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Optimizing Least-Squares Rational Filters for SolvingInterior Eigenvalue Problems
Jan Winkelmann, Edoardo Di Napoli{winkelmann,dinapoli}@aices.rwth-aachen.de
Aachen Institute for Advanced Studies in Computational Engineering Science
July 6, 2016
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1 Introduction and Motivation
2 Least-Squares Filters
3 Comparison to State of the Art
4 Optimization of Least-Squares Filters
5 Conclusion
6 Constrained LSOP
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Filtered Subspace Iteration
Consider the interior hermitian eigenvalue problem
Ax = λx , with A = AH ∈ Cn×n sparse and λ ∈ I = [λmin, λmax ]
Can be extended to generalized interior (non-hermitian) eigenproblems
Algorithm Sketch1 Filter: X̂n+1 := ρ(A)Xn
2 Rayleigh-Ritz:[Q,Λ] := eig(X̂H
n+1 · A · X̂n+1)
Xn+1 := X̂n+1 · Q
X0 initial guess of eigenvectorsρ(A) spectral projector
Example (Gauss filter)
−2 −1 0 1 2
0
0.2
0.4
0.6
0.8
1
Real axis
Filte
rva
lue
Π(x)Gauss
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Rational filter methods
Classic rational filter methods use numerical integration of a contour
ρ(A)X =12πı
∮C
(zI − A)−1Xdz ≈N∑i=1
ωi (zi I − A)−1X
We can use any function that yieldscheap system solves. For Example:
ρ(A) =N∑i=1
αi (zi I − A)−1
Previous WorkSS-Method 1
FEAST 2
Zolotarev Filters 3
Least-Squares 4,5
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Rational filter methods
Classic rational filter methods use numerical integration of a contour
ρ(A)X =12πı
∮C
(zI − A)−1Xdz ≈N∑i=1
ωi (zi I − A)−1X
We can use any function that yieldscheap system solves. For Example:
ρ(A) =N∑i=1
αi (zi I − A)−1
Previous WorkSS-Method 1
FEAST 2
Zolotarev Filters 3
Least-Squares 4,5
1Sakurai, Sugiura. Journal of Computational and Applied Mathematics; 20032Polizzi. Physical Review B; 20093Güttel, et.al. SIAM Journal on Scientific Computing; 20154Barel. Linear Algebra and its Applications; 20155Xi, Saad, Y. Preprint; 2015
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Least-Squares based filters
The idea:
minαi ,zi 1≤i≤N
∫ ∞−∞
∣∣∣∣∣∏(x)−N∑i=1
αi
zi − x
∣∣∣∣∣2
dx
Note that:∏(x) is the the indicator
function on [−1, 1].Coefficients αi , and poles zi arecomplex-valued.We optimize coefficients andpoles of the rational function.
Potential Benefits:Faster convergenceIncreased flexibilityProblem specific filters
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First Results
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.2
0
0.2
0.4
0.6
0.8
1
1.2
Real axis
Filte
rva
lue
GaussLSOPΠ
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First Results
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Real axis
Filte
rva
lue
GaussLSOP
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Our Goals
Filtered Subspace Iteration is sensitive to the spectrumAppropriate choice of subspace size and degree requiredDifferent filter types have different strengthsEven with some prior knowledge:Optimal choice of parameters is difficult.This is a difficult problem ⇒ We will not change this
The filter should be "smart"Take advantage of the spectrumIdeally, this would work automatically
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State of the Art Filters
Gauss FilterFrom Gauss(-Legendre) QuadratureBenefits from larger search subspace (≈ 1.5)
Zolotarev FilterFrom Cauer, Elliptical, or Zolotarev FiltersDoes not benefit from larger subspacesMay have better convergence or load-balancing
Choosing optimal filter and parameters is difficultFEAST 3.0 as Benchmark toolAll Timings are for single threaded execution
Maximum of 40 iterationsTermination criterion: Residual ≤ 10−14
Filter–type, –degree and search subspace as indicated
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Filter "Comparison", 4 Poles per Quadrant
0 0.5 1 1.5 2 2.5 310−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Real axis
Filte
rva
lue
ZolotarevLSOPGauss
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2D FEM: CNT Spectrum Visualization
10 20 30 40 50 60 70 80 90 100
−60
−40
−20
0
Arbitrary Index
Eige
nval
ue
Eigenvaluesλ minλ max
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2D FEM: CNT, 4 Poles per Quadrant
100 120 140 1600
10
20
30
% of target subspace
Iter
atio
ns
GaussZolotarev
LSOP
100 120 140 1600
10
20
30
% of target subspace
Tim
e(s)
GaussZolotarev
LSOP
Gauss filter benefits from larger subspacesLeast-Squares filter requires less iterations than ZolotarevSomewhat benefits from larger subspace
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2D FEM: CNT, 1 Pole per Quadrant
100 120 140 1600
10
20
30
40
% of target subspace
Iter
atio
ns
GaussLSOP
100 120 140 1600
10
20
30
% of target subspace
Tim
e(s)
GaussLSOP
Zolotarev does not converge within 40 iterations
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2D FEM: CNT, Complete Timings
100 120 140 160
10
20
30
% of target subspace
Tim
e(s)
Gauss-8Zolotarev-8LSOP-8Gauss-4
Zolotarev-4LSOP-4Gauss-1LSOP-1
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2D FEM: CNT, Complete Timings
100 120 140 160
10
20
30
% of target subspace
Tim
e(s)
Gauss-8Zolotarev-8LSOP-8Gauss-4
Zolotarev-4LSOP-4Gauss-1LSOP-1
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2D FEM: CNT, Complete Timings
100 120 140 160
10
20
30
% of target subspace
Tim
e(s)
Gauss-8Zolotarev-8LSOP-8Gauss-4
Zolotarev-4LSOP-4Gauss-1LSOP-1
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3D FEM: Caffeinep2 Spectrum Visualization
20 40 60 80 100 120 140 160 180 200−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
·10−16
Arbitrary Index
Eige
nval
ue
Eigenvaluesλ minλ max
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3D FEM: Caffeinep2, 4 Poles per Quadrant
100 120 140 160
10
15
20
% of target subspace
Iter
atio
ns
ZolotarevLSOP
100 120 140 160600
800
1,000
1,200
% of target subspace
Tim
e(s)
ZolotarevLSOP
Gauss does not converge(recall the large cluster at the edge of the interval)Zolotarev outperforms Least-Squares filters for any number of poles
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Recall: Zolotarev vs LSOP Filters
0.9 0.95 1 1.05 1.1 1.15 1.210−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Real axis
Filte
rva
lue
ZolotarevLSOP
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Wrap-up: Non Problem-Specific Filters
Results for unconstrained Least-Squares filtersWorks well even with low degreesThis is not the take away today
The take-away should be:1 Gather spectrum information (separate topic)2 Use appropriate filter and parameters3 Facilitated via custom Least-Squares filters
Least-Squares filters are very flexibleConstraints for asymmetry, steepness, decay possibleFast generation of filters possible
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1 Introduction and Motivation
2 Least-Squares Filters
3 Comparison to State of the Art
4 Optimization of Least-Squares Filters
5 Conclusion
6 Constrained LSOP
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Approximating Π(x)
minγi ,zi 1≤i≤N/4
∫ ∞−∞
∣∣∣∏(x)− f (x , z , γ)∣∣∣2 dx , with
f (x , z , γ) =
N/4∑i=1
γizi − x
+γi
zi − x− γi
zi + x− γi
zi + x
Reflection Symmetry: f (−x , z , γ) = f (x , z , γ)
Complex Conjugation: f (x , z , γ) = f (x , z , γ)
Symmetries force evenness and real-nessAnalogous to existing integration rulesFor simplicity we disallow poles on the axesAsymmetric filters lack reflection symmetry
We require cheap computation of:Least-Squares ResidualGradients w.r.t. αi , zi
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Approximating Π(x)
minγi ,zi 1≤i≤N/4
∫ ∞−∞
∣∣∣∏(x)− f (x , z , γ)∣∣∣2 dx , with
f (x , z , γ) =
N/4∑i=1
γizi − x
+γi
zi − x− γi
zi + x− γi
zi + x
Reflection Symmetry: f (−x , z , γ) = f (x , z , γ)
Complex Conjugation: f (x , z , γ) = f (x , z , γ)
Symmetries force evenness and real-nessAnalogous to existing integration rulesFor simplicity we disallow poles on the axesAsymmetric filters lack reflection symmetry
We require cheap computation of:Least-Squares ResidualGradients w.r.t. αi , zi
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Calculating Least-Squares Residuals
∫ ∞−∞
w(x)
∣∣∣∣∣∣∏
(x)−N/4∑i=1
γizi − x
+γi
zi − x− γi
zi + x− γi
zi + x
∣∣∣∣∣∣2
dx
An appropriate w(x) simplifies the calculation:
w(x) =
0, if |x | > a
β, if |x | ≤ 11, else
With β ≈ 0.4, and a ≈ 5.Exact 1.0 in [−1, 1] not required⇒ Small β increases sharpness at -1,1’Cut-off’ after a ⇒ Definite integrals
We can provide a (matrix) form for the residual
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Optimization
We can derive Gradients w.r.t. zi , and αi
Use of symmetries ⇒ Cheaper GradientsFast filter generation via Levenberg-MarquardtDetails in forthcoming publication
Descent methods require a starting positionVery relevant for non-convex optimization
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Local Minima for 4 Poles per Quadrant
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.5
1
Real axis
Filte
rva
lue
The filter is non-convex in the poles ziResulting filter depends on the starting positionMany approaches do not yield good results
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Local Minima for 4 Poles per Quadrant
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.5
1
Real axis
Filte
rva
lue
The filter is non-convex in the poles ziResulting filter depends on the starting positionMany approaches do not yield good results
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Use Existing Methods as Starting Points
Existing filters already have good residuals
1 2 3 4 5 6 7 8 9 10
100
10−2
10−4
10−6
Poles per Quadrant
Leas
t-Sq
uare
sRes
idua
l
GaussZolotarev
⇒ Good results as starting positionsBoth produce the same results for ≤ 7 poles per quadrantSo far, these are the best residuals we know of
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Use Existing Methods as Starting Points
Existing filters already have good residuals
1 2 3 4 5 6 7 8 9 10
100
10−2
10−4
10−6
Poles per Quadrant
Leas
t-Sq
uare
sRes
idua
l
GaussZolotarev
LSOP (Gauss)LSOP (Zolotarev)
⇒ Good results as starting positionsBoth produce the same results for ≤ 7 poles per quadrantSo far, these are the best residuals we know of
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Conclusion
Filters are important for contour-based eigensolversNo single filter type is best
We present Least-Squares based filters:Rich framework for filter generationPotential for custom filtersCompetetive results for unconstrained Least-Squares filtersFast generation of filters
No theoretical insights on generated filters(i.e. worst-case convergence ratio)Systematic comparisons of filters is tricky
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Future Work
Parallelism and convergence of linear system solves
Further details in forthcoming publication:
More constrained optimization of filters
Systematic comparisons of filters
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The End
Thank you for your attentionQuestions?
Financial support from the DeutscheForschungsgemeinschaft (GermanResearch Association) through grantGSC 111 is gratefully acknowledged.
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Constrained LSOP
Poles near the real axis ⇒ Almost real shift zi of MatrixN/4∑i=1
γiIzi − A
+γi
I zi − A− γi
Izi + A− γi
I zi + A
X = B
The shifted (zi I − A) may become (almost) singular
Larger imaginary part on the poles can prevent this "problem"
minγi ,zi 1≤i≤N/4
F (x), such that
=(zi ) ≥ 0.1, 1 ≤ i ≤ N/4
F (x) =
∫ ∞−∞
w(x)
∣∣∣∣∣∣∏
(x)−N/4∑i=1
γizi − x
+γi
zi − x− γi
zi + x− γi
zi + x
∣∣∣∣∣∣2
dx
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Constrained LSOP
Poles near the real axis ⇒ Almost real shift zi of MatrixN/4∑i=1
γiIzi − A
+γi
I zi − A− γi
Izi + A− γi
I zi + A
X = B
The shifted (zi I − A) may become (almost) singular
Larger imaginary part on the poles can prevent this "problem"
minγi ,zi 1≤i≤N/4
F (x), such that
=(zi ) ≥ 0.1, 1 ≤ i ≤ N/4
F (x) =
∫ ∞−∞
w(x)
∣∣∣∣∣∣∏
(x)−N/4∑i=1
γizi − x
+γi
zi − x− γi
zi + x− γi
zi + x
∣∣∣∣∣∣2
dx
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Zolotarev vs constrained LSOP, 4 Poles per Quadrant
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.2
0
0.2
0.4
0.6
0.8
1
1.2Zolotarev
LSOP
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Zolotarev vs constrained LSOP, 4 Poles per Quadrant
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210−7
10−6
10−5
10−4
10−3
10−2
10−1
100
ZolotarevLSOP
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Zolotarev vs constrained LSOP, 4 Poles per Quadrant
0.2 0.4 0.6 0.8 1
10−3
10−2
10−1
100
Zolotarev PolesLSOP Poles
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CAFF timings
100 120 140 160
600
800
1,000
1,200
1,400
1,600
% of target subspace
Tim
e(s)
Zolotarev-3LSOP-3
Zolotarev-4LSOP-4
Zolotarev-8LSOP-8
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