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This book explores four guiding themes – reduced order modelling, high dimensional problems, efficient algorithms, and applications – by reviewing recent algorithmic and mathematical advances and the development of new research directions for uncertainty quantification in the context of partial differential equations with random inputs. Highlighting the most promising approaches for (near-) future improvements in the way uncertainty quantification problems in the partial differential equation setting are solved, and gathering contributions by leading international experts, the book’s content will impact the scientific, engineering, financial, economic, environmental, social, and commercial sectors.



Effect of Load Path on Parameter Identification for Plasticity Models Using Bayesian Methods

To evaluate the cyclic behavior under different loading conditions using the kinematic and isotropic hardening theory of steel, a Chaboche viscoplastic material model is employed. The parameters of a constitutive model are usually identified by minimization of the distance between model response and experimental data. However, measurement errors and differences in the specimens lead to deviations in the determined parameters. In this article the Chaboche model is used and a stochastic simulation technique is applied to generate artificial data which exhibit the same stochastic behavior as experimental data. Then the model parameters are identified by applying an estimation using Bayes’s theorem. The Gauss–Markov–Kalman filter using functional approximation is introduced and employed to estimate the model parameters in the Bayesian setting. Identified parameters are compared with the true parameters in the simulation, and the efficiency of the identification method is discussed. At the end, the effect of the load path on the parameter identification is investigated.
Ehsan Adeli, Bojana Rosić, Hermann G. Matthies, Sven Reinstädler

A Compressive Spectral Collocation Method for the Diffusion Equation Under the Restricted Isometry Property

We propose a compressive spectral collocation method for the numerical approximation of Partial Differential Equations (PDEs). The approach is based on a spectral Sturm-Liouville approximation of the solution and on the collocation of the PDE in strong form at randomized points, by taking advantage of the compressive sensing principle. The proposed approach makes use of a number of collocation points substantially less than the number of basis functions when the solution to recover is sparse or compressible. Focusing on the case of the diffusion equation, we prove that, under suitable assumptions on the diffusion coefficient, the matrix associated with the compressive spectral collocation approach satisfies the restricted isometry property of compressive sensing with high probability. Moreover, we demonstrate the ability of the proposed method to reduce the computational cost associated with the corresponding full spectral collocation approach while preserving good accuracy through numerical illustrations.
Simone Brugiapaglia

Surrogate-Based Ensemble Grouping Strategies for Embedded Sampling-Based Uncertainty Quantification

The embedded ensemble propagation approach introduced in Phipps et al. (SIAM J. Sci. Comput. 39(2):C162, 2017) has been demonstrated to be a powerful means of reducing the computational cost of sampling-based uncertainty quantification methods, particularly on emerging computational architectures. A substantial challenge with this method however is ensemble-divergence, whereby different samples within an ensemble choose different code paths. This can reduce the effectiveness of the method and increase computational cost. Therefore grouping samples together to minimize this divergence is paramount in making the method effective for challenging computational simulations. In this work, a new grouping approach based on a surrogate for computational cost built up during the uncertainty propagation is developed and applied to model advection-diffusion problems where computational cost is driven by the number of (preconditioned) linear solver iterations. The approach is developed within the context of locally adaptive stochastic collocation methods, where a surrogate for the number of linear solver iterations, generated from previous levels of the adaptive grid generation, is used to predict iterations for subsequent samples, and group them based on similar numbers of iterations. The effectiveness of the method is demonstrated by applying it to highly anisotropic advection-dominated diffusion problems with a wide variation in solver iterations from sample to sample. It extends the parameter-based grouping approach developed in D’Elia et al. (SIAM/ASA J. Uncertain. Quantif. 6:87, 2017) to more general problems without requiring detailed knowledge of how the uncertain parameters affect the simulation’s cost, and is also less intrusive to the simulation code.
M. D’Elia, E. Phipps, A. Rushdi, M. S. Ebeida

Conservative Model Order Reduction for Fluid Flow

In the past decade, model order reduction (MOR) has been successful in reducing the computational complexity of elliptic and parabolic systems of partial differential equations (PDEs). However, MOR of hyperbolic equations remains a challenge. Symmetries and conservation laws, which are a distinctive feature of such systems, are often destroyed by conventional MOR techniques which result in a perturbed, and often unstable reduced system. The importance of conservation of energy is well-known for a correct numerical integration of fluid flow. In this paper, we discuss model reduction, that exploits skew-symmetry of conservative and centered discretization schemes, to recover conservation of energy at the level of the reduced system. Moreover, we argue that the reduced system, constructed with the new method, can be identified by a reduced energy that mimics the energy of the high-fidelity system. Therefore, the loss in energy, associated with the model reduction, remains constant in time. This results in an, overall, correct evolution of the fluid that ensures robustness of the reduced system. We evaluate the performance of the proposed method through numerical simulation of various fluid flows, and through a numerical simulation of a continuous variable resonance combustor model.
Babak Maboudi Afkham, Nicolò Ripamonti, Qian Wang, Jan S. Hesthaven

Piecewise Polynomial Approximation of Probability Density Functions with Application to Uncertainty Quantification for Stochastic PDEs

The probability density function (PDF) associated with a given set of samples is approximated by a piecewise-linear polynomial constructed with respect to a binning of the sample space. The kernel functions are a compactly supported basis for the space of such polynomials, i.e. finite element hat functions, that are centered at the bin nodes rather than at the samples, as is the case for the standard kernel density estimation approach. This feature naturally provides an approximation that is scalable with respect to the sample size. On the other hand, unlike other strategies that use a finite element approach, the proposed approximation does not require the solution of a linear system. In addition, a simple rule that relates the bin size to the sample size eliminates the need for bandwidth selection procedures. The proposed density estimator has unitary integral, does not require a constraint to enforce positivity, and is consistent. The proposed approach is validated through numerical examples in which samples are drawn from known PDFs. The approach is also used to determine approximations of (unknown) PDFs associated with outputs of interest that depend on the solution of a stochastic partial differential equation.
Giacomo Capodaglio, Max Gunzburger

Analysis of Probabilistic and Parametric Reduced Order Models

Stochastic models share many characteristics with generic parametric models. In some ways they can be regarded as a special case. But for stochastic models there is a notion of weak distribution or generalised random variable, and the same arguments can be used to analyse parametric models. Such models in vector spaces are connected to a linear map, and in infinite dimensional spaces are a true generalisation. Reproducing kernel Hilbert space and affine- / linear- representations in terms of tensor products are directly related to this linear operator. This linear map leads to a generalised correlation operator, and representations are connected with factorisations of the correlation operator. The fitting counterpart in the stochastic domain to make this point of view as simple as possible are algebras of random variables with a distinguished linear functional, the state, which is interpreted as expectation. The connections of factorisations of the generalised correlation to the spectral decomposition, as well as the associated Karhunen-Loève- or proper orthogonal decomposition will be sketched. The purpose of this short note is to show the common theoretical background and pull some lose ends together.
Hermann G. Matthies

Reduced Order Isogeometric Analysis Approach for PDEs in Parametrized Domains

In this contribution, we coupled the isogeometric analysis to a reduced order modelling technique in order to provide a computationally efficient solution in parametric domains. In details, we adopt the free-form deformation method to obtain the parametric formulation of the domain and proper orthogonal decomposition with interpolation for the computational reduction of the model. This technique provides a real-time solution for any parameter by combining several solutions, in this case computed using isogeometric analysis on different geometrical configurations of the domain, properly mapped into a reference configuration. We underline that this reduced order model requires only the full-order solutions, making this approach non-intrusive. We present in this work the results of the application of this methodology to a heat conduction problem inside a deformable collector pipe.
Fabrizio Garotta, Nicola Demo, Marco Tezzele, Massimo Carraturo, Alessandro Reali, Gianluigi Rozza

Uncertainty Quantification Applied to Hemodynamic Simulations of Thoracic Aorta Aneurysms: Sensitivity to Inlet Conditions

In this work, the numerical simulation of the blood flow inside a patient specific aorta in presence of an aneurysm is considered. A systematic sensitivity analysis of numerical predictions to the shape of the inlet flow rate waveform is carried out. In particular, two parameters are selected to describe the inlet waveform: the stroke volume and the period of the cardiac cycle. In order to limit the number of hemodynamic simulations required, we used a stochastic method based on the generalized polynomial chaos (gPC) approach, in which the selected parameters are considered as random variables with a given probability distribution. The uncertainty is propagated through the numerical model and a continuous response surface of the output quantities of interest in the parameter space can be recovered through a “surrogate” model. For both selected uncertain parameters, we first assumed uniform Probability Density Functions (PDFs) on a given variation range, and then we used clinical data to construct more accurate beta PDFs. In all cases, the two input parameters appeared to have a significant influence on wall shear stresses, confirming the need of using patient-specific inlet conditions.
Alessandro Boccadifuoco, Alessandro Mariotti, Katia Capellini, Simona Celi, Maria Vittoria Salvetti

Cavitation Model Parameter Calibration for Simulations of Three-Phase Injector Flows

A stochastic sensitivity analysis and calibration of the cavitation model parameters in the URANS simulations of a configuration representative of high-pressure injectors for automotive applications is carried out. A popular homogeneous-flow cavitation model is considered, in which the mass transfer due to cavitation is given by the Schnerr–Sauer model together with the classical Rayleigh–Plesset equation. A stochastic approach based on the generalized Polynomial Chaos (gPC) expansion is adopted, which allows continuous response surfaces of the quantities of interest in the parameter space to be obtained starting from a few deterministic simulations. The considered uncertain parameters are the so-called scaling factors. The calibration of these parameters is carried out by using the gPC response surfaces for a axisymmetric simplified geometry against the experimental value of the critical cavitation point, i.e. the condition at which the injector is choked. The procedure is carried out for two different turbulence models, viz. the k − ω SST and RSM models. The so-obtained optimal parameter set-ups are then validated for the real three-dimensional geometry. The k − ω SST optimal set-up gives very accurate predictions also in the three-dimensional case. Finally, the results obtained with this optimal set-up are compared to those given by standard values, confirming that the predictions of the different flow regimes occurring in high-pressure injectors are highly sensitive to cavitation model parameters.
Alessandro Anderlini, Maria Vittoria Salvetti, Antonio Agresta, Luca Matteucci

Non-intrusive Polynomial Chaos Method Applied to Full-Order and Reduced Problems in Computational Fluid Dynamics: A Comparison and Perspectives

In this work, Uncertainty Quantification (UQ) based on non-intrusive Polynomial Chaos Expansion (PCE) is applied to the CFD problem of the flow past an airfoil with parameterized angle of attack and inflow velocity. To limit the computational cost associated with each of the simulations required by the non-intrusive UQ algorithm used, we resort to a Reduced Order Model (ROM) based on Proper Orthogonal Decomposition (POD)-Galerkin approach. A first set of results is presented to characterize the accuracy of the POD-Galerkin ROM developed approach with respect to the Full Order Model (FOM) solver (OpenFOAM). A further analysis is then presented to assess how the UQ results are affected by substituting the FOM predictions with the surrogate ROM ones.
Saddam Hijazi, Giovanni Stabile, Andrea Mola, Gianluigi Rozza

A Practical Example for the Non-linear Bayesian Filtering of Model Parameters

In this tutorial we consider the non-linear Bayesian filtering of static parameters in a time-dependent model. We outline the theoretical background and discuss appropriate solvers. We focus on particle-based filters and present Sequential Importance Sampling (SIS) and Sequential Monte Carlo (SMC). Throughout the paper we illustrate the concepts and techniques with a practical example using real-world data. The task is to estimate the gravitational acceleration of the Earth g by using observations collected from a simple pendulum. Importantly, the particle filters enable the adaptive updating of the estimate for g as new observations become available. For tutorial purposes we provide the data set and a Python implementation of the particle filters.
Matthieu Bulté, Jonas Latz, Elisabeth Ullmann


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