Approximation and Uncertainty Quantification of Systems with Arbitrary Parameter Distributions Using Weighted Leja Interpolation
Abstract
:1. Introduction
2. Theory and Methodology
2.1. Stochastic Parametric Models
2.2. Univariate Interpolation Schemes
2.2.1. Univariate Lagrange Interpolation
2.2.2. Hierarchical Univariate Interpolation
2.2.3. Interpolatory Univariate Quadrature
2.3. Leja Interpolation Nodes
2.3.1. Unweighted Leja Nodes
2.3.2. Weighted Leja Nodes
2.4. Sparse Adaptive Leja Interpolation
2.4.1. Generalized Sparse Grid Interpolation
2.4.2. Adaptive Anisotropic Leja Interpolation
2.4.3. Post-Processing
3. Results
3.1. Error Metrics
3.2. Accuracy versus Costs
3.3. Borehole Model
- The parameter originally follows the normal distribution . The distribution is now truncated to the range ;
- The parameter r originally follows the log-normal distribution . Therefore, the parameter r has a mean value equal to and a variance equal to . The corresponding truncated normal distribution is defined by these mean and variance values, as well as by the truncation range l
- The remaining parameters are originally uniformly distributed, such that , , , , , and . Assuming a uniform distribution with support in , the corresponding truncated normal distribution is given as , i.e., the mean value and variance of the normal distribution correspond to those of the original uniform distribution, while the truncation limits coincide with the uniform distribution’s support boundaries.
3.4. Steel Column Limit State Function
3.5. Meromorphic Function
3.6. Dielectric Inset Waveguide
4. Discussion
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Loukrezis, D.; De Gersem, H. Approximation and Uncertainty Quantification of Systems with Arbitrary Parameter Distributions Using Weighted Leja Interpolation. Algorithms 2020, 13, 51. https://doi.org/10.3390/a13030051
Loukrezis D, De Gersem H. Approximation and Uncertainty Quantification of Systems with Arbitrary Parameter Distributions Using Weighted Leja Interpolation. Algorithms. 2020; 13(3):51. https://doi.org/10.3390/a13030051
Chicago/Turabian StyleLoukrezis, Dimitrios, and Herbert De Gersem. 2020. "Approximation and Uncertainty Quantification of Systems with Arbitrary Parameter Distributions Using Weighted Leja Interpolation" Algorithms 13, no. 3: 51. https://doi.org/10.3390/a13030051