p-Refined Multilevel Quasi-Monte Carlo for Galerkin Finite Element Methods with Applications in Civil Engineering
Civil engineering applications are often characterized by a large uncertainty on the material parameters. Discretization of the underlying equations is typically done by means of the Galerkin Finite Element method. The uncertain material parameter can be expressed as a random field represented by, f...
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MDPI AG
2020-04-01
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| Series: | Algorithms |
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| Online Access: | https://www.mdpi.com/1999-4893/13/5/110 |
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| author | Philippe Blondeel Pieterjan Robbe Cédric Van hoorickx Stijn François Geert Lombaert Stefan Vandewalle |
| author_facet | Philippe Blondeel Pieterjan Robbe Cédric Van hoorickx Stijn François Geert Lombaert Stefan Vandewalle |
| author_sort | Philippe Blondeel |
| collection | DOAJ |
| description | Civil engineering applications are often characterized by a large uncertainty on the material parameters. Discretization of the underlying equations is typically done by means of the Galerkin Finite Element method. The uncertain material parameter can be expressed as a random field represented by, for example, a Karhunen–Loève expansion. Computation of the stochastic responses, i.e., the expected value and variance of a chosen quantity of interest, remains very costly, even when state-of-the-art Multilevel Monte Carlo (MLMC) is used. A significant cost reduction can be achieved by using a recently developed multilevel method: p-refined Multilevel Quasi-Monte Carlo (p-MLQMC). This method is based on the idea of variance reduction by employing a hierarchical discretization of the problem based on a p-refinement scheme. It is combined with a rank-1 Quasi-Monte Carlo (QMC) lattice rule, which yields faster convergence compared to the use of random Monte Carlo points. In this work, we developed algorithms for the p-MLQMC method for two dimensional problems. The p-MLQMC method is first benchmarked on an academic beam problem. Finally, we use our algorithm for the assessment of the stability of slopes, a problem that arises in geotechnical engineering, and typically suffers from large parameter uncertainty. For both considered problems, we observe a very significant reduction in the amount of computational work with respect to MLMC. |
| first_indexed | 2024-03-10T20:10:56Z |
| format | Article |
| id | doaj.art-e18c78f4880647ab98f9b8dfeee7a63c |
| institution | Directory Open Access Journal |
| issn | 1999-4893 |
| language | English |
| last_indexed | 2024-03-10T20:10:56Z |
| publishDate | 2020-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Algorithms |
| spelling | doaj.art-e18c78f4880647ab98f9b8dfeee7a63c2023-11-19T22:56:45ZengMDPI AGAlgorithms1999-48932020-04-0113511010.3390/a13050110p-Refined Multilevel Quasi-Monte Carlo for Galerkin Finite Element Methods with Applications in Civil EngineeringPhilippe Blondeel0Pieterjan Robbe1Cédric Van hoorickx2Stijn François3Geert Lombaert4Stefan Vandewalle5Department of Computer Science, KU Leuven, Celestijnenlaan 200A, 3001 Leuven, BelgiumDepartment of Computer Science, KU Leuven, Celestijnenlaan 200A, 3001 Leuven, BelgiumDepartment of Civil Engineering, KU Leuven, Kasteelpark Arenberg 40, 3001 Leuven, BelgiumDepartment of Civil Engineering, KU Leuven, Kasteelpark Arenberg 40, 3001 Leuven, BelgiumDepartment of Civil Engineering, KU Leuven, Kasteelpark Arenberg 40, 3001 Leuven, BelgiumDepartment of Computer Science, KU Leuven, Celestijnenlaan 200A, 3001 Leuven, BelgiumCivil engineering applications are often characterized by a large uncertainty on the material parameters. Discretization of the underlying equations is typically done by means of the Galerkin Finite Element method. The uncertain material parameter can be expressed as a random field represented by, for example, a Karhunen–Loève expansion. Computation of the stochastic responses, i.e., the expected value and variance of a chosen quantity of interest, remains very costly, even when state-of-the-art Multilevel Monte Carlo (MLMC) is used. A significant cost reduction can be achieved by using a recently developed multilevel method: p-refined Multilevel Quasi-Monte Carlo (p-MLQMC). This method is based on the idea of variance reduction by employing a hierarchical discretization of the problem based on a p-refinement scheme. It is combined with a rank-1 Quasi-Monte Carlo (QMC) lattice rule, which yields faster convergence compared to the use of random Monte Carlo points. In this work, we developed algorithms for the p-MLQMC method for two dimensional problems. The p-MLQMC method is first benchmarked on an academic beam problem. Finally, we use our algorithm for the assessment of the stability of slopes, a problem that arises in geotechnical engineering, and typically suffers from large parameter uncertainty. For both considered problems, we observe a very significant reduction in the amount of computational work with respect to MLMC.https://www.mdpi.com/1999-4893/13/5/110Multilevel Monte CarloMultilevel Quasi-Monte Carloh- and p-refinementuncertainty quantificationstructural engineeringgeotechnical engineering |
| spellingShingle | Philippe Blondeel Pieterjan Robbe Cédric Van hoorickx Stijn François Geert Lombaert Stefan Vandewalle p-Refined Multilevel Quasi-Monte Carlo for Galerkin Finite Element Methods with Applications in Civil Engineering Algorithms Multilevel Monte Carlo Multilevel Quasi-Monte Carlo h- and p-refinement uncertainty quantification structural engineering geotechnical engineering |
| title | p-Refined Multilevel Quasi-Monte Carlo for Galerkin Finite Element Methods with Applications in Civil Engineering |
| title_full | p-Refined Multilevel Quasi-Monte Carlo for Galerkin Finite Element Methods with Applications in Civil Engineering |
| title_fullStr | p-Refined Multilevel Quasi-Monte Carlo for Galerkin Finite Element Methods with Applications in Civil Engineering |
| title_full_unstemmed | p-Refined Multilevel Quasi-Monte Carlo for Galerkin Finite Element Methods with Applications in Civil Engineering |
| title_short | p-Refined Multilevel Quasi-Monte Carlo for Galerkin Finite Element Methods with Applications in Civil Engineering |
| title_sort | p refined multilevel quasi monte carlo for galerkin finite element methods with applications in civil engineering |
| topic | Multilevel Monte Carlo Multilevel Quasi-Monte Carlo h- and p-refinement uncertainty quantification structural engineering geotechnical engineering |
| url | https://www.mdpi.com/1999-4893/13/5/110 |
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