| dc.contributor.advisor | Luke, Russell Prof. Dr. | |
| dc.contributor.author | Hermer, Neal | |
| dc.date.accessioned | 2019-03-21T13:32:40Z | |
| dc.date.available | 2019-03-21T13:32:40Z | |
| dc.date.issued | 2019-03-21 | |
| dc.identifier.uri | http://hdl.handle.net/11858/00-1735-0000-002E-E5DD-D | |
| dc.identifier.uri | http://dx.doi.org/10.53846/goediss-7357 | |
| dc.language.iso | eng | de |
| dc.rights.uri | http://creativecommons.org/licenses/by-nc-nd/4.0/ | |
| dc.subject.ddc | 510 | de |
| dc.title | Random Function Iterations for Stochastic Feasibility Problems | de |
| dc.type | doctoralThesis | de |
| dc.contributor.referee | Luke, Russell Prof. Dr. | |
| dc.date.examination | 2019-01-24 | |
| dc.description.abstracteng | The aim of this thesis is to develop a theory that describes errors in fixed point iterations
stochastically, treating the iterations as a Markov chain and analyzing them for
convergence in distribution. These particular Markov chains are also called iterated random
functions. The convergence theory for iterated random averaged operators turns out
to be simple in $\mathbb{R}^n$: If an invariant measure for the Markov operator exists, the chain
converges to an invariant measure, which may depend on the initial distribution. The
stochastic fixed point problem is hence to find invariant measures of the Markov operator.
We formulate different error models and study whether the corresponding Markov
operator possesses an invariant measure; in some cases also rates of convergence w.r.t.
metrics on the space of probability measures can be computed (geometric rates).
There occur two major types of convergence. Weak convergence of the distributions of
the iterates (or their average) and almost sure convergence. The stochastic fixed point
problem can be seen as either consistent or inconsistent stochastic feasibility problem,
where almost sure convergence is observed in the former and weak convergence
in the latter. The type of convergence turns out to determine the consistency of the
problem. We give conditions for which we can expect convergence in the above terms for
general assumptions on the underlying metric space, and nonexpansive, paracontractive
or averaged mappings.
Since the focus of this thesis is probabilistic, when applied to algorithms for optimization,
convergence is in distribution and the fixed points are measures. This perspective is
particularly useful when the underlying problem models systems with measurement errors,
or even when the problem is deterministic, but the algorithm for its numerical solution is
implemented on conventional computers with finite-precision arithmetic. | de |
| dc.contributor.coReferee | Sturm, Anja Prof. Dr. | |
| dc.subject.eng | averaged mappings | de |
| dc.subject.eng | nonexpansive mappings | de |
| dc.subject.eng | stochastic feasibility | de |
| dc.subject.eng | stochastic fixed point problem | de |
| dc.subject.eng | iterated random functions | de |
| dc.subject.eng | convergence of markov chain | de |
| dc.identifier.urn | urn:nbn:de:gbv:7-11858/00-1735-0000-002E-E5DD-D-7 | |
| dc.affiliation.institute | Fakultät für Mathematik und Informatik | de |
| dc.subject.gokfull | Mathematik (PPN61756535X) | de |
| dc.identifier.ppn | 1666649406 | |