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    An Augmented Incomplete Factorization Approach for Computing the Schur Complement in Stochastic Optimization

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    Type
    Article
    Authors
    Petra, Cosmin G.
    Schenk, Olaf
    Lubin, Miles
    Gäertner, Klaus
    Date
    2014-01
    Permanent link to this record
    http://hdl.handle.net/10754/597514
    
    Metadata
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    Abstract
    We present a scalable approach and implementation for solving stochastic optimization problems on high-performance computers. In this work we revisit the sparse linear algebra computations of the parallel solver PIPS with the goal of improving the shared-memory performance and decreasing the time to solution. These computations consist of solving sparse linear systems with multiple sparse right-hand sides and are needed in our Schur-complement decomposition approach to compute the contribution of each scenario to the Schur matrix. Our novel approach uses an incomplete augmented factorization implemented within the PARDISO linear solver and an outer BiCGStab iteration to efficiently absorb pivot perturbations occurring during factorization. This approach is capable of both efficiently using the cores inside a computational node and exploiting sparsity of the right-hand sides. We report on the performance of the approach on highperformance computers when solving stochastic unit commitment problems of unprecedented size (billions of variables and constraints) that arise in the optimization and control of electrical power grids. Our numerical experiments suggest that supercomputers can be efficiently used to solve power grid stochastic optimization problems with thousands of scenarios under the strict "real-time" requirements of power grid operators. To our knowledge, this has not been possible prior to the present work. © 2014 Society for Industrial and Applied Mathematics.
    Citation
    Petra CG, Schenk O, Lubin M, Gäertner K (2014) An Augmented Incomplete Factorization Approach for Computing the Schur Complement in Stochastic Optimization. SIAM Journal on Scientific Computing 36: C139–C162. Available: http://dx.doi.org/10.1137/130908737.
    Sponsors
    We acknowledge Jeff Hammond for his guidance in BG/Pimplementation and informative discussions. We also acknowledge Edward Rothbergof Gurobi for an early discussion on the use of the augmented approach. K. G¨artneralso acknowledges King Abdullah University of Science and Technology (KAUST) forproviding access to a BG/P platform.
    Publisher
    Society for Industrial & Applied Mathematics (SIAM)
    Journal
    SIAM Journal on Scientific Computing
    DOI
    10.1137/130908737
    ae974a485f413a2113503eed53cd6c53
    10.1137/130908737
    Scopus Count
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