Stochastic Gradient Descent and its Application for Parametrized Boundary Value Problems under Uncertainties

Wolf, Florian

Stochastic Gradient Descent and its Application for Parametrized Boundary Value Problems under Uncertainties

Dateien

Wolf_2-1as0cw2bsjlzs7.pdfGröße: 1.64 MBDownloads: 579

Datum

2021

Open Access-Veröffentlichung

Open Access Green

Sammlungen

Mathematik und Statistik

Publikationstyp

Bachelorarbeit

Publikationsstatus

Published

Zusammenfassung

In this thesis we want to give a theoretical and practical introduction to stochastic gradient descent (SGD) methods. In the theoretical part, we prove two fundamental convergence results that hold under certain assumptions, like a strongly convex objective function. The first result covers the convergence behaviour of SGD running with a fixed step size sequence and is expanded to the second result, which deals with SGD running with a diminishing step size sequence. For both cases, we provide an upper bound for the expected optimality gap. At the expense of a concrete convergence rate, we then generalize both results to non-convex objective functions. The practical part of this thesis deals with the application of SGD as a convincing and stable optimizer for parametrized boundary value problems under uncertainties. Firstly, we discretize an ordinary differential equation (ODE) Dirichlet problem using finite differences (FD) and improve the results by using preconditioning techniques and a weighted norm. Secondly, we generalize the results to an elliptic partial differential equation (PDE) Dirichlet problem and aim for a weak solution using a finite element (FE) discretization. For both problems, the SGD algorithm convinces with stable results and provides convergence in expectation.

Fachgebiet (DDC)

510 Mathematik

Schlagwörter

Stochastic Gradient Descent, Boundary Value Problems,

Zitieren

ISO 690

WOLF, Florian, 2021. Stochastic Gradient Descent and its Application for Parametrized Boundary Value Problems under Uncertainties [Bachelor thesis]. Konstanz: Universität Konstanz

BibTex

@mastersthesis{Wolf2021Stoch-54423,
  year={2021},
  title={Stochastic Gradient Descent and its Application for Parametrized Boundary Value Problems under Uncertainties},
  address={Konstanz},
  school={Universität Konstanz},
  author={Wolf, Florian}
}

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Hochschulschriftenvermerk

Konstanz, Universität Konstanz, Bachelorarbeit, 2021

Universitätsbibliographie

Ja