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Scientific Computing Seminar

Scientific Computing Seminar. AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues *. * Work supported by NSF under grants NSF/ITR-0082094, NSF/ACI-0305120 and by the Minnesota Supercomputing Institute . C. Bekas: SC Seminar . Introduction and Motivation. Target Problem

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Scientific Computing Seminar

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  1. Scientific Computing Seminar AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues* *Work supported by NSF under grants NSF/ITR-0082094, NSF/ACI-0305120 and by the Minnesota Supercomputing Institute

  2. C. Bekas: SC Seminar Introduction and Motivation • Target Problem • Compute a large number of the smallest eigenvalues of large sparse matrices • Numerous important applications, including: • structural engineering • computational materials science (electronic structure calculations) • Signal/Image processing and Control Shift and Invert techniques can be very successful (Grimes et al 94, MSC.NASTRAN)… BUT quickly become impractical to use: • For very large problem sizes (…N>106) a supercomputer is needed • When we need to compute several hundreds or thousands of eigenvalues (deep in the spectrum) reorthogonalization costs dominate and become prohibitive! AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  3. C. Bekas: SC Seminar Introduction and Motivation Component Mode Synthesis (CMS) (Hurty ’60, Graig-Bampton ’68) Well known alternative. Used for many years in Structural Engineering. But it too suffers from limitations due to problem size… AMLS, (Bennighof, Lehoucq, Kaplan and collaborators)  Multilevel CMS method (solves the dimensionality problem)  Automatic computation of substructures (easy application)  Approximation: Truncated Congruence Transformation  Builds very large projection basis withoutreorthogonalization  Successful in computing thousands of eigenvalues in vibro-acoustic analysis (N>107) in a few hours on workstations (Kropp–Heiserer, 02) Accuracy issues  AMLS accuracy is adequate in Structural Eng. (in the order of the discretization error) , but  higher accuracy is needed in other applications, (i.e. electronic structure calculations)  AMLS is an one shot approach: no (iterative) refinement is done AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  4. C. Bekas: SC Seminar Recent Advances The success of AMLS in Struct. Eng. and its potential for other applications has sparked several new research initiatives. Some include: - Bekas and Saad Purely algebraic analysis of AMLS 1) Approximation to a nonlinear (Schur) eigenvalue problem 2) Approximation of the resolvent (A- I)-1 by a careful projection 3) Improvements: a) 2nd order expansions, b) Krylov projections and combinations - Yang et al Algebraic substructuring. Careful selection of added eigenvectors for improved accuracy. - Elssel and Voss A priori error bounds for algebraic substructuring. AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  5. C. Bekas: SC Seminar In this talk We will describe - Approximation mechanism behind AMLS. We will review our recent purely algebraic analysis of AMLS (Bekas – Saad, 04). - Iterative application of AMLS 1) Refinement of approximated eigenvalues 2) Multiple shifts in AMLS 3) Computation of eigenvalues in an interval - Numerical examples AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  6. 2 1 Subdivide  into 2 subdomains: 1 and 2 C. Bekas: SC Seminar Component Mode Synthesis: a model problem Consider the model problem: Y on the unit square . We wish to compute smallest eigenvalues. • Component Mode Synthesis • Solve problem on each i • “Combine” partial solutions X AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  7. CMS: Approximation procedure. Ignore coupling! C. Bekas: SC Seminar Component Mode Synthesis: Approximation • CMS: Approximation procedure (2) • Solve B v =  v • Remember that B is block diagonal • Thus: we have to solve smaller decoupled eigenproblems • Then, CMS approximates the coupling among the subdomains by the application of a carefully selected operator on the interface unknowns AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  8. Block Gaussian eliminator matrix: such that: Where S = C – E> B-1 E is the Schur Complement Equivalent Generalized Eigenproblem : UTAUu= UTU u C. Bekas: SC Seminar Recent CMS method: AMLS AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  9. Solve decoupled problems Form basis FINAL APPROXIMATION: C. Bekas: SC Seminar AMLS Approximation AMLS: Approximation procedure. Ignore coupling! AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  10. B1 E1 S3 B2 In the following we analyze the approximation mechanism of one step of AMLS, adopting a purely algebraic setting. S2 This will naturally lead to improved versions of the method E1* S1 C. Bekas: SC Seminar AMLS: Multilevel application Scheme applied recursively. Resulting to thousands of subdomains. Successful in computing thousands smallest eigenvalues in vibro-acoustic analysis with problem size N>107 AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  11. Leads to: Substitute equation (1) in equation (2) to give equivalent non-linear problem: C. Bekas: SC Seminar AMLS: approximation to a nonlinear eigenproblem! AMLS: Approximation procedure. DO NOT ignore coupling this time AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  12. Resolvent: Basic Property We can show that: C. Bekas: SC Seminar AMLS: approximation to a nonlinear eigenproblem! Define: AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  13. Truncate the series. Keep the first two terms only. Then… Approximate problem: Remember: AMLS: C. Bekas: SC Seminar AMLS: approximation to a nonlinear eigenproblem! Neumann Series of the Resolvent: AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  14. We can show that if: Then,  is eigenvalue of (1) and Respective eigenvector C. Bekas: SC Seminar AMLS: The projection view-point Initial problem: AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  15. 1. AMLS solves approximately (truncation) 2. AMLS, change of basis: 3. Thus: C. Bekas: SC Seminar AMLS: The projection view-point 4. Remedy: augment the space of approximants by eigenvectors of B. Why? AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  16. Then, this difference is: Thus: the difference is large in eigenvectors of B with eigenvalues close to the smallest eigenvalues of A. C. Bekas: SC Seminar AMLS: The projection view-point We examine the difference: Expansion in terms of eigenvectors of B: AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  17. Let: mB “smallest” eigenvectors of B XB: restriction to the B-part (upper part) of the space of approximants. Then: C. Bekas: SC Seminar AMLS: The projection view-point Therefore: augmenting the space of approximants with ‘’smallest’’eigenvectors of B is well justified. Can we bound the error? AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  18. C. Bekas: SC Seminar Improvements The algebraic framework we have described leads to improvements: Nonlinear Schur complement problem: • Introduce an additional term of the truncated Neumann series • Utilize corresponding 2nd order projection Approximation of the resolvent (B- I)-1: • Approximate with Krylov subspaces on B-1 and • Utilize corresponding projection Combinations of the above improvements lead to hybrid algorithms with enhanced stability and robustness AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  19. Therefore it is natural to consider the Krylov subspace or in general the block Krylov subspace many j The columns of US are eigenvectors of the nonlinear Schur complement problem Vk is an orthonormal block Krylov basis C. Bekas: SC Seminar Augmenting with Krylov Subspaces We need to approximate AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  20. AMLS approximation 2nd order approximation Quadratic Eigenvalue Problem: C. Bekas: SC Seminar Second Order Approximation Neumann Series of the Resolvent: Can we do better? AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  21. Linearization leads to equivalent generalized eigenproblem: Quadratic Eigenvalue Problem: C. Bekas: SC Seminar Second Order Approximation: Solving the QED AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  22. Better approx. by the QEP We can add more eigenvectors of B…or add “second order” vectors: ADD THE VECTORS Construct basis Solve C. Bekas: SC Seminar Second Order Approximation: Projection view point Eigenvector of A: AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  23. Let: mB smallest eigenvectors of B XB: restriction to the B-part (upper part) of the space of approximants. Then: C. Bekas: SC Seminar Second Order Approximation: Error bounds Augmenting the space of approximants with eigenvectors of B and “second order vectors” leads to quadratic error bounds compared to adding just eigenvectors AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  24. C. Bekas: SC Seminar Improvements The algebraic framework we have described leads to improvements: Our previous work Nonlinear Schur complement problem: • Introduce an additional term of the truncated Neumann series • Approximation of the resolvent (B- I)-1: • Approximate with Krylov subspaces on B-1 and • Utilize corresponding projection Recent Work • Framework for the iterative refinement of the AMLS approximations • Utilization of many different shifts (A-k I)-1 that can be used for… • …computation of eigenvalues deep in the spectrum • More robust method. Currently examining connections with shift-invert Lanczos (Grimes et al) and Rational Krylov (Ruhe) AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  25. C. Bekas: SC Seminar Iterative Refinement (1/3) AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  26. AMLS APPROXIMATION R: AMLS REMAINDER C. Bekas: SC Seminar Iterative Refinement (2/3) NON-LINEAR SCHUR COMPLEMENT OR WITH SOME REARRANGEMENT… AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  27. C. Bekas: SC Seminar Iterative Refinement (3/3) AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  28. C. Bekas: SC Seminar Using Multiple Shifts in AMLS (1/3) MATRIX: BCSSTK11 1st shift: 3, close to eigenvalues in the interval [2.9,3] 2nd shift: 69, close to eigenvalues in the interval [68,70] CAN WE COMBINE THESE TWO SHIFTS? AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  29. Smallest eigenvectors of Smallest eigenvectors of PROJECTED PROBLEM C. Bekas: SC Seminar Using Multiple Shifts in AMLS (2/3) AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  30. C. Bekas: SC Seminar Using Multiple Shifts in AMLS (3/3) AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  31. C. Bekas: SC Seminar Numerical Experiments AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  32. Bef. reord. /1 =0.87 After reord. /1 =0.07 Matrix: BCSSTK11(N=1473), Reorder using Nes. Disec., size of Schur complement NS=94 C. Bekas: SC Seminar Numerical Experiments: Standard v.s. Krylov Thus: Small /1favors the Krylov version AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  33. C. Bekas: SC Seminar Numerical Experiments: Standard v.s. Krylov REORDERED VERSION • Number of Schur eigenvectors: 5 • Number of added vectors: • AMLS: 30 eigenvectors of B (left) and 40 eigenvectors of B (right) • Krylov: 30=5 x 6 (left) and 40=5 x 8 (right) AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  34. C. Bekas: SC Seminar Numerical Experiments: Standard v.s. Krylov NOT REORDERED VERSION • Number of Schur eigenvectors: 5 • Number of added vectors: • AMLS: 30 eigenvectors of B (left) and 40 eigenvectors of B (right) • Krylov: 30=5 x 6 (left) and 40=5 x 8 (right) AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  35. C. Bekas: SC Seminar Second order method Matrix: BCSSTK11 Approximate S() u= u with a Quadratic eigenvalue problem instead of the original generalized eigenvalue problem of AMLS AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  36. C. Bekas: SC Seminar Second order method 2nd order: 10 Schur vectors: 20 added Krylov vectors. 30 vectors in total AMLS: 10 Schur vectors: 20, 40, 60 and 100 added eigenvectors of B Matrix: BCSSTK11 Compare 2nd order AMLS (20 added vectors) v.s. standard AMLS with increasing number of added eigenvectors of B (mB=20,40,60,100) AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  37. REMEDY: Augment subspace with eigenvectors of B too! VB: eigenvectors of B VK: Krylov basis US: Schur eigenvectors C. Bekas: SC Seminar Combine AMLS with Krylov AMLS Matrix: 5-point stencil discretization of the Laplacian Use only the 2 smallest Schur vectors. Then, for larger eigenvalues the Krylov version can fail. AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  38. REMEDY: Augment subspace with eigenvectors of B too! VB: eigenvectors of B VK: Krylov subspace US: Schur eigenvectors C. Bekas: SC Seminar Combine AMLS with Krylov AMLS Matrix: 5-point stencil discretization of the Laplacian Use only 2 smallest Schur vectors. Then, for larger eigenvalues the Krylov version can fail. AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  39. C. Bekas: SC Seminar Iterative Refinement Next shift k is 1st smallest approximate eigenvalue at step k-1 Next shift k is 5th smallest approximate eigenvalue at step k-1 Matrix: 5-point stencil discretization of the Laplacian AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  40. C. Bekas: SC Seminar Multiple Shifts Dimension of each Qi : 40 Dimension of each Qi : 60 Matrix: 5-point stencil discretization of the Laplacian AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

  41. C. Bekas: SC Seminar Conclusions AMLS is a promising alternative to Shift-Invert methods for very large problems • In this work we have presented an analysis of AMLS based on a completely algebraic framework: • AMLS is a nonlinear (spectral) Schur complement method that utilizes… • Projection on carefully selected eigenspaces to approximate the solution • Improvements • Based on the algebraic framework we have proposed Krylov projection subspaces and second order approximations (and combinations) with significant improvements. • Iterative Refinement • Approximations can be iteratively refined, allowing for very good accuracy • Many different shifts are combined and thus we can compute eigenvalues deep in the spectrum • Current Work • Investigate strategies to compute all eigenvalues in a (large) interval [a,b] AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

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