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Author:Laaksonen, Mikael
Title:Finite Element Methods for Stochastic Eigenvalue Problems
Elementtimenetelmä stokastisille ominaisarvotehtäville
Publication type:Master's thesis
Publication year:2014
Pages:56      Language:   eng
Department/School:Perustieteiden korkeakoulu
Main subject:Mekaniikka   (Mat-5)
Supervisor:Stenberg, Rolf
Instructor:Hakula, Harri
Electronic version URL: http://urn.fi/URN:NBN:fi:aalto-201507013707
OEVS:
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Location:P1 Ark Aalto  1471   | Archive
Keywords:stochastic collocation
stochastic Galerkin method
stochastic eigenvalue problem
Rayleigh quotient iteration
Smolyak construction
sFEM
stokastinen kollokaatio
stokastinen Galerkinin elementtimenetelmä
stokastinen ominaisarvotehtävä
Rayleighin iteraatio
Smolyakin konstruktio
Abstract (eng): In this thesis we consider finite element methods for stochastic eigenvalue problems.
As a model problem we will consider the eigenvalue problem of an elliptic diffusion operator, where the diffusion coefficient is assumed to be a random field.
We discuss the fundamental theory of discretizing equations of this kind and consider methods of approximately solving them.

We present two numerical schemes of solving the model problem.
The first one is a specific combination of the stochastic Galerkin method, stochastic collocation method, and the Rayleigh quotient iteration.
The second approach is a pure collocation method, where we use Smolyak sparse grids to reduce the number of collocation points.

We illustrate the convergence and functionality of the presented methods by applying them to the model problem.
The stochastic collocation method is found to be a reliable choice.
The Rayleigh quotient iteration scheme also seems to have potential, although it significantly overestimates the variance of the solution.
Abstract (fin): Tässä diplomityössä tarkastellaan stokastisten ominaisarvotehtävien ratkaisemista elementtimenetelmällä.
Malliongelmaksi asetetaan elliptisen diffuusio-operaattorin ominaisarvotehtävä, jossa diffuusiokertoimen oletetaan olevan satunnaiskenttä.
Työssä käsitellään tämän tyyppisten tehtävien diskretointiin tarvittavaa teoriaa ja tarkastellaan numeerisia menetelmiä niiden ratkaisemiseksi.

Malliongelman ratkaisemiseksi esitetään kaksi menetelmää.
Ensimmäinen on eräänlainen yhdistelmä stokastisesta Galerkinin menetelmästä, stokastisesta kollokaatiomenetelmästä sekä Rayleighin iteraatiosta.
Toinen menetelmä on puhdas kollokaatiomenetelmä, jossa käytetään Smolyakin konstruktiota kollokaatiopisteiden määrän vähentämiseksi.

Esitettyjen menetelmien suppenemista ja toimivuutta havainnollistetaan soveltamalla niitä malliongelmaan.
Stokastinen kollokaatiomenetelmä todetaan luotettavaksi vaihtoehdoksi.
Myös Rayleighin iteraatioon perustuva menetelmä vaikuttaa potentiaaliselta, vaikka se antaakin selvästi liian suuria arvioita ratkaisun varianssille.
ED:2014-06-26
INSSI record number: 49341
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