Sakurai-Sugiura algorithm based eigenvalue solver for Siesta. Georg Huhs
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1 Sakurai-Sugiura algorithm based eigenvalue solver for Siesta Georg Huhs
2 Motivation Timing analysis for one SCF-loop iteration: left: CNT/Graphene, right: DNA
3 Siesta Specifics High fraction of EVs needed (e.g. 30%) Sparse matrices Actual solver: ScaLAPACK Dense solver Limited scaling
4 Siesta Specifics High fraction of EVs needed (e.g. 30%) Sparse matrices Actual solver: ScaLAPACK Dense solver Limited scaling
5 Characteristics SS-method, Tetsuya Sakurai and Hiroshi Sugiura, 2002 Solver for GEV A x = λ B x A, B C n n Projection method, based on complex contour integration Meant for finding EVs in a certain domain Suitable for large, sparse matrices parallelisation
6 Basic steps Divide domain of interest into subdomains, in each: 1 Solve a set of linear systems 2 Use results for numerical integrations 3 Construct subspace 4 Find eigenvalues/-vectors in subspace and backtransform 5 Select correct eigenpairs
7 Parallelisation Three levels of parallelisation 1 Handling subdomains in parallel 2 Solving N linear systems in parallel 3 Use parallel solver for each linear system
8 Parallelisation Three levels of parallelisation 1 Handling subdomains in parallel 2 Solving N linear systems in parallel 3 Use parallel solver for each linear system
9 Parallelisation Three levels of parallelisation 1 Handling subdomains in parallel 2 Solving N linear systems in parallel 3 Use parallel solver for each linear system
10 Timing expectations Example: DNA, 776 atoms, n = 7752
11 Using GPUs Implemented: Solving the linear systems with CUSP. Algorithm: gmres without preconditioner! Gain: Speedup of 3-4 compared to WSMP (for the smaller examples) Timing solving all lin. systems in parallel (per subdomain) Problem Setup Solve Comm total ScaLAPACK water pt (n = 3280) DNA (n = 7752) hemoglobina (n = 16707) All times in seconds
12 Conclusions Issues Inefficient in serial mode Memory parallelisation only in linear solver Benefits Robust algorithm, parameterisation can be hidden from user Actual problems: SS can be faster than ScaLAPACK, depending on scaling of lin. solver Future problems: high potential for larger numbers of processors Possibility to use many GPUs efficiently
13 Conclusions Issues Inefficient in serial mode Memory parallelisation only in linear solver Benefits Robust algorithm, parameterisation can be hidden from user Actual problems: SS can be faster than ScaLAPACK, depending on scaling of lin. solver Future problems: high potential for larger numbers of processors Possibility to use many GPUs efficiently
14 Thanks to Alberto Garcı a, Pablo Ordejo n (Siesta) Tetsuya Sakurai (Algorithm) Anshul Gupta (WSMP) BSC support team Jose Marı a Cela (BSC)
15 More details (1) 1 Define EV-range γ, ρ and a set of vectors V C n L. 2 Set quatrature points: z j = γ + ρe (2πi(j+ 1 2) 1 N ), j = 0,..., N 1 3 Solve the linear systems (z j B A) Y j = B V N systems, L right hand sides each independent!
16 More details (2) 4 Numerical integration S k = 1 N N/2 1 j=0 ( ) k+1 zj γ Y j, k = 0,..., M 1 ρ Matrix S = [S 0, S 1,..., S M 1 ] C n ML 5 Find the rank K of S with an SVD and shrink it to n K 6 QR-decomposition of S orthogonal basis Q 7 à = Q T (A γb) Q B = Q T B Q Ã, B C K K 8 Eigenpairs of à x = λ B x
17 More details (3) 8 Eigenpairs of à x = λ B x 9 Eigenpairs of the original problem λ j = λ j + γ x j = Q x j 10 Remove exterior eigenvalues and ghosts.
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