# On Numerical Solution of Multiparametric Eigenvalue Problems

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**On Numerical Solution of Multiparametric Eigenvalue Problems.** / Laaksonen, Mikael.

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*On Numerical Solution of Multiparametric Eigenvalue Problems*. Aalto University.

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TY - THES

T1 - On Numerical Solution of Multiparametric Eigenvalue Problems

AU - Laaksonen, Mikael

PY - 2018

Y1 - 2018

N2 - In this thesis, numerical methods for solving multiparametric eigenvalue problems, i.e., eigenvalue problems of operators that depend on a countable number of parameters, are considered. Such problems arise, for instance, in engineering applications, where a single deterministic problem may depend on a number of design parameters, or through parametrization of random inputs in physical systems with data uncertainty. The focus in this work is on approaches based on the stochastic Galerkin finite element method. In particular, we suggest a novel and efficient algorithm, the spectral inverse iteration, for computing approximate eigenpairs in the case of simple eigenvalues. This algorithm is also extended to a spectral subspace iteration, which allows computation of approximate invariant subspaces associated to eigenvalues of higher multiplicity. A step-by-step analysis is presented on the asymptotic convergence of the spectral inverse iteration and the results of this analysis are verified by a series of detailed numerical experiments. Convergence of the spectral subspace iteration is also illustrated in the numerical experiments, specifically for problems with eigenvalue crossings within the parameter space. Sparse stochastic collocation algorithms are used as reference when validating the output of the two algorithms. As an application of our algorithms we consider solving mechanical vibration problems with uncertain inputs. A hybrid method is suggested for computing eigenmodes of structures with randomness in both geometry and the elastic modulus. Furthermore, two different strategies are presented for computing the eigenmodes for a shell of revolution: one based on dimension reduction and separation of the eigenmodes by wavenumber, and another based on applying the algorithm of spectral subspace iteration directly to the original problem.

AB - In this thesis, numerical methods for solving multiparametric eigenvalue problems, i.e., eigenvalue problems of operators that depend on a countable number of parameters, are considered. Such problems arise, for instance, in engineering applications, where a single deterministic problem may depend on a number of design parameters, or through parametrization of random inputs in physical systems with data uncertainty. The focus in this work is on approaches based on the stochastic Galerkin finite element method. In particular, we suggest a novel and efficient algorithm, the spectral inverse iteration, for computing approximate eigenpairs in the case of simple eigenvalues. This algorithm is also extended to a spectral subspace iteration, which allows computation of approximate invariant subspaces associated to eigenvalues of higher multiplicity. A step-by-step analysis is presented on the asymptotic convergence of the spectral inverse iteration and the results of this analysis are verified by a series of detailed numerical experiments. Convergence of the spectral subspace iteration is also illustrated in the numerical experiments, specifically for problems with eigenvalue crossings within the parameter space. Sparse stochastic collocation algorithms are used as reference when validating the output of the two algorithms. As an application of our algorithms we consider solving mechanical vibration problems with uncertain inputs. A hybrid method is suggested for computing eigenmodes of structures with randomness in both geometry and the elastic modulus. Furthermore, two different strategies are presented for computing the eigenmodes for a shell of revolution: one based on dimension reduction and separation of the eigenmodes by wavenumber, and another based on applying the algorithm of spectral subspace iteration directly to the original problem.

KW - Eigenvalue problems

KW - uncertainty quantification

KW - sFEM

KW - stochastic Galerkin method

KW - stochastic collocation

KW - sparse tensor approximation

KW - mechanical vibration

KW - shells of revolution

KW - ominaisarvotehtävät

KW - epävarmuuden kvantifiointi

KW - stokastinen Galerkinin menetelmä

KW - stokastinen kollokaatio

KW - harva tensoriapproksimaatio

KW - mekaaninen värähtely

KW - pyörähdyskuori

KW - Eigenvalue problems

KW - uncertainty quantification

KW - sFEM

KW - stochastic Galerkin method

KW - stochastic collocation

KW - sparse tensor approximation

KW - mechanical vibration

KW - shells of revolution

M3 - Doctoral Thesis

SN - 978-952-60-8068-0

T3 - Aalto University publication series DOCTORAL DISSERTATIONS

PB - Aalto University

ER -

ID: 30202714