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Abstract
The systematic quantification of the uncertainties affecting dynamical systems and the characterization of the uncertainty of their outcomes is critical for engineering design and analysis, where risks must be reduced as much as possible. Uncertainties stem naturally from our limitations in measurements, predictions and manufacturing, and we can say that any dynamical system used in engineering is subject to some of these uncertainties.
The first part of this work presents an overview of the mathematical framework used in Uncertainty Quantification (UQ) analysis and introduces the spectral tensortrain (STT) decomposition, a novel highorder method for the effective propagation of uncertainties which aims at providing an exponential convergence rate while tackling the curse of dimensionality. The curse of dimensionality is a problem that afflicts many methods based on metamodels, for which the computational cost increases exponentially with the number of inputs of the approximated function – which we will call dimension in the following.
The STTdecomposition is based on the Polynomial Chaos (PC) approximation and the lowrank decomposition of the function describing the Quantity of Interest of the considered problem. The lowrank decomposition is obtained through the discrete tensortrain decomposition, which is constructed using an optimization algorithm for the selection of the relevant points on which the function needs to be evaluated. The selection of these points is informed by the approximated function and thus it is able to adapt to its features. The number of function evaluations needed for the construction grows only linearly with the dimension and quadratically with the rank.
In this work we will present and use the functional counterpart of this lowrank decomposition and, after proving some auxiliary properties, we will apply PC on it, obtaining the STTdecomposition. This will allow the decoupling of each dimension, leading to a much cheaper construction of the PC surrogate. In the associated paper, the capabilities of the STTdecomposition are checked on commonly used test functions and on an elliptic problem with random inputs.
This work will also present three active research directions aimed at improving the efficiency of the STTdecomposition. In this context, we propose three new strategies for solving the ordering problem suffered by the tensortrain decomposition, for computing better estimates with respect to the norms usually employed in UQ and for the anisotropic adaptivity of the method.
The second part of this work presents engineering applications of the UQ framework. Both the applications are characterized by functions whose evaluation is computationally expensive and thus the UQ analysis of the associated systems will benefit greatly from the application of methods which require few function evaluations.
We first consider the propagation of the uncertainty and the sensitivity analysis of the nonlinear dynamics of railway vehicles with suspension components whose characteristics are uncertain. These analysis are carried out using mostly PC methods, and resorting to random sampling methods for comparison and when strictly necessary.
The second application of the UQ framework is on the propagation of the uncertainties entering a fully nonlinear and dispersive model of water waves. This computationally challenging task is tackled with the adoption of stateoftheart software for its numerical solution and of efficient PC methods. The aim of this study is the construction of stochastic benchmarks where to test UQ methodologies before being applied to fullscale problems, where efficient methods are necessary with today’s computational resources.
The outcome of this work was also the creation of several freely available Python modules for Uncertainty Quantification, which are listed and described in the appendix.
The first part of this work presents an overview of the mathematical framework used in Uncertainty Quantification (UQ) analysis and introduces the spectral tensortrain (STT) decomposition, a novel highorder method for the effective propagation of uncertainties which aims at providing an exponential convergence rate while tackling the curse of dimensionality. The curse of dimensionality is a problem that afflicts many methods based on metamodels, for which the computational cost increases exponentially with the number of inputs of the approximated function – which we will call dimension in the following.
The STTdecomposition is based on the Polynomial Chaos (PC) approximation and the lowrank decomposition of the function describing the Quantity of Interest of the considered problem. The lowrank decomposition is obtained through the discrete tensortrain decomposition, which is constructed using an optimization algorithm for the selection of the relevant points on which the function needs to be evaluated. The selection of these points is informed by the approximated function and thus it is able to adapt to its features. The number of function evaluations needed for the construction grows only linearly with the dimension and quadratically with the rank.
In this work we will present and use the functional counterpart of this lowrank decomposition and, after proving some auxiliary properties, we will apply PC on it, obtaining the STTdecomposition. This will allow the decoupling of each dimension, leading to a much cheaper construction of the PC surrogate. In the associated paper, the capabilities of the STTdecomposition are checked on commonly used test functions and on an elliptic problem with random inputs.
This work will also present three active research directions aimed at improving the efficiency of the STTdecomposition. In this context, we propose three new strategies for solving the ordering problem suffered by the tensortrain decomposition, for computing better estimates with respect to the norms usually employed in UQ and for the anisotropic adaptivity of the method.
The second part of this work presents engineering applications of the UQ framework. Both the applications are characterized by functions whose evaluation is computationally expensive and thus the UQ analysis of the associated systems will benefit greatly from the application of methods which require few function evaluations.
We first consider the propagation of the uncertainty and the sensitivity analysis of the nonlinear dynamics of railway vehicles with suspension components whose characteristics are uncertain. These analysis are carried out using mostly PC methods, and resorting to random sampling methods for comparison and when strictly necessary.
The second application of the UQ framework is on the propagation of the uncertainties entering a fully nonlinear and dispersive model of water waves. This computationally challenging task is tackled with the adoption of stateoftheart software for its numerical solution and of efficient PC methods. The aim of this study is the construction of stochastic benchmarks where to test UQ methodologies before being applied to fullscale problems, where efficient methods are necessary with today’s computational resources.
The outcome of this work was also the creation of several freely available Python modules for Uncertainty Quantification, which are listed and described in the appendix.
Original language  English 

Place of Publication  Kgs. Lyngby 

Publisher  Technical University of Denmark 
Number of pages  321 
Publication status  Published  2015 
Series  DTU Compute PHD2014 

Number  359 
ISSN  09093192 
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Projects
 1 Finished

Uncertainty Quantification for advanced engineering applications
Bigoni, D., EngsigKarup, A. P., Hesthaven, J., True, H., Sørensen, M. P., Le Maitre, O. P. & Funfschilling, C.
Technical University of Denmark
15/12/2011 → 19/03/2015
Project: PhD