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Report from LBNL

Report from LBNL. TOPS Meeting 01-25-2002. Investigators. Staff Members: Parry Husbands Sherry Li Osni Marques Esmond G. Ng Chao Yang New Postdocs: Laura Grigori Ali Pinar. Eigenvalue Calculations. Collaboration with SLAC in electromagnetic simulations Parry Husbands, Chao Yang

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Report from LBNL

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

  2. Investigators • Staff Members: • Parry Husbands • Sherry Li • Osni Marques • Esmond G. Ng • Chao Yang • New Postdocs: • Laura Grigori • Ali Pinar

  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. 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. 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. 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. 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. 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. Preconditioning • 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. 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. 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

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