Abstract
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 stepbystep 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.
Original language  English 

Qualification  Doctor's degree 
Awarding Institution 

Supervisors/Advisors 

Publisher  
Print ISBNs  9789526080680 
Electronic ISBNs  9789526080697 
Publication status  Published  2018 
MoE publication type  G5 Doctoral dissertation (article) 
Keywords
 Eigenvalue problems
 uncertainty quantification
 sFEM
 stochastic Galerkin method
 stochastic collocation
 sparse tensor approximation
 mechanical vibration
 shells of revolution
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Laaksonen, M. (2018). On Numerical Solution of Multiparametric Eigenvalue Problems. Aalto University.