## Seminar in Numerical Analysis: Olaf Schenk (Universita della Svizzera italiana)

**Performance Engineering and Sparse Matrices: Introduction, applications and supercomputing**

We will review the state-of-the art techniques in the parallel direct solution of linear systems of equations and present several recent new research directions. This includes (i) fast methods for evaluating certain selected elements of a matrix function that can be used for solving the Kohn-Sham-equation without explicit diagonalization and (ii) stochastic optimization problems under uncertainty from power grid problems from electrical power grid systems. Several algorithmic and performance engineering advances are discussed to sove the underlying sparse linear algebra problems. The new developments include novel incomplete augmented multicore sparse factorizations, multicore- and GPU-based dense matrix implementations, and communication-avoiding Krylov solvers. We also improve the interprocess communication on Cray systems to solve e.g. 24-hour horizon power grid problems from electrical power grid systems of realistic size with up to 1.95 billion decision variables and 1.94 billion constraints. Full-scale results are reported on Cray XC30 and BG/Q, where we observe very good parallel efficiencies and solution times within a operationally defined time interval. To our knowledge, "real-time"-compatible performance on a broad range of architectures for this class of problems has not been possible prior to present work.

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