p-Refined Multilevel Quasi-Monte Carlo for Galerkin Finite Element Methods with Applications in Civil Engineering
Abstract
:1. Introduction
2. p-Refined Multilevel Quasi-Monte Carlo
2.1. The Multilevel and Multilevel Quasi Monte Carlo Methods
2.2. Mesh Hierarchies
- 1.
- a closed subset with nonempty interior and piecewise smooth boundary, i.e., the element domain,
- 2.
- a finite-dimensional space of functions on T, i.e., the space of shape functions, and
- 3.
- a basis for, i.e., the set of nodal variables.
2.3. Algorithm
Algorithm 1: p-MLQMC. |
Algorithm 2: Point Set Generation |
2.4. Cost Complexity Theorem
- 1.
- ,
- 2.
- and
- 3.
- .
2.5. Modeling the Spatial Variability
2.6. Incorporating the Uncertainty in the Model
3. Model Problems and Numerical Results
3.1. Model Problem 1
3.1.1. Description
3.1.2. Finite Element Method
3.1.3. Mesh Discretization
3.1.4. Modeling the Spatial Variability
3.1.5. Quantity of Interest
3.2. Numerical Results for Model Problem 1
3.2.1. Rates
3.2.2. Number of Samples
3.2.3. Uncertainty Propagation in the Solution
3.2.4. Runtime
3.2.5. Level Adaptivity
3.3. Model Problem 2
3.3.1. Description
3.3.2. Finite Element Method
3.3.3. Mesh Discretization
3.3.4. Modeling the Spatial Variability
3.3.5. Quantity of Interest
3.4. Numerical Results for Model Problem 2
3.4.1. Rates
3.4.2. Number of Samples
3.4.3. Uncertainty Propagation in the Solution
3.4.4. Runtime
3.4.5. Level Adaptivity
4. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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h-ML(Q)MC | p-ML(Q)MC | |||||||
---|---|---|---|---|---|---|---|---|
Level | Nel | DOF | Order | Nquad | Nel | DOF | Order | Nquad |
0 | 80 | 210 | 1 | 9 | 80 | 210 | 1 | 9 |
1 | 320 | 738 | 1 | 9 | 80 | 738 | 2 | 25 |
2 | 1280 | 2754 | 1 | 9 | 80 | 1586 | 3 | 49 |
3 | 5120 | 10,626 | 1 | 9 | 80 | 2754 | 4 | 81 |
4 | 20,480 | 41,730 | 1 | 9 | / | / | / | / |
h-ML(Q)MC | p-ML(Q)MC | |||||||
---|---|---|---|---|---|---|---|---|
Level | Nel | DOF | Order | Nquad | Nel | DOF | Order | Nquad |
0 | 33 | 48 | 1 | 7 | 33 | 48 | 1 | 7 |
1 | 132 | 160 | 1 | 7 | 33 | 338 | 3 | 16 |
2 | 528 | 582 | 1 | 7 | 33 | 892 | 5 | 28 |
3 | 2112 | 2218 | 1 | 7 | 33 | 1720 | 7 | 37 |
4 | 8448 | 8658 | 1 | 7 | / | / | / | / |
p-ML(Q)MC | ||||
---|---|---|---|---|
Level | Nel | DOF | Order | Nquad |
0 | 80 | 210 | 1 | 25 |
1 | 80 | 738 | 2 | 81 |
2 | 80 | 1586 | 3 | 121 |
3 | 80 | 2754 | 4 | 225 |
4 | 80 | 4242 | 5 | 289 |
5 | 80 | 6050 | 6 | 441 |
6 | 80 | 8178 | 7 | 625 |
7 | 80 | 10,626 | 8 | 729 |
p-ML(Q)MC | ||||
---|---|---|---|---|
Level | Nel | DOF | Order | Nquad |
0 | 33 | 48 | 1 | 7 |
1 | 33 | 160 | 2 | 13 |
2 | 33 | 338 | 3 | 19 |
3 | 33 | 582 | 4 | 25 |
4 | 33 | 892 | 5 | 28 |
5 | 33 | 1268 | 6 | 33 |
6 | 33 | 1720 | 7 | 37 |
7 | 33 | 2218 | 8 | 61 |
8 | 33 | 2792 | 9 | 73 |
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Blondeel, P.; Robbe, P.; Van hoorickx, C.; François, S.; Lombaert, G.; Vandewalle, S. p-Refined Multilevel Quasi-Monte Carlo for Galerkin Finite Element Methods with Applications in Civil Engineering. Algorithms 2020, 13, 110. https://doi.org/10.3390/a13050110
Blondeel P, Robbe P, Van hoorickx C, François S, Lombaert G, Vandewalle S. p-Refined Multilevel Quasi-Monte Carlo for Galerkin Finite Element Methods with Applications in Civil Engineering. Algorithms. 2020; 13(5):110. https://doi.org/10.3390/a13050110
Chicago/Turabian StyleBlondeel, Philippe, Pieterjan Robbe, Cédric Van hoorickx, Stijn François, Geert Lombaert, and Stefan Vandewalle. 2020. "p-Refined Multilevel Quasi-Monte Carlo for Galerkin Finite Element Methods with Applications in Civil Engineering" Algorithms 13, no. 5: 110. https://doi.org/10.3390/a13050110
APA StyleBlondeel, P., Robbe, P., Van hoorickx, C., François, S., Lombaert, G., & Vandewalle, S. (2020). p-Refined Multilevel Quasi-Monte Carlo for Galerkin Finite Element Methods with Applications in Civil Engineering. Algorithms, 13(5), 110. https://doi.org/10.3390/a13050110