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Report from LBNL TOPS Meeting 01-25-2002
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TOPS/01-25-02 – 2Investigators Staff Members: Parry Husbands Sherry Li Osni Marques Esmond G. Ng Chao Yang New Postdocs: Laura Grigori Ali Pinar
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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
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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?
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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.
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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
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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)
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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
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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
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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
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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
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