A data-driven method for parametric PDE Eigenvalue Problems using Gaussian Process with different covariance functions
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866913372218327040 |
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| author | Alghamdi, Moataz Bertrand, Fleurianne Boffi, Daniele Halim, Abdul |
| author_facet | Alghamdi, Moataz Bertrand, Fleurianne Boffi, Daniele Halim, Abdul |
| contents | We use a Gaussian Process Regression (GPR) strategy that was recently developed [3,16,17] to analyze different types of curves that are commonly encountered in parametric eigenvalue problems. We employ an offline-online decomposition method. In the offline phase, we generate the basis of the reduced space by applying the proper orthogonal decomposition (POD) method on a collection of pre-computed, full-order snapshots at a chosen set of parameters. Then, we generate our GPR model using four different Matérn covariance functions. In the online phase, we use this model to predict both eigenvalues and eigenvectors at new parameters. We then illustrate how the choice of each covariance function influences the performance of GPR. Furthermore, we discuss the connection between Gaussian Process Regression and spline methods and compare the performance of the GPR method against linear and cubic spline methods. We show that GPR outperforms other methods for functions with a certain regularity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_18064 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A data-driven method for parametric PDE Eigenvalue Problems using Gaussian Process with different covariance functions Alghamdi, Moataz Bertrand, Fleurianne Boffi, Daniele Halim, Abdul Numerical Analysis 65N25, 65N30, 35B30, 78M34, 35P15 We use a Gaussian Process Regression (GPR) strategy that was recently developed [3,16,17] to analyze different types of curves that are commonly encountered in parametric eigenvalue problems. We employ an offline-online decomposition method. In the offline phase, we generate the basis of the reduced space by applying the proper orthogonal decomposition (POD) method on a collection of pre-computed, full-order snapshots at a chosen set of parameters. Then, we generate our GPR model using four different Matérn covariance functions. In the online phase, we use this model to predict both eigenvalues and eigenvectors at new parameters. We then illustrate how the choice of each covariance function influences the performance of GPR. Furthermore, we discuss the connection between Gaussian Process Regression and spline methods and compare the performance of the GPR method against linear and cubic spline methods. We show that GPR outperforms other methods for functions with a certain regularity. |
| title | A data-driven method for parametric PDE Eigenvalue Problems using Gaussian Process with different covariance functions |
| topic | Numerical Analysis 65N25, 65N30, 35B30, 78M34, 35P15 |
| url | https://arxiv.org/abs/2303.18064 |