Numerical Analysis Seminar

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Date, time & place Tuesday 16:30 - 18:00 002Room #002 (Graduate School of Math. Sci. Bldg.)
Organizer(s) Norikazu Saito, Takahito Kashiwabara

2016/06/13

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Atsushi Suzuki (Osaka University)
Dissection : A direct solver with kernel detection for finite element matrices
(日本語)
[ Abstract ]
Large-scale sparse matrices are solved in finite element analyses of elasticity and/or flow problems. In some cases, the matrix may be singular, e.g. due to pressure ambiguity of the Navier-Stokes equations, or due to rigid body movements of sub-domain elasticity problems by a domain decomposition method. Therefore, it is better the linear solver understands rank-deficiency of the matrix.
By assuming the matrix is factorized into LDU form with a symmetric partial permutation, and by introducing a threshold to postpone factorization for pseudo null pivots, solvability of the last Schur complement matrix will be examined. Usual procedure for rank-deficiency problem is based on computation of eigenvalues or singular values and an introduced threshold determines the null space. However, developed new algorithm in DOI:10.1002/nme.4729 is based on computation of residuals combined with orthogonal projections onto supposed image spaces and there is no necessary to introduce a threshold for understanding zero value in floating point. The algorithm uses higher precision arithmetic, e.g. quadruple precision, to distinguish numerical round-off errors that occurred during factorization of the whole sparse matrix from ones during the kernel detection procedure itself.
This is joint work with François-Xavier Roux (LJLL, UPMC/ONERA).