Approximate Selective Matrix Inversion Algorithms for Computing the Diagonal Entries of the Inverse Matrix

Functions of entries of inverses of matrices like all diagonal entries of a sparse matrix inverse or its trace arise in several important computational applications such as density functional theory [2], covariance matrix analysis in uncertainty quantification [1] or when evaluating Green's functions in computational nanolelectronics [4].

We will present algorithms for (approximately) computing selective parts of the matrix inverse such as the diagonal part. The core part of this algorithm uses an approximate version of the SelInv method by Lin et al [3]. We will discuss modifications of this algorithm to make it applicable for incomplete factorization methods. Among these are inverse awareness of the compute incomplete triangular factor as well as eigenvector deflation techniques. Its overall performance will be demonstrated for selected numerical examples, in particular for symmetric and indefinite application problems which frequently arise from practical applications.

References.

[1] C. Bekas, A. Curioni, and I. Fedulova. Low-cost high performance uncertainty quantification. Concurrency and Computation: Practice and Experience, 2011.

[2] W. Kohn, L. Sham, et al. Self-consistent equations including exchange and correlation effects. Phys. Rev, 140(4A):A1133-A1138, 1965.

[3] L. Lin, C. Yang, J.C. Meza, J. Lu, L. Ying, and W. E. SelInv - an algorithm for selected inversion of a sparse symmetric matrix. ACM Transactions on Mathematical Software}, 37(4):40:1-40:19, 2011.

[4] M. Luisier, T. Boykin, G. Klimeck, and W. Fichtner. Atomistic nanoelectronic device engineering with sustained performances up to 1.44 pflop/s. In High Performance Computing, Networking, Storage and Analysis (SC), 2011 International Conference for, pages 1-11. IEEE, 2011.

Current affiliation: professor for numerical mathematics at TU Braunschweig (since 2006). Habilitation: venia legendi for Mathematics at TU Berlin, 2003. PhD: PhD in Mathematics at TU Chemnitz, 1998.
Research field: Numerical analysis, preconditioning methods in particular those based on incomplete factorizations, or approximate inverses or multilevel-based, scientific computing, model order reduction methods, hierarchical matrices.
Co-authored software: ILUPACK (multilevel incomplete factorization package), JADAMILU, (symmetric eigenvalue computation package based on preconditioned Jacobi-Davidson methods).