1. Report from LBNL TOPS Meeting 01-25-2002

2. TOPS/01-25-02 – 2Investigators  Staff Members:  Parry Husbands  Sherry Li  Osni Marques  Esmond G. Ng  Chao Yang  New Postdocs:  Laura Grigori  Ali Pinar

3. TOPS/01-25-02 – 3 Eigenvalue Calculations  Collaboration with SLAC in electromagnetic simulations  Parry Husbands, Chao Yang  Ported Omega3P (a generalized eigensolver) to Cray T3E and IBM SP at NERSC  Started to analyze and understand the convergence property of “inexact” shift-invert Lanczos (ISIL) algorithm in Omega3P  Seek ways to improve ISIL  Compare ISIL with exact shift-invert Lanczos

4. TOPS/01-25-02 – 4 Analyzing ISIL  Yong Sun (Stanford): Implemented “inexact” shift-invert Lanczos algorithm in Omega3P  Work well on some problems, but not on others. Why?  Need to solve Ax = b, where A = K – sM may be indefinite  Use PCG + localized symmetric Gauss-Seidel to solve Ax = b Currently solved using Aztec “Local” means matrix splitting on distributed submatrix; splitting yields a matrix B, which is used to construct a preconditioner P Apply CG to PAx = Pb CG convergence tolerance: 10 -2  Issues to be investigated: In terms of eigenvectors of A, is it OK to have large error in the direction associated with the smallest eigenvalues of A? Is it OK not to have a Krylov subspace?

5. TOPS/01-25-02 – 5 Improving accuracy of ISIL  Suppose an approximate eigenpair ( ,q) is computed (perhaps from ISIL)  One can seek a correction pair ( ,z) such that (  + ,q+z) is a better approximation to the generalized eigenvalue problem.  Yong: If q and z are orthogonal, the refinement can be obtained by solving a second order corrector equation, which is nonsymmetric.  If q and z are M-orthogonal, then the second order corrector equation will be symmetric.  Implementation underway.

6. TOPS/01-25-02 – 6 Parallel exact shift-invert Lanczos  Provide a reference point for other eigenvalue calculation methods  Effective and reliable for small to medium sized problems (0.5 – 1 M unknowns); possible on NERSC IBM SP  PARPACK (Sorensen’s Implicitly Restarted Lanczos/Arnoldi)  Lanczos/Arnoldi vectors are distributed  Projected problem (tridiagonal/Hessenberg) replicated  Need sparse LU factorizations Incorporated distributed-memory SuperLU  Symbolic processing is sequential and requires a fully assembled matrix  Solution vector and right-hand side are not distributed yet Considering Raghavan’s DSCPACK for real symmetric matrices  Use AZTEC to perform parallel (mass) matrix-vector multiplications

7. TOPS/01-25-02 – 7 Eigenvalue Opportunities  Supernovae Project, Tony Mezzacappa (ORNL)  Large, sparse eigenvalue problems  Matrices never formed explicitly  Each matrix is a function of 0-1 matrices  Fusion Project, Mitch Pindzola (Auburn)  Currently solving small dense Hermitian eigenvalue problems using ScaLapack from ACTS Toolkit  Eventually will be dealing with large complex symmetric eigenvalue problems  A number of chemistry projects  Piotr Piecuch (Michigan StateU)  Russ Pitzer (Ohio State U)  Peter Taylor (UC San Diego)

8. TOPS/01-25-02 – 8 Eigenvalue Opportunities  One-day meeting between TOPS/eigenvalue and apps planned.  Endorsed by several apps  Details to be worked out  All TOPS/eigenvalue folks to be invited LBNL, UCB, ANL  Mitch Pindzola has requested an eigenvalue short course be given in the summer

9. TOPS/01-25-02 – 9Preconditioning  Scalable preconditioning using incomplete factorization  Padma Raghavan, Keita Teranishi, Esmond Ng  Parallel implementation of incomplete Cholesky factorization  Use of selective inversion to improve scalability of parallel application of incomplete factors during iterations  Performance studied  Paper completed and submitted to Numerical Linear Algebra and Applications

10. TOPS/01-25-02 – 10 Sparse Direct Methods  Distributed memory SuperLU (SuperLU_DIST)  Sherry Li  Working with Argonne folks to interface distributed-memory SuperLU code with PETSc  Finishing 2 papers One on distributed-memory SuperLU Another on on a new ordering algorithm for unsymmetric LU factorization  Next milestone is to provide distributed matrix input for distributed-memory SuperLU

11. TOPS/01-25-02 – 11 Sparse Direct Methods  Sparse Gaussian elimination with low storage requirements  Alan George, Esmond Ng  Attempt to break the storage bottleneck  Based on “throw-away” ideas Discard portion of factors after it is computed Recompute missing portion of factors when needed  Reduce storage requirement substantially, but increase solution time … can control how much storage to use  Sequential implementation for symmetric positive definite matrices completed  Parallel implementation to follow  Extension to general nonsymmetric matrices to be investigated