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Coping with Complexity: Model Reduction and Data Analysis pp 63–89Cite as

Universal Algorithms, Mathematics of Semirings and Parallel Computations

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Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 75))

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This isaut]Grigory L. Litvinovaut]Victor P. Maslovaut]Anatoly Ya. Rodionovaut]Andrei N. Sobolevskii a survey paper on applications of mathematics of semirings to numerical analysis and computing. Concepts of universal algorithm and generic program are discussed. Relations between these concepts and mathematics of semirings are examined. A very brief introduction to mathematics of semirings (including idempotent and tropical mathematics) is presented. Concrete applications to optimization problems, idempotent linear algebra and interval analysis are indicated. It is known that some nonlinear problems (and especially optimization problems) become linear over appropriate semirings with idempotent addition (the so-called idempotent superposition principle). This linearity over semirings is convenient for parallel computations.

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Acknowledgements

This work is partially supported by the RFBR grant 08-01-00601. The authors are grateful to A. G. Kushner for his kind help.

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Authors and Affiliations

  1. The J. V. Poncelet Laboratory (UMI 2615 CNRS), Independent University of Moscow, Moscow, Russia

    Grigory L. Litvinov & Andrei N. Sobolevski

  2. D. V. Skobeltsyn Research Institute for Nuclear Physics, Moscow State University, Moscow, Russia

    Grigory L. Litvinov & Anatoly Ya. Rodionov

  3. Faculty of Physics, Moscow State University, Moscow, Russia

    Victor P. Maslov

  4. A. A. Kharkevich Institute for Information Transmission Problems, Moscow, Russia

    Andrei N. Sobolevski

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  1. Grigory L. Litvinov
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  2. Victor P. Maslov
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  3. Anatoly Ya. Rodionov
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  4. Andrei N. Sobolevski
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Correspondence to Grigory L. Litvinov .

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  1. Dept. Mathematics, University of Leicester, University Road, Leicester, LE1 7RH, United Kingdom

    Alexander N. Gorban

  2. Dept. Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200a, Leuven, 3001, Belgium

    Dirk Roose

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Litvinov, G.L., Maslov, V.P., Rodionov, A.Y., Sobolevski, A.N. (2011). Universal Algorithms, Mathematics of Semirings and Parallel Computations. In: Gorban, A., Roose, D. (eds) Coping with Complexity: Model Reduction and Data Analysis. Lecture Notes in Computational Science and Engineering, vol 75. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14941-2_4

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