ILAS 2017 July 25, 2017

Transcription

1 Polynomial and rational filtering for eigenvalue problems and the EVSL project Yousef Saad Department of Computer Science and Engineering University of Minnesota ILAS 217 July 25, 217

2 Large eigenvalue problems in applications Challenge in eigenvalue problems: extract large number of eigenvalues & vectors of very large matrices (quantum physics/ chemistry,...) - often in the middle of spectrum. Example: Excited states involve transitions much more complex computations than for DFT (ground states) Large matrices, *many* eigen-pairs to compute Illustration: Hamiltonian of size n 1 6 get 1% of bands 2 ILAS July 25, 217

3 Solving large interior eigenvalue problems Three broad approaches: 1. Shift-invert (real shifts) 2. Polynomial filtering 3. Rational filtering (Cauchy, + others). Issues with shift-and invert (and related approaches) Issue 1: factorization may be too expensive Can use iterative methods? Issue 2: Iterative techniques often fail Reason: Highly indefinite problems. First Alternative: Spectrum slicing with Polynomial filtering 3 ILAS July 25, 217

4 Spectrum Slicing Situation: very large number of eigenvalues to be computed Goal: compute spectrum by slices by applying filtering Apply Lanczos or Subspace iteration to problem: 1.8 φ(λ i ) Pol. of degree 32 approx δ(.5) in [ 1 1] φ(a)u = µu φ(t) a polynomial or rational function that enhances wanted eigenvalues φ ( λ ) λ i.2 λ 4 ILAS July 25, 217

5 Rationale. Eigenvectors on both ends of wanted spectrum need not be orthogonalized against each other : Idea: Get the spectrum by slices or windows [e.g., a few hundreds or thousands of pairs at a time] Can use polynomial or rational filters 5 ILAS July 25, 217

6 Hypothetical scenario: large A, *many* wanted eigenpairs Assume A has size 1M... and you want to compute 5, eigenvalues/vectors (huge for numerical analysits, not for physicists) in the lower part of the spectrum - or the middle. By (any) standard method you will need to orthogonalize at least 5K vectors of size 1M. Then: Space needed: b = 4TB *just for the basis* Orthogonalization cost: = 5 PetaOPS. At step k, each orthogonalization step costs 4kn This is 2, n for k close to 5,. 6 ILAS July 25, 217

7 Illustration: All eigenvalues in [, 1] of a 49 3 Laplacean 35 3 Computing all 1,971 e.v. s. in [, 1] Matvec Orth Total 25 2 Time Number of slices Note: This is a small pb. in a scalar environment. Effect likely much more pronounced in a fully parallel case. 7 ILAS July 25, 217

8 How do I slice my spectrum?.25 Slice spectrum into 8 with the DOS.2.15 Answer: Use the DOS..1 DOS We must have: ti+1 t i φ(t)dt = 1 n slices b a φ(t)dt 8 ILAS July 25, 217

9 Polynomial filtering Apply Lanczos or Subspace iteration to: M = ρ(a) where ρ(t) is a polynomial Each matvec y = Av is replaced by y = ρ(a)v. Eigenvalues in high part of filter will be computed first. Old (forgotten) idea. But new context is *very* favorable 9 ILAS July 25, 217

10 What polynomials? LS approximations to δ-dirac functions Pol. of degree 32 approx δ(.5) in [ 1 1] Obtain the LS approximation to the δ Dirac function Centered at some point (TBD) inside the interval. φ ( λ ) φ(λ i ) λ i λ W ll express everything in the interval [ 1, 1] 1 ILAS July 25, 217

11 Theory The Chebyshev expansion of δ γ is k { 1 j = ρ k (t) = µ j T j (t) with µ j = 2 cos(j cos 1 (γ)) j > j= Recall: The delta Dirac function is not a function we can t properly approximate it in least-squares sense. However: Proposition Let ˆρ k (t) be the polynomial that minimizes r(t) w over all polynomials r of degree k, such that r(γ) = 1, where. w represents the Chebyshev L 2 -norm. Then ˆρ k (t) = ρ k (t)/ρ k (γ). 11 ILAS July 25, 217

12 A few technical details. Issue # one: balance the filter To facilitate the selection of wanted eigenvalues [Select λ s such that ρ(λ) > bar] we need to find γ so that ρ(ξ) == ρ(η) ρ k ( λ ).6.4 ρ k ( λ ) ξ η ξ η λ λ Procedure: Solve the equation ρ γ (ξ) ρ γ (η) = with respect to γ, accurately. Use Newton or eigenvalue formulation. 12 ILAS July 25, 217

13 Issue # two: Determine degree & polynomial (automatically) Start low then increase degree until value (s) at the boundary (ies) become small enough - Exple for [.833,.97..] 1.2 Degree = 3; Sigma damping 1.2 Degree = 4; Sigma damping 1.2 Degree = 7; Sigma damping Degree = 13; Sigma damping 1.2 Degree = 2; Sigma damping 1.2 Degree = 23; Sigma damping ILAS July 25, 217

14 Polynomial filtered Lanczos: No-Restart version Degree = 23; Sigma damping Use Lanczos with full reorthogonalization on ρ(a). Eigenvalues of ρ(a): ρ(λ i ) Accept if ρ(λ i ) bar Ignore if ρ(λ i ) < bar ρ(λ) Unwanted eigenvalues Wanted 14 ILAS July 25, 217

15 Polynomial filtered Lanczos: Thick-Restart version PolFilt Thick-Restart Lanczos in a picture: 1.8 φ(λ i ) Pol. of degree 32 approx δ(.5) in [ 1 1] If accurate then lock else add to Thick Restart set. φ ( λ ).6.4 Reject.2 λ i λ Due to locking, no more candidates will show up in wanted area after some point Stop. 15 ILAS July 25, 217

16 TR Lanczos: The 3 types of basis vectors Basis vectors Matrix representation Locked vectors Thick restart vectors Lanczos vectors 16 ILAS July 25, 217

17 Experiments: Hamiltonian matrices from PARSEC Matrix n nnz [a, b] [ξ, η] ν [ξ,η] Ge 87 H , M [ 1.21, 32.76] [.64,.53] 212 Ge 99 H 1 112, M [ 1.22, 32.7] [.65,.96] 25 Si 41 Ge 41 H , M [ 1.12, 49.82] [.64,.28] 218 Si 87 H 76 24, M [ 1.19, 43.7] [.66,.3] 213 Ga 41 As 41 H , M [ 1.25, 131] [.64,.] ILAS July 25, 217

18 Results: (No-Restart Lanczos) Matrix deg iter matvec CPU time (sec) matvec orth. total max residual Ge 87 H ,2 26, Ge 99 H ,9 28, Si 41 Ge 41 H , Si 87 H ,1 29, Ga 41 As 41 H , Demo with Si1H16 [n = 17, 77, nnz(a) = 446, 5] 18 ILAS July 25, 217

19 RATIONAL FILTERS

20 Why use rational filters? Consider a spectrum like this one: 1 9 Polynomial filtering utterly ineffective for this case Second issue: situation when Matrix-vector products are expensive Generalized eigenvalue problems. 2 ILAS July 25, 217

21 Alternative is to use rational filters: φ(z) = j α j z σ j φ(a) = j α j(a σ j I) 1 We now need to solve linear systems Tool: Cauchy integral representations of spectral projectors P = 1 2iπ Γ (A si) 1 ds Numer. integr. P P Use Krylov or S.I. on P Sakurai-Sugiura approach [Krylov] FEAST [Subs. iter.] (E. Polizzi) 21 ILAS July 25, 217

22 What makes a good filter real(σ) =[.]; imag(σ) =[ 1.1]; pow = 3 Opt. pole 3 Single pole Single pole 3 Gauss 2 poles Gauss 2 poles real(σ) =[.]; imag(σ) =[ 1.1]; pow = 3 Opt. pole 3 Single pole Single pole 3 Gauss 2 poles Gauss 2 poles Assume subspace iteration is used with above filters. Which filter will give better convergence? Simplest and best indicator of performance of a filter is the magnitude of its derivative at -1 (or 1) 22 ILAS July 25, 217

23 The Gauss viewpoint: Least-squares rational filters Given: poles σ 1, σ 2,, σ p Related basis functions φ j (z) = 1 z σ j Find φ(z) = p j=1 α jφ j (z) that minimizes w(t) h(t) φ(t) 2 dt h(t) = step function χ [ 1,1]. w(t)= weight function. For example a = 1, β =.1 w(t) = if t > a β if t 1 1 else 23 ILAS July 25, 217

24 Advantages: Can select poles far away from real axis faster iterative solvers Very flexible can be adapted to many situations Can repeat poles (!) Implemented in EVSL.. [Interfaced to SuiteSparse as a solver] 24 ILAS July 25, 217

25 Spectrum Slicing and the EVSL project Newly released EVSL uses polynomial and rational filters Each can be appealing in different situations. Spectrum slicing: cut the overall interval containing the spectrum into small sub-intervals and compute eigenpairs in each sub-interval independently φ ( λ ) λ 25 ILAS July 25, 217

26 Levels of parallelism Macro task 1 Slice 1 Domain 1 Slice 2 Slice 3 Domain 2 Domain 3 Domain 4 The two main levels of parallelism in EVSL 26 ILAS July 25, 217

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28 EVSL Main Contributors (version 1.1.) + support Ruipeng Li LLNL Yuanzhe Xi Post-doc (UMN) Luke Erlandson UG Intern (UMN) Work supported by DOE [ending this summer] and by NSF [going forward] 28 ILAS July 25, 217

29 EVSL: current status & plans Version _1. Released in Sept. 216 Matrices in CSR format (only) Standard Hermitian problems (no generalized) Spectrum slicing with KPM (Kernel Polynomial Meth.) Trivial parallelism across slices with OpenMP Methods: Non-restart Lanczos polynomial & rational filters Thick-Restart Lanczos polynomial & rational filters Subspace iteration polynomial & rational filters 29 ILAS July 25, 217

30 Version _1.1.x V_1.1. Due for release end of July general matvec [passed as function pointer] Ax = λbx Fortran (3) interface. Spectrum slicing by Lanczos and KPM Efficient Spectrum slicing for Ax = λbx (no solves with B). Version _1.2.x V_1.2. Early 218 (?) Fully parallel version [MPI + openmp] Challenge application in earth sciences [in progress] 3 ILAS July 25, 217

31 Conclusion Polynomial Filtering appealing when # of eigenpairs to be computed is large and Matvecs are not too expensive Somewhat costly for generalized eigenvalue problems Will not work well for spectra with large outliers. Alternative: Rational filtering Both approaches implemented in EVSL Current focus: provide as many interfaces as possible. EVSL code available here: EVSL Also on github (development) 31 ILAS July 25, 217