An Augmented Incomplete Factorization Approach for Computing the Schur Complement in Stochastic Optimization
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Petra-2014-An augmented incompl.pdf
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Author(s)
Petra, Cosmin G.
Schenk, Olaf
Gäertner, Klaus
Lubin, Miles C
Date Issued
January 2014
Journal
SIAM Journal on Scientific Computing
Publisher
Society for Industrial and Applied Mathematics
Citation
Petra, Cosmin G., Olaf Schenk, Miles Lubin, and Klaus Gäertner. “An Augmented Incomplete Factorization Approach for Computing the Schur Complement in Stochastic Optimization.” SIAM Journal on Scientific Computing 36, no. 2 (January 2014): C139–C162. © 2014, Society for Industrial and Applied Mathematics.
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Final published version
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 high-performance 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.
MIT Department
Massachusetts Institute of Technology. Operations Research Center
Sloan School of Management
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1137/130908737
