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Solving Linear Systems with Polynomial Parameter Dependency with Application to the Verified Solution of Problems in Structural Mechanics

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Optimization, Simulation, and Control

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 76))

Abstract

We give a short survey on methods for the enclosure of the solution set of a system of linear equations where the coefficients of the matrix and the right hand side depend on parameters varying within given intervals. Then we present a hybrid method for finding such an enclosure in the case that the dependency is polynomial or rational. A general-purpose parametric fixed-point iteration is combined with efficient tools for range enclosure based on the Bernstein expansion of multivariate polynomials. We discuss applications of the general-purpose parametric method to linear systems obtained by standard finite element analysis of mechanical structures and illustrate the efficiency of the new parametric solver.

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Notes

  1. 1.

    Preliminary results were presented at the 2nd International Conference on Uncertainty in Structural Dynamics, Sheffield, UK, June 15–17, 2009.

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Acknowledgements

This work has been supported by the State of Baden-Württemberg, Germany.

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Authors and Affiliations

  1. University of Applied Sciences / HTWG Konstanz, Postfach 100543, D-78405, Konstanz, Germany

    Jürgen Garloff & Andrew P. Smith

  2. Institute of Mathematics & Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev str., Bldg. 8, BG-1113, Sofia, Bulgaria

    Evgenija D. Popova

Authors
  1. Jürgen Garloff
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  2. Evgenija D. Popova
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  3. Andrew P. Smith
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Corresponding author

Correspondence to Jürgen Garloff .

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Editors and Affiliations

  1. , Institute of Mathematics, National University of Mongolia, University Street 1, Ulaanbaatar, 210646, Erdenet, Mongolia

    Altannar Chinchuluun

  2. , Department of Industrial & Systems Engin, University of Florida, Weil Hall 401, Gainesville, 32611, Florida, USA

    Panos M. Pardalos

  3. The School of Economic Studies, Dept.of Mathematics & Computer Science, National University of Mongolia, Ulaanbaatar, 210646, Mongolia

    Rentsen Enkhbat

  4. , Centre for Process Systems Engineering, Imperial College London, South Kensington Campus, London, SW7 2AZ, United Kingdom

    E. N. Pistikopoulos

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Garloff, J., Popova, E.D., Smith, A.P. (2013). Solving Linear Systems with Polynomial Parameter Dependency with Application to the Verified Solution of Problems in Structural Mechanics. In: Chinchuluun, A., Pardalos, P., Enkhbat, R., Pistikopoulos, E. (eds) Optimization, Simulation, and Control. Springer Optimization and Its Applications, vol 76. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5131-0_19

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