Linear and Nonlinear SP 2 Methods for Large Scale Eigenvalue - PowerPoint PPT Presentation

Linear and Nonlinear “ SP 2 ” Methods for Large Scale Eigenvalue Calculations Zhaojun Bai University of California, Davis Workshop on Recent Advances in Numerical Methods for Eigenvalue Problems NCTS, Taiwan, January 4-8, 2008

Linear and Nonlinear “ SP 2 ” Methods for Large Scale Eigenvalue Calculations (SP 2 = Structure-Preserving Subspace Projection) Zhaojun Bai University of California, Davis Workshop on Recent Advances in Numerical Methods for Eigenvalue Problems NCTS, Taiwan, January 4-8, 2008

Two parts of this talk • Part I: Linear problems Algebraic Substructuring for eigenvalue and frequency response calculations • Part II: Nonlinear problems Nonlinear Rayleigh-Ritz ITerative (NRRIT) technique for solving nonlinear eigenvalue problems

Part I Algebraic Substructuring for Eigenvalue and Frequency Response Calculations joint work with X. S. Li, C. Yang, P. Husbands, E. Ng LBNL W. Gao Fudan University, China J. H. Ko Konkuk University, South Korea

EIG and FRA • EIG (eigenvalue problem) σ K σ q = λ σ Mq Kq = λMq − → where K σ = K − σM, λ σ = λ − σ • FRA (frequency response calculation) H ( ω ) = l T ( K − ω 2 M + i ωD ) − 1 b = l T [ γ 1 ( ω ) K σ + γ 2 ( ω, σ ) M ] − 1 b ↑ σ, D = αK + βM where γ 1 ( ω ) = 1 + i ωα γ 2 ( ω, σ ) = − ω 2 + σ + i ω ( σα + β ) ,

Substructuring methods • Eigenvalue and frequency response calculations are ubiquitous. • Substructuring techniques dates back to the 1960s (CMS) • Substructuring holds great promoise for solving extremely large scale problems, e.g., AMLS • Issues as the techniques extended for broader applications: – Extraction of arbitrary eigenmodes – High frequency response calculations – Accuracy enhancement – Performance tuning

Outline of Part I 1. Substructure partition and reduction (projection) 2. ASEIG : eigenvalue calculation 3. ASFRA : frequency response calculation 4. Remarks

Substructure partition I.1 • Single-level ⎡ ⎤ ⎡ ⎤ K σ K σ M 11 M 13 ⎢ 11 13 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ K σ = ⎢ K σ 22 K σ ⎥ ⎢ ⎥ M = M 22 M 23 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 23 ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ K σ 31 K σ 32 K σ M 31 M 32 M 33 33 • Multi-level (nested dissection) �� 1 �� 2 �� 3 �� 4 �� 5 �� 15 �� 6 �� 7 �� 8 �� 7 14 �� 9 �� 10 �� 11 �� 6 �� 3 10 13 �� 12 �� 13 �� 14 �� 11 12 1 2 4 5 8 9 �� 15 �� e.g., by M E T I S of Karypis and Kumar

Substructure reduction (projection) I.1 • Block elimination matrix L ⎡ ⎤ ⎡ ⎤ K σ � M 11 M 13 ⎢ 11 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ K σ = L − T K σ L − 1 = M = L − T ML − 1 = ⎢ K σ ⎥ ⎢ ⎥ � � � ⎦ , M 22 M 23 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 22 ⎢ ⎥ ⎢ ⎥ ⎣ ⎣ ⎦ K σ � � � � M 31 M 32 M 33 33 • Extraction of local modes ⎡ ⎤ ( K σ S 1 ← − 11 , M 11 ) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ( K σ S m = S 2 ← − 22 , M 22 ) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ K σ � � S 3 ← − ( 33 , M 33 ) • Subspace projection onto span { S m } ⎡ ⎤ ⎡ ⎤ k σ m 11 m 13 � ⎢ 11 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ K σ m = S T K σ S m = M m = S T ⎢ k σ ⎥ ⎢ ⎥ � � ⎦ , MS m = m 22 m 23 ⎢ ⎥ ⎢ � ⎥ ⎢ ⎥ ⎢ ⎥ m 22 m ⎢ ⎥ ⎢ ⎥ ⎣ ⎣ ⎦ k σ � m 31 m 32 m 33 � � � 33

Eigenvalue calculation I.2 • Projected (smaller) eigenvalue problem: K σ m Φ = M m ΦΘ σ , • “Global” eigenpairs: Θ σ = diag ( θ σ ) = diag (Θ σ l , Θ σ n , Θ σ r ) � Φ l Φ n Φ r � Φ = ( φ ) = • Retained modes: Φ n ( determined by “global cutoff values” ) � Φ l Φ r � Truncated modes: Φ t = • Ritz pairs (Θ σ n + σ, L − 1 S m Φ n ) ≈ ( λ, q ) ’s of ( K, M ) • Why does it work, ... [Yang et al ], [Voss et al ], ...

AS algorithm for eigenvalue computation I.2 0. Substructure partition, if necessary 1. Block elimination and congruence transformation 2. Mode selection/computation for sub-structures and separators 3. Subspace assembling 4. Projection calculation 5. Projected eigenvalue problem

AS implementation I.2 • Major operations: 1. Congruence transformations and projection 2. projected eigenvalue problem • Cost 1. flops: more than a single sparse Cholesky factorization 2. storage: Block Cholesky factors + projected matrices + ... But no triangular solvers, no (re)-orthogonalization ... vs. SIL • ASEIG package [ACM TOMS ’08] 1. interleaves steps 1-4 computations 2. recomputes some of intermediate matrix blocks instead of storing 50% of memory saving trade with 15% recompute time

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