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

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2013
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Popova, Evgenija D.
Smith, Andrew P.
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CHINCHULUUN, Altannar, ed., Panos M. PARDALOS, ed., Rentsen ENKHBAT, ed., E. N. PISTIKOPOULOS, ed.. Optimization, Simulation, and Control. New York, NY: Springer New York, 2013, pp. 301-318. Springer Optimization and Its Applications. 76. ISBN 978-1-4614-5130-3. Available under: doi: 10.1007/978-1-4614-5131-0_19
Zusammenfassung

We give a short survey on methods for the enclosure of the solution set of a system of linear equations where the coeffficients 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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Fachgebiet (DDC)
510 Mathematik
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parametric systems of linear equations, range enclosure, Bernstein polynomials
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ISO 690GARLOFF, Jürgen, Evgenija D. POPOVA, Andrew P. SMITH, 2013. Solving Linear Systems with Polynomial Parameter Dependency with Application to the Verified Solution of Problems in Structural Mechanics. In: CHINCHULUUN, Altannar, ed., Panos M. PARDALOS, ed., Rentsen ENKHBAT, ed., E. N. PISTIKOPOULOS, ed.. Optimization, Simulation, and Control. New York, NY: Springer New York, 2013, pp. 301-318. Springer Optimization and Its Applications. 76. ISBN 978-1-4614-5130-3. Available under: doi: 10.1007/978-1-4614-5131-0_19
BibTex
@incollection{Garloff2013Solvi-25018,
  year={2013},
  doi={10.1007/978-1-4614-5131-0_19},
  title={Solving Linear Systems with Polynomial Parameter Dependency with Application to the Verified Solution of Problems in Structural Mechanics},
  number={76},
  isbn={978-1-4614-5130-3},
  publisher={Springer New York},
  address={New York, NY},
  series={Springer Optimization and Its Applications},
  booktitle={Optimization, Simulation, and Control},
  pages={301--318},
  editor={Chinchuluun, Altannar and Pardalos, Panos M. and Enkhbat, Rentsen and Pistikopoulos, E. N.},
  author={Garloff, Jürgen and Popova, Evgenija D. and Smith, Andrew P.}
}
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