08:30
Uncertainty quantification of stochastic dynamical systems through efficient reduced order models using new error bounds
Md. Nurtaj Hossain | Indian Institute of Science, Bangalore | India
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Authors:
Md. Nurtaj Hossain | Indian Institute of Science, Bangalore | India
Debraj Ghosh | Indian Institute of Science, Bangalore | India
Uncertainty quantification of a stochastic dynamical system requires solving computationally expensive higher dimensional model (HDM) at multiple parameter values, for instance, in a Monte Carlo simulation. Therefore, in this circumstance, replacement of the original HDM with a computationally cheaper reduced order model (ROM) can reduce the computational time significantly. The proper orthogonal decomposition (POD) based ROM which can be applied to both linear and nonlinear dynamical systems, has been successful in various applications. However, since the POD based ROM is trained at a few arbitrary parameter values and constructed using a lower dimensional subspace, it not only lacks robustness but also entails error in the solution. To address these issues, two a posteriori error bounds are derived here one for linear, and another for nonlinear dynamical systems. These error bounds are combined with a greedy search to propose an algorithm for uncertainty quantification. Here, a multi-frequency vibrational particle swarm optimization is used for the greedy search. To accelerate the convergence at each iteration of the greedy search, the ROM is updated at the global maxima and at a few local maxima. To test the performance of proposed error bounds and the algorithm, three different numerical examples --- (i) Burger’s equation, (ii) bladed disk assembly, and (iii) beam on nonlinear Winkler foundation are chosen. A significant speed-up is achieved in all three examples.
08:50
Parametric Low-fidelity Model Design for Bi-fidelity Approximation
Felix Newberry | University of Colorado Boulder | United States
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Felix Newberry | University of Colorado Boulder | United States
Alireza Doostan | University of Colorado Boulder | United States
A universal challenge in uncertainty quantification is the alleviation of computational expense necessary to achieve a desired solution with acceptable accuracy. In solving complex systems of PDEs, researchers frequently have access to models of differing fidelities. High-fidelity models yield physically accurate predictions at significant computational cost, while low-fidelity models are cheap to evaluate but produce a poor approximation of the problem physics. Bi-fidelity models seek to exploit these characteristics and achieve accuracy close to the high-fidelity model with expense close to the low-fidelity model. In this work we present a method of parametric low-fidelity model design to reduce the bi-fidelity error, the measured discrepancy between a high-fidelity and bi-fidelity solution. We employ an a posteriori bi-fidelity error bound that estimates the bi-fidelity error for a given pair of high and low-fidelity models. Through deviation of low-fidelity model parameter settings from their nominal values, we determine the parameter choice that minimizes the error bound and examine whether this minimum corresponds to a similar improvement in bi-fidelity error. The method is applied to test cases from both fluid and structural mechanics. We find that an informed choices of low-fidelity model parameters are capable of reducing the bi-fidelity model error by 5-10 times.
09:10
A reduced basis method for elliptic PDEs with random data based on adaptive snapshots
Christopher Müller | TU Darmstadt | Germany
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Christopher Müller | TU Darmstadt | Germany
Jens Lang | TU Darmstadt | Germany
We consider model order reduction by a greedy reduced basis method for elliptic boundary value problems with parametrized random and deterministic inputs. When the deterministic parameters are fixed, the solution of the remaining problem can be approximated by a stochastic Galerkin finite element (SGFE) method. The dimension of the associated block-structured system of equations can, however, become large due to the curse of dimensionality. In order to alleviate this issue and to keep the dimension of the discrete spaces small, we rely on an adaptive SGFE discretization. In the general case, there will thus be a different SGFE space for every computed snapshot. This interferes with the standard reduced basis approach which relies on the fact that all snapshots originate from the same discretization space. We discuss the adjustments that have to be made such that adaptive snapshots can be treated and derive a computable error estimator for the greedy construction of the reduced space. Eventually, we illustrate the results based on a convection-diffusion-reaction test case.
09:30
- CANCELED - A Sampling Method for Kriging Surrogate Model with QOI Filling Space Based On Max-Min Distance
Yanjin Wang | Institute of Applied Physics and Computational Mathematics, Beijing | China
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Yanjin Wang | Institute of Applied Physics and Computational Mathematics, Beijing | China
Surrogate models have been widely used in Complex engineering system designs to mitigate the computational burden. Constructing surrogate model general includes two parts: design of experiments and surrogate model construction. In this paper, a sampling method based on space-filling is proposed to refine the Kriging surrogate model. The sampling method considers filling QOI space with Max-Min distance. The accuracy, efficiency and robustness of the presented sampling method are verified by several test examples.
09:50
Predictive Accuracy of Dynamic Mode Decomposition
Hannah Lu | Stanford University | United States
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Hannah Lu | Stanford University | United States
Daniel Tartakovsky | Stanford University | United States
Dynamic mode decomposition (DMD), within the family of singular-value decompositions (SVD), is a popular tool of data-driven regression. While multiple numerical tests demonstrated the power and efficiency of DMD in representing data (i.e., in the interpolation mode), applications of DMD as a predictive tool (i.e., in the extrapolation mode) are scarce. This is due, in part, to the lack of rigorous error estimators for DMD-based predictions. We provide a theoretical error estimator for DMD extrapolation of numerical solutions to linear and nonlinear parabolic equations. This error analysis allows one to monitor and control the errors associated with DMD-based temporal extrapolation of numerical solutions to parabolic differential equations. We use several computational experiments to verify the robustness of our error estimators and to compare the predictive ability of DMD with that of proper orthogonal decomposition (POD), another member of the SVD family. Our analysis demonstrates the importance of a proper selection of observables, as predicted by the Koopman operator theory. In all the tests considered, DMD outperformed POD in terms of efficiency due to its iteration-free feature. In some of these experiments, POD proved to be more accurate than DMD. This suggests that DMD is preferable for obtaining a fast prediction with slightly lower accuracy, while POD should be used if the accuracy is paramount.
10:10
Surrogate Model Construction with Regression Element Trees
Daniel Meyer | ETH Zurich | Switzerland
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Daniel Meyer | ETH Zurich | Switzerland
Yu Wang | Boston University | United States
This work is concerned with a simple tree-based regression method for quantitative responses and uncensored predictors that offers test error levels comparable to ensemble-based methods such as boosting and random forest, but at the same time provides a single updatable tree. Our method is based on recursive binary splitting, which is driven by statistical tests that inspect the homoscedasticity and normality of the residuals. Our method applies parametric regression models in the tree nodes, which are determined based on least-squares and which lead to short trees. We assess the performance of the new regressor in different case studies and compare against existing methods.