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# Proper Orthogonal Decomposition Analysis for a Non-Linear Spatiotemporal SIR-Model

Type of Publication: | Diploma thesis |

Publication status: | Published |

URI (citable link): | http://nbn-resolving.de/urn:nbn:de:bsz:352-2-1tm1zuh8yjxn99 |

Author: | Nguyen Pham, Hai Dang |

Year of publication: | 2022 |

Summary: |
This thesis shows how Proper Orthogonal Decomposition (POD) techniques and the Discrete Empirical Interpolation Method (DEIM) can be applied to decrease the computational effort in solving a Semi-linear Parabolic Reaction-Diffusion System, a class of Partial Differential Equa- tions (PDE), which describes the system dynamics of a time-space-dependent SIR-compartment- model.
We investigate this model, which is widely-known as a mathematical tool for describing the course of an epidemic by quantifying the number of individuals of a population, which are classified in susceptible (S), infected (I) and removed (R) compartments. We introduce assumptions on the SIR-model and establish initial conditions to specify the dynamics of the model. This leads to a PDE, based on the SIR-model, which is solved using the Finite Element (FE) method and then Finite Difference (FD) schemes. We formulate a model order reduced prob- lem, obtained by the POD-method, in order to approximate the FE-solution and to decrease the computational effort. To overcome unnecessary back and forth projections between the full and reduced space in the non-linear part, we introduce the DEIM. For verifying theoretical results we formulate an SIR-problem on a domain with the shape of the state of Berlin. Based on that, we conduct detailed numerical investigations for the Semi-implicit Euler, implicit Euler and the Crank-Nicolson method, which belong to the class of FD-schemes. Investigations on the convergence and parameter perturbation analysis of the DEIM-solution are made for validation purpose. Last but not least, we conclude that the reduced method is advantageous in terms of computational effort, since the speedup of the computation time is improved significantly, while the error is in a satisfactory range. |

Dissertation note: | Master thesis, Universität Konstanz |

Subject (DDC): | 510 Mathematics |

Keywords: | PDE, SIR model, pandemic, simulation, FEM, Finite Element, Proper Orthogonal Decomposition, POD, DEIM, Finite Difference, Berlin, Semi implicit Euler, Crank-Nicolson, convergence, error estimate, optimal control, parameter sensitivity, gramian |

Link to License: | In Copyright |

Bibliography of Konstanz: | Yes |

Checksum:
MD5:38ef24effdd7b30cd364a42a3f4876f1

NGUYEN PHAM, Hai Dang, 2022. Proper Orthogonal Decomposition Analysis for a Non-Linear Spatiotemporal SIR-Model [Master thesis]. Konstanz: Universität Konstanz

@mastersthesis{NguyenPham2022Prope-58026, title={Proper Orthogonal Decomposition Analysis for a Non-Linear Spatiotemporal SIR-Model}, year={2022}, address={Konstanz}, school={Universität Konstanz}, author={Nguyen Pham, Hai Dang} }

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