Current solving strategy

The set of generalized eigenproblems P(1) ...P(ℓ)P(ℓ+1) ...P(N) is handled as a set of disjoint problems N ×P; Each problem P(ℓ) is solved independently using a direct solver as a black-box from a standard library (i.e. ScaLAPACK).

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Sequences of Eigenproblems

Adjacent iteration cycles

ITERATION (ℓ) P(ℓ)

k1

(X(ℓ)

k1 ,Λ(ℓ) k1 )

P(ℓ)

k2

(X(ℓ)

k2 ,Λ(ℓ) k2 )

P(ℓ)

kN

(X(ℓ)

kN ,Λ(ℓ) kN )

X ≡ {x1,...,xn}

direct solver direct solver direct solver

ITERATION (ℓ+1) P(ℓ+1)

k1

(X(ℓ+1)

k1

,Λ(ℓ+1)

k1

) P(ℓ+1)

k2

(X(ℓ+1)

k2

,Λ(ℓ+1)

k2

) P(ℓ+1)

kN

(X(ℓ+1)

kN

,Λ(ℓ+1)

kN

) Λ ≡ diag(λ1,...,λn)

direct solver direct solver direct solver

Next cycle

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Sequences of Eigenproblems

Adjacent iteration cycles

ITERATION (ℓ) P(ℓ)

k1

(X(ℓ)

k1 ,Λ(ℓ) k1 )

P(ℓ)

k2

(X(ℓ)

k2 ,Λ(ℓ) k2 )

P(ℓ)

kN

(X(ℓ)

kN ,Λ(ℓ) kN )

X ≡ {x1,...,xn}

direct solver direct solver direct solver

ITERATION (ℓ+1) P(ℓ+1)

k1

(X(ℓ+1)

k1

,Λ(ℓ+1)

k1

) P(ℓ+1)

k2

(X(ℓ+1)

k2

,Λ(ℓ+1)

k2

) P(ℓ+1)

kN

(X(ℓ+1)

kN

,Λ(ℓ+1)

kN

) Λ ≡ diag(λ1,...,λn)

direct solver direct solver direct solver

Next cycle

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Sequences of Eigenproblems

Adjacent iteration cycles

ITERATION (ℓ) P(ℓ)

k1

(X(ℓ)

k1 ,Λ(ℓ) k1 )

P(ℓ)

k2

(X(ℓ)

k2 ,Λ(ℓ) k2 )

P(ℓ)

kN

(X(ℓ)

kN ,Λ(ℓ) kN )

X ≡ {x1,...,xn}

direct solver direct solver direct solver

ITERATION (ℓ+1) P(ℓ+1)

k1

(X(ℓ+1)

k1

,Λ(ℓ+1)

k1

) P(ℓ+1)

k2

(X(ℓ+1)

k2

,Λ(ℓ+1)

k2

) P(ℓ+1)

kN

(X(ℓ+1)

kN

,Λ(ℓ+1)

kN

) Λ ≡ diag(λ1,...,λn)

direct solver direct solver direct solver

Next cycle

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Correlation between eigenproblems

Definition and solving strategies

Definition: Correlation

Two adjacent problems P(ℓ+1) and P(ℓ) are said to be correlated when the eigenpairs (X(ℓ+1),Λ(ℓ+1)) are in some relation with the eigenpairs (X(ℓ),Λ(ℓ)).

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Correlation between eigenproblems

Definition and solving strategies

Definition: Correlation

Two adjacent problems P(ℓ+1) and P(ℓ) are said to be correlated when the eigenpairs (X(ℓ+1),Λ(ℓ+1)) are in some relation with the eigenpairs (X(ℓ),Λ(ℓ)).

Alternative solving strategy

Consider the entire sequence of eigenproblems P(1),...,P(N) as one self-contained problem; Exploit information provided from the correlation; In FLAPW, correlation appears in the way angles b/w eigenvectors of adjacent eigenproblems evolve [EDN, Blügel and Bientinesi 2012] Θ(ℓ)

ki ≡ {θ1,...,θn} = diag

• 1 −X(ℓ−1)

ki

, ˜ X(ℓ)

ki

• for fixed ki

θ(2)

j

θ(3)

j

··· θ(N)

j

: θ(2)

j

≫ θ(N)

j

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An alternative solving strategy

Adjacent cycles

ITERATION (ℓ) P(ℓ)

k1

(X(ℓ)

k1 ,Λ(ℓ) k1 )

P(ℓ)

k2

(X(ℓ)

k2 ,Λ(ℓ) k2 )

P(ℓ)

kN

(X(ℓ)

kN ,Λ(ℓ) kN )

X ≡ {x1,...,xn}

iterative solver iterative solver iterative solver

ITERATION (ℓ+1) P(ℓ+1)

k1

(X(ℓ+1)

k1

,Λ(ℓ+1)

k1

) P(ℓ+1)

k2

(X(ℓ+1)

k2

,Λ(ℓ+1)

k2

) P(ℓ+1)

kN

(X(ℓ+1)

kN

,Λ(ℓ+1)

kN

) Λ ≡ diag(λ1,...,λn)

iterative solver iterative solver iterative solver

Next cycle

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Outline

Sequences of correlated eigenproblems The algorithm: Chebyshev Filtered Sub-space Iteration method (ChFSI) ChFSI parallelization and numerical tests

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Chebyshev Filtered Subspace Iteration method (ChFSI)

Properties and algorithm evolution

Iterative solver musts input: the full set of multiple starting vectors Z0 ≡ X(ℓ−1)

ki

(:,1 : NEV); needed: it can efficiently use dense linear algebra kernels (i.e. xGEMM); needed: it avoids stalling when facing small clusters of eigenvalues; Chebyshev Subspace Iteration Firstly introduced in [Rutishauser 1969] A version (called CheFSI) tailored to electronic structure computation in [Zhou, Saad, Tiago and Chelikowski 2006] for sparse eigenvalue problems. Our ChFSI: 1) is tailored for dense eigenproblem sequences, 2) introduces a locking mechanism, 3) contains a refining inner loop, and 4) optimizes the polynomial degree.

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ChFSI pseudocode

INPUT: Hamiltonian, approximate eigenvectors – Z0, extreme eigenvalues {λ1,λNEV}, TOL, DEG. OUTPUT: NEV wanted eigenpairs (Λ,W).

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ChFSI pseudocode

INPUT: Hamiltonian, approximate eigenvectors – Z0, extreme eigenvalues {λ1,λNEV}, TOL, DEG. OUTPUT: NEV wanted eigenpairs (Λ,W).

1 Lanczos step. Identify the bounds for the eigenspectrum interval

corresponding to the wanted eigenspace.

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ChFSI pseudocode

INPUT: Hamiltonian, approximate eigenvectors – Z0, extreme eigenvalues {λ1,λNEV}, TOL, DEG. OUTPUT: NEV wanted eigenpairs (Λ,W).

1 Lanczos step. Identify the bounds for the eigenspectrum interval

corresponding to the wanted eigenspace. REPEAT UNTIL CONVERGENCE:

2 Chebyshev filter. Filter a block of vectors W ←

− Z0.

3 Re-orthogonalize the vectors outputted by the filter; W = QR. 4 Compute the Rayleigh quotient G = Q†HQ. 5 Compute the primitive Ritz pairs (Λ,Y) by solving for GY = YΛ. 6 Compute the approximate Ritz pairs (Λ,W ← QY). 7 Check which one among the Ritz vectors converged. 8 Deflate and lock the converged vectors.

END REPEAT

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The core of the algorithm: Chebyshev filter

The basic principle

Theorem

Let |γ| > 1 and Pm denote the set of polynomials of degree smaller or equal to m. Then the extremum

min

p∈Pm,p(γ)=1 max t∈[−1,1]|p(t)|

is reached by

pm(t) . = Cm(t) Cm(γ).

where Cm is the Chebyshev polynomial of the first kind of order m, defined as

Cm(t) = cos(marccos(t)), t ∈ [−1,1]; cosh(marccosh(t)), |t| > 1.

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The core of the algorithm: Chebyshev filter

Chebyshev polynomials

A generic vector v = ∑n

i=1 sixi is very quickly aligned in the direction of the

eigenvector corresponding to the extremal eigenvalue λ1

vm = pm(A)v =

n

∑

i=1

si pm(A)xi =

n

∑

i=1

si pm(λi)xi = s1x1 +

n

∑

i=2

si Cm( λi−c

e )

Cm( λ1−c

e

) xi ∼ s1x1

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The core of the algorithm: Chebyshev filter

In practice

Three-terms recurrence relation

Cm+1 (t) = 2xCm (t)−Cm−1 (t); m ∈ N, C0 (t) = 1, C1 (t) = x Zm . = pm( ˜ H) Z0 with ˜ H = H −cIn FOR: i = 1 → DEG −1 Zi+1 ← 2σi+1 e ˜ H × Zi −σi+1σi Zi−1 xGEMM END FOR.

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Polynomial degree optimization

Convergence ratio and residuals

Definition

The convergence ratio for the eigenvector xi corresponding to eigenvalue λi / ∈ [α,β] is defined as τ(λi) = |ρi|−1 = min

±

• λi −c

e ± λi −c e 2 −1

• .

The further away λi is from the interval [α,β] the smaller is |ρi|−1 and the faster the convergence to xi is.

For a set of input vectors V = {v1,v2,...,vnev}

Residuals are a function of m and |ρ|

Res(vm

i ) ∼

Const×

• 1

ρi

• m

1 ≤ i ≤ k. Res(vm+m0

i

) ≈ Res(vm0

i )

• 1

ρi

• m

⇒ mi ≥ ln

• TOL

Res(v

m0 i

)

• /lnρi

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ChFSI Single Optimization pseudocode

1 Chebyshev filter. Initial filter W ←

− Z0. with DEG= m0.

2 Re-orthogonalize W = QR & compute the Rayleigh quotient G = Q†HQ. 3 Solve the reduced problem GY = YΛ and compute the approximate Ritz

pairs (Λ,W ← QY) and store their residuals Res(wi).

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ChFSI Single Optimization pseudocode

1 Chebyshev filter. Initial filter W ←

− Z0. with DEG= m0.

2 Re-orthogonalize W = QR & compute the Rayleigh quotient G = Q†HQ. 3 Solve the reduced problem GY = YΛ and compute the approximate Ritz

pairs (Λ,W ← QY) and store their residuals Res(wi). REPEAT UNTIL CONVERGENCE:

4 Optimizer. Compute the polynomial degrees mi ≥ ln

• TOL

Res(wi)

• /lnρi.

5 Chebyshev filter. Filter W ←

− Z0 with DEG= mi.

6 Re-orthogonalize W = QR & compute the Rayleigh quotient G = Q†HQ. 7 Solve the reduced problem GY = YΛ and compute the approximate Ritz

pairs (Λ,W ← QY).

8 lock the converged vectors. 9 Store the residuals Res(wi) of the unconverged vectors.

END REPEAT

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Outline

Sequences of correlated eigenproblems The algorithm: Chebyshev Filtered Sub-space Iteration method (ChFSI) ChFSI parallelization and numerical tests

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Experimental tests setup

C++ implementation of ChFSI

OMPChFSI – OpenMP parallelization for shared memory EleChFSI – Elemental (MPI) parallelization for distributed memory Matrix sizes: 2,600 ÷ 29,500. Tests were performed on the JUROPA cluster. 2 Intel Xeon 5570 (Nehalem-EP) quad-core processors/node, 2.93GHz; 24 GB/node;

THEORETICAL PEAK PERFORMANCE/CORE=11.71 GFLOPS;

Minimum absolute tolerance of residuals Res(xi) = 10−08 ÷10−10; All numerical data are MEDIAN values over 12 distinct measurements.

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Distributed memory: “Elemental” framework

The core of the library is the two-dimensional cyclic element-wise matrix distribution. p MPI processes are logically viewed as a two-dimensional r ×c process grid with p = r ×c. The matrix A = [aij] is distributed over the grid in such a way that the process (s,t) owns the matrix. As,t =    aγ,δ aγ,δ+c ... aγ+r,δ aγ+r,δ+c ... . . . . . .   , γ ≡ (s+σr) mod r and δ ≡ (t +σc) mod c, σr and σc are arbitrarily chosen alignment parameters.

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Distributed memory: “Elemental” framework

The overall performance is influenced by grid shape and algorithmic block size. Grid Shape

For a given number p > 1 of processors there are several possible choices for r and c making a grid shape (r,c) = r ×c. It can have a significant impact on the overall performance and is usually influenced by the algorithm.

Algorithmic block size

Algorithmic block size: size of blocks of input data, correlated to the square root of the L2 cache. Not only depends on the algorithm itself, but it is also affected by the architecture.

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Distributed memory: “Elemental” framework

ℓ = 20, size n = 13,379, and number of sought after eigenpairs nev = 972. The problem was solved with EleChFSI using 16, 32, and 64 cores, all possible grid shapes (r, c) and three distinct algorithmic block sizes.

( 1 , 1 6 ) ( 2 , 8 ) ( 4 , 4 ) ( 8 , 2 ) ( 1 6 , 1 ) ( 1 , 3 2 ) ( 2 , 1 6 ) ( 4 , 8 ) ( 8 , 4 ) ( 1 6 , 2 ) ( 3 2 , 1 ) ( 1 , 6 4 ) ( 2 , 3 2 ) ( 4 , 1 6 ) ( 8 , 8 ) ( 1 6 , 4 ) ( 3 2 , 2 ) ( 6 4 , 1 ) Grid Shape 100 200 300 400 500 Time [sec] 16 Cores 32 Cores 64 Cores The interplay between algorithmic block size and grid shape. Algorithmic blocksize 64 Algorithmic blocksize 128 Algorithmic blocksize 256

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ChFSI time profile

As a function of iteration cycles

Time spent in each stage of the algorithm as a function of the iteration index ℓ for a system of size n = 9,273. 3 6 9 12 iteration index ℓ 5 10 15 20 25 time [s] convergence filter Rayleigh - Ritz QR

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Speed-up

Speed-up =

CPU time (input random vectors) CPU time (input approximate eigenvectors)

3 7 11 15 19 23 Iteration Index

• 1

2 3 4 Speed-up Au98Ag10 - n =13,379 - 128 cores.

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Scalability

Strong Scalability (the size of the eigenproblems are kept fixed while the number of cores is progressively increased) for EleChFSI over two systems of size n = 13,379−12,455−9,273 respectively. 16 32 64 128 256 number of cores 10 102 103 time [s] 196.0 138.0 36.0 102.0 71.0 19.2 55.7 38.2 10.8 384.0 269.0 69.8 30.2 21.4 6.5 Au98Ag10 TiO2 Na15Cl14Li

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EleChFSI versus direct solvers (parallel MRRR)

For the size of eigenproblems here tested the ScaLAPACK implementation

• f BXINV or MRRR is on par of worse than EleMRRR. For this reason a

direct comparison with ScaLAPACK is not included. 3 6 9 12 eigensystem index ℓ 5 10 15 20 25 30 time [s] EleMRRR EleChFSI, 1e-10 EleChFSI OPT, 1e-10 64 cores 128 cores

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EleChFSI versus direct solvers (parallel MRRR)

For the size of eigenproblems here tested the ScaLAPACK implementation

• f BXINV or MRRR is on par of worse than EleMRRR. For this reason a

direct comparison with ScaLAPACK is not included.

620 1358 2037 25 50 75 100 125 150 175 200 time [s] ℓ =4 620 1358 2037 number of eigenpairs sought after ℓ =11 620 1358 2037 ℓ =18 EleMRRR ChFSI, 1e-10 ChFSI, 1e-08

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Some numerical instabilities

10 20 30 40 50 60 10

−11

10

−10

10

−9

10

−8

10

−7

10

−6

Evolution of residuals for the lowest 100 eigenpairs for 10−9 perturbation Polynomial degree (5 −> 60) Eigenpairs residuals

Y0 = Y

sol + 1e−09 * randn1

TOL = 1e−10 nev = 100 nex = 20 10 20 30 40 50 60 10

−11

10

−10

10

−9

10

−8

10

−7

10

−6

Evolution of residuals for the lowest 100 eigenpairs for 10−9 perturbation Polynomial degree (5 −> 60) Eigenpairs residuals

Y0 = Y

sol + 1e−09 * randn2

TOL = 1e−10 nev = 100 nex = 20 10 20 30 40 50 60 10

−16

10

−14

10

−12

10

−10

10

−8

10

−6

Evolution of residuals for the lowest 100 eigenpairs for 10−9 perturbation Polynomial degree (5 −> 60) Eigenpairs residuals

Y0 = Y

sol + 1e−09 * randn1

TOL = 1e−14 nev = 100 nex = 20 5 10 15 20 25 30 35 10

−16

10

−14

10

−12

10

−10

10

−8

10

−6

Evolution of residuals for the lowest 100 eigenpairs for 10−9 perturbation Polynomial degree (5 −> 37) Eigenpairs residuals

Y0 = Y

sol + 1e−09 * randn1

TOL = 1e−10 nev = 100 nex = 40

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Some numerical instabilities

5 10 15 20 25 30 35 40 45 50 55 60 10

−11

10

−10

10

−9

10

−8

10

−7

10

−6 Evolution of residuals for 4 eigenpairs among the lowest 100

Polynomial degree (5 −> 60) Eigenpairs residuals

Res(vm) = a * 1 / |i|m + b * |1|m/|i|m

5 10 15 20 25 30 35 40 45 50 55 60 10

−11

10

−10

10

−9

10

−8

10

−7

10

−6 Evolution of residuals for 4 eigenpairs among the lowest 100

Polynomial degree (5 −> 60) Eigenpairs residuals

Res(vm) = a * 1 / |i|m * sqrt(1 + b * |1|2m )

There are probably two effects that contribute to these instabilities: Early locking deprives the search space of needed components for the convergence of the remaining eigenpairs In the re-orthogonalization there are inexact cancellations against small vanishing values. Such “approximate” zeros are multiplied by |ρ1|m

|ρi|m ≫ 1

In the above example: b ∼ 10−10 or b ∼ 10−13

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Conclusions and future work

Algorithmic strategy

Sequences of “correlated” eigenproblems ⇒ Tailored algorithms Exploiting the correlation of the eigenproblem sequence to speedup the solution of each P(ℓ) is a successful strategy; Combining iterative methods with kernels for dense linear algebra can pay off. The parallelization shows great potential for scalability and parallel efficiency; Uncovering information can lead to further algorithmic optimizations; ONGOING AND FUTURE WORK

1 Fine-tuning filter optimization by profiling eigenvector degrees to

convergence along the sequence so as to further reduce complexity;

2 Exploiting different architectures for parallelization (GPUs, Xeon Phi).

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References

1 EDN and M.Berliafa An Optimized and Scalable Eigensolver for Sequences of Eigenvalue Problems Under review for Concurrency and Computation: Practice and Experience. 2 EDN, and M. Berljafa A Parallel and Scalable Iterative Solver for Sequences of Dense Eigenproblems Arising in FLAPW Parallel Processing and Applied Mathematics, Lecture Notes in Computer Science, Vol 8385. [arXiv:1305.5120] 3 EDN, and M. Berljafa Block Iterative Eigensolvers for Sequences of Correlated Eigenvalue Problems

• Comp. Phys. Comm. 184 (2013), pp. 2478-2488, [arXiv:1206.3768].

4 EDN, P . Bientinesi, and S. Blügel, Correlation in sequences of generalized eigenproblems arising in Density Functional Theory,

• Comp. Phys. Comm. 183 (2012), pp. 1674-1682, [arXiv:1108.2594].

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