Journal ArticleDOI
Data-driven reduced order modeling for parametric PDE eigenvalue problems using Gaussian process regression
TLDR
In this paper , a data-driven reduced basis (RB) method for parametric eigenvalue problems is proposed, which is based on the offline and online paradigms.Abstract:
In this article, we propose a data-driven reduced basis (RB) method for the approximation of parametric eigenvalue problems. The method is based on the offline and online paradigms. In the offline stage, we generate snapshots and construct the basis of the reduced space, using a POD approach. Gaussian process regressions (GPR) are used for approximating the eigenvalues and projection coefficients of the eigenvectors in the reduced space. All the GPR corresponding to the eigenvalues and projection coefficients are trained in the offline stage, using the data generated in the offline stage. The output corresponding to new parameters can be obtained in the online stage using the trained GPR. The proposed algorithm is used to solve affine and non-affine parameter-dependent eigenvalue problems. The numerical results demonstrate the robustness of the proposed non-intrusive method.read more
Citations
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Journal ArticleDOI
A reduced order model for the finite element approximation of eigenvalue problems
TL;DR: In this paper , a reduced order method for the approximation of the eigensolutions of the Laplace problem with Dirichlet boundary condition was proposed, where a time continuation technique that consists in the introduction of a fictitious time parameter was used.
Journal ArticleDOI
On the effect of different samplings to the solution of parametric PDE Eigenvalue Problems
TL;DR: In this article , the authors apply reduced order techniques for the approximation of parametric eigenvalue problems and investigate the effect of the choice of sampling points on the performance of sampling.
Journal ArticleDOI
A data-driven method for parametric PDE Eigenvalue Problems using Gaussian Process with different covariance functions
TL;DR: In this article , a data-driven method for approximating both eigenvalues and eigenvectors in parametric eigenvalue problems is proposed, where the basis of the reduced space is generated by applying the proper orthogonal decomposition (POD) approach on a collection of pre-computed full-order snapshots at a chosen set of parameters.
References
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Book ChapterDOI
Gaussian processes in machine learning
TL;DR: In this paper, the authors give a basic introduction to Gaussian Process regression models and present the simple equations for incorporating training data and examine how to learn the hyperparameters using the marginal likelihood.
Book
Mixed Finite Element Methods and Applications
TL;DR: In this paper, the authors discuss the algebraic aspects of saddle point problems in Hilbert spaces and approximate saddle point approximations in Finite Element Methods (FEM) in function spaces.
Journal ArticleDOI
Balanced Model Reduction via the Proper Orthogonal Decomposition
Karen Willcox,Jaime Peraire +1 more
TL;DR: A new method for performing a balanced reduction of a high-order linear system is presented, which combines the proper orthogonal decomposition and concepts from balanced realization theory and extends to nonlinear systems.
Book
Certified Reduced Basis Methods for Parametrized Partial Differential Equations
TL;DR: In this article, the authors provide a thorough introduction to the mathematical and algorithmic aspects of certified reduced basis methods for parametrized partial differential equations, including model construction, error estimation and computational efficiency.
Journal ArticleDOI
Galerkin Proper Orthogonal Decomposition Methods for a General Equation in Fluid Dynamics
Karl Kunisch,Stefan Volkwein +1 more
TL;DR: Error estimates for Galerkin proper orthogonal decomposition (POD) methods for nonlinear parabolic systems arising in fluid dynamics are proved and the backward Euler scheme is considered.