Adaptive Multi-Level Monte Carlo and Stochastic Collocation Methods for Hyperbolic Partial Differential Equations with Random Data on Networks
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| Title: | Adaptive Multi-Level Monte Carlo and Stochastic Collocation Methods for Hyperbolic Partial Differential Equations with Random Data on Networks |
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| Authors: | Strauch, Elisa |
| Publisher Information: | 2023 |
| Document Type: | Electronic Resource |
| Abstract: | In this thesis, we develop reliable and fully error-controlled uncertainty quantification methods for hyperbolic partial differential equations (PDEs) with random data on networks. The goal is to combine adaptive strategies in the stochastic and physical space with a multi-level structure in such a way that a prescribed accuracy of the simulation is achieved while the computational effort is reduced. First, we consider hyperbolic PDEs on networks excluding any type of uncertainty. We introduce a model hierarchy with decreasing fidelity which can be obtained by simplifications of complex model equations. This hierarchy allows to apply more accurate models in regions of the network of complex dynamics and simplified models in regions of low dynamics. Next, we extend the network problem by uncertain initial data and uncertain conditions posed at the boundary and at inner network components. In order to predict the behavior of the considered system despite the uncertainties, we want to approximate relevant output quantities and their statistical properties, like the expected value and variance. For the study of the influence of the uncertainties, we focus on two sampling-based approaches: the widely used Monte Carlo (MC) method and the stochastic collocation (SC) method which is a promising alternative and therefore of main interest in this work. These approaches allow to reuse existing numerical solvers of the deterministic problem such that the implementation is simplified. We develop an adaptive single-level (SL) approach for both methods where we efficiently combine adaptive strategies in the stochastic space with adaptive physical approximations. The physical approximations are computed with a sample-dependent resolution in space, time and model hierarchy. The extension to a multi-level (ML) structure is realized by coupling physical approximations with different accuracies such that the computational cost is minimized. Due to a posteriori error indicators, we can |
| Index Terms: | Ph.D. Thesis, NonPeerReviewed, info:eu-repo/semantics/doctoralThesis |
| URL: | |
| Availability: | Open access content. Open access content CC BY-SA 4.0 International - Creative Commons, Attribution ShareAlike info:eu-repo/semantics/openAccess |
| Note: | text English |
| Other Numbers: | DETUD oai:tuprints.ulb.tu-darmstadt.de:23310 Strauch, Elisa <http://tuprints.ulb.tu-darmstadt.de/view/person/Strauch=3AElisa=3A=3A.html> (2023)Adaptive Multi-Level Monte Carlo and Stochastic Collocation Methods for Hyperbolic Partial Differential Equations with Random Data on Networks. Technische Universität Darmstadtdoi: 10.26083/tuprints-00023310 <https://doi.org/10.26083/tuprints-00023310> Ph.D. Thesis, Primary publication, Publisher's Version 1372646490 |
| Contributing Source: | TECHNISCHE UNIV DARMSTADT From OAIster®, provided by the OCLC Cooperative. |
| Accession Number: | edsoai.on1372646490 |
| Database: | OAIster |
Nájsť tento článok vo Web of Science | Abstract: | In this thesis, we develop reliable and fully error-controlled uncertainty quantification methods for hyperbolic partial differential equations (PDEs) with random data on networks. The goal is to combine adaptive strategies in the stochastic and physical space with a multi-level structure in such a way that a prescribed accuracy of the simulation is achieved while the computational effort is reduced. First, we consider hyperbolic PDEs on networks excluding any type of uncertainty. We introduce a model hierarchy with decreasing fidelity which can be obtained by simplifications of complex model equations. This hierarchy allows to apply more accurate models in regions of the network of complex dynamics and simplified models in regions of low dynamics. Next, we extend the network problem by uncertain initial data and uncertain conditions posed at the boundary and at inner network components. In order to predict the behavior of the considered system despite the uncertainties, we want to approximate relevant output quantities and their statistical properties, like the expected value and variance. For the study of the influence of the uncertainties, we focus on two sampling-based approaches: the widely used Monte Carlo (MC) method and the stochastic collocation (SC) method which is a promising alternative and therefore of main interest in this work. These approaches allow to reuse existing numerical solvers of the deterministic problem such that the implementation is simplified. We develop an adaptive single-level (SL) approach for both methods where we efficiently combine adaptive strategies in the stochastic space with adaptive physical approximations. The physical approximations are computed with a sample-dependent resolution in space, time and model hierarchy. The extension to a multi-level (ML) structure is realized by coupling physical approximations with different accuracies such that the computational cost is minimized. Due to a posteriori error indicators, we can |
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