Peters, Michael. Numerical methods for boundary value problems on random domains. 2014, Doctoral Thesis, University of Basel, Faculty of Science.

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Official URL: http://edoc.unibas.ch/diss/DissB_11087
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Abstract
In this thesis, we consider the numerical solution of
elliptic boundary value problems on random domains.
The underlying domain is modelled
via a random vector field which is given by its mean
and its covariance.
Having these statistics of the random perturbation at
hand, we aim at determining the related statistics of
the random solution.
To that end, we propose the domain mapping method
on the one hand and the perturbation method on the
other hand.
For the domain mapping method, we have to compute the
random vector field's KarhunenLoève expansion.
For this purpose, we compare cluster methods, namely
the adaptive cross approximation and the fast multipole
method, and the pivoted Cholesky decomposition.
After this, we show regularity results for the random
solution dependent on the decay of the random vector
field's KarhunenLoève expansion. These results are used
to employ a QuasiMonte Carlo quadrature for the
approximation of mean and variance.
For the perturbation method, we linearize the random
solution's dependence on the vector field by means of
a shape Taylor expansion. This approach yields a single
partial differential equation for the approximation of
the mean and a tensor product partial differential
equation for the approximation of the covariance. The latter
is solved efficiently with the aid of the sparse tensor
product combination technique.
elliptic boundary value problems on random domains.
The underlying domain is modelled
via a random vector field which is given by its mean
and its covariance.
Having these statistics of the random perturbation at
hand, we aim at determining the related statistics of
the random solution.
To that end, we propose the domain mapping method
on the one hand and the perturbation method on the
other hand.
For the domain mapping method, we have to compute the
random vector field's KarhunenLoève expansion.
For this purpose, we compare cluster methods, namely
the adaptive cross approximation and the fast multipole
method, and the pivoted Cholesky decomposition.
After this, we show regularity results for the random
solution dependent on the decay of the random vector
field's KarhunenLoève expansion. These results are used
to employ a QuasiMonte Carlo quadrature for the
approximation of mean and variance.
For the perturbation method, we linearize the random
solution's dependence on the vector field by means of
a shape Taylor expansion. This approach yields a single
partial differential equation for the approximation of
the mean and a tensor product partial differential
equation for the approximation of the covariance. The latter
is solved efficiently with the aid of the sparse tensor
product combination technique.
Advisors:  Harbrecht, Helmut 

Committee Members:  Schwab, Christoph 
Faculties and Departments:  05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Computational Mathematics (Harbrecht) 
UniBasel Contributors:  Peters, Michael and Harbrecht, Helmut 
Item Type:  Thesis 
Thesis Subtype:  Doctoral Thesis 
Thesis no:  11087 
Thesis status:  Complete 
Number of Pages:  127 p. 
Language:  English 
Identification Number: 

Last Modified:  22 Jan 2018 15:52 
Deposited On:  13 Jan 2015 14:23 
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