Using automatic differentiation for compressive sensing in uncertainty quantification

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Design of new materials is quite a difficult task owing to various time and length scales and affiliated uncertainties. The major challenge is to include all these in a conventional model. Hyperparameter models in machine learning can be used to overcome these issues. In this paper, an artificial neural network (ANN) model is developed to estimate the effective elastic parameters of unidirectional fiber reinforced composites using representative volume elements (RVE) considering uncertainty in the fiber diameter. The diameter probability distribution is constructed from the acquired gray images by employing image processing operations. The generalized Polynomial Chaos (gPC) expansion is then used to represent the distribution as a random input parameter for finite element analysis, from where the effective parameters are realized. Similarly, the outputs of the FE model, i.e., elastic parameters, are approximated by gPC expansions having unknown deterministic coefficients and random orthogonal Hermite polynomials. A set of collocation points are generated from roots of the random polynomials; from there, the unknown coefficients are estimated. The realization samples are utilized to train an ANN algorithm based on supervised deep learning. The developed ANN model is later tested and validated for a new sample set of data. It is shown that the ANN model with few hidden layers and neurons has a high accuracy for estimation of the elastic parameters directly from the information on the distribution of fiber diameters.

Uncertainties in dynamic response of buildings with non-linear base-isolators

A Bayesian mixed shrinkage prior procedure for spatial–stochastic basis selection and evaluation of gPC expansions: Applications to elliptic SPDEs

We analyze the complexity of the sparse-grid interpolation and sparse-grid quadrature of countably-parametric functions which take values in separable Banach spaces with unconditional bases. Assuming a suitably quantified holomorphic dependence on the parameters, we establish dimension-independent convergence rate bounds for sparse-grid approximation schemes. Analogous results are shown in the case that the parametric families are obtained as approximate solutions of corresponding parametric-holomorphic, nonlinear operator equations as considered in [A. Cohen and A. Chkifa and Ch. Schwab: Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs, J. Math. Pures Appl. 103 (2015) 400–428], for example by means of stable, finite-dimensional approximations. We discuss in detail nonlinear Petrov–Galerkin projections. Error and convergence rate bounds for constructive and explicit multilevel, sparse tensor approximation schemes combining sparse-grid interpolation in the parameter space and general, multilevel discretization schemes in the physical domain are proved. The present results unify and generalize earlier works in terms of the admissible multilevel approximations in the physical domain (comprising general stable Petrov–Galerkin and discrete Petrov–Galerkin schemes, collocation and stable domain approximations) and in terms of the admissible operator equations (comprising smooth, nonlinear locally well-posed operator equations). Additionally, a novel computational strategy to localize sequences of nested index sets for the anisotropic Smolyak interpolation in parameter space is developed which realizes best [Formula: see text]-term benchmark convergence rates. We also consider Smolyak-type quadratures in this general setting, for which we establish improved convergence rates based on cancellations in the integrands’ gpc expansions by symmetries of quadratures and the probability measure [J.Z̃ech and Ch.S̃chwab: Convergence rates of high dimensional Smolyak quadrature, Report 2017-27, SAM ETH Zürich (2017)]. Several examples illustrating the abstract theory include domain uncertainty quantification (UQ) for general, linear, second-order, elliptic advection–reaction–diffusion equations on polygonal domains, where optimal convergence rates of FEM are known to require local mesh refinement near corners. Further applications of the presently developed theory comprise evaluations of posterior expectations in Bayesian inverse problems.

The electromagnetic properties of SRF cavities are mostly determined by their shape. Due to fabrication tolerances, tuning and limited resolution of measurement systems, the exact shape remains uncertain. In order to make assessments for the real life behaviour it is important to quantify how these geometrical uncertainties propagate through the mathematical system and influence certain electromagnetic properties, like the resonant frequencies of the structure’s eigenmodes. This can be done by using non-intrusive straightforwardmethodslikeMonte-Carlo(MC)simulations. However, such simulations require a large number of deterministic problem solutions to obtain a sufficient accuracy. In order to avoid this scaling behaviour, the so-called generalized polynomial chaos (gPC) expansion is used. This technique allows for the relatively fast computation of uncertainty propagation for few uncertain parameters in the case of computationally expensive deterministic models. In this paper we use the gPC expansion to quantify the propagation of uncertain geometry on the example of single cell cavities used for BESSY VSR as well as to compare the obtained results with the MC simulation.

RF-MEMS devices area complex systems governed by the interaction of a variety of forces, including electrostatics, solid deformation, fluid damping and contact. The performance and reliability of these devices is strongly dependent on device geometry and composition, and also on material microstructure and related properties. In this paper, we consider multiscale simulation of RF MEMS switched. At the device level, we introduce a comprehensive integrated numerical framework to simulate the major governing physics and their interactions. At the micron scale, we develop a mesoscale contact model to describe the history-dependent force-displacement relationships in terms of the surface roughness, the long-range attractive interaction between the two surfaces, and the repulsive interaction between contacting asperities (including elastic and plastic deformation). The inputs to this model are obtained from atomic level simulations and nanoscale surface topography characterization. The mesoscale contact model is integrated in the device-level simulation to predict the pull-in and pull-out behavior of these switches. The uncertainties associated with the simulation are quantified and propagated using a non-intrusive collocation method based on generalized Polynomial Chaos (gPC) expansions. With such a framework, we are able to predict the PDFs of pull-in and pull-out voltage, identify the critical factors that have the most influence on the quantities of interest, and therefore guide resource allocation and risk-informed decision-making.

Fast and robust parameter estimation with uncertainty quantification for the cardiac function

Sparse polynomial chaos expansions have recently emerged in uncertainty quantification analysis as a tool to solve high dimensional problems, e.g. stochastic problems involving a few dozens to a few hundred random variables. Based on penalized regression analysis and the so-called least angle regression algorithm the method has proven efficiency in a number of applications. The approach was so far rather limited to scalar output quantities, e.g. quantities of interest that are post-processed from the solution of a stochastic partial differential equation (SPDE). In this paper we extend this approach to vector output quantities in order to obtain the complete solution field. This is carried out by using principal component analysis before computing the PC expansions of the various components. As a whole a complete non intrusive framework is obtained that is only based on a set of deterministic solutions of the underlying deterministic problem. The approach is illustrated by the computation of the displacement field of a tension rod with lognormal, spatially variable Young’s modulus. The problem exhibits 62 stochastic dimensions.

Uncertainty quantification in LES of a turbulent bluff-body stabilized flame

Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies

In the framework of process safety studies, consequence assessment of accidental scenarios is a crucial step affecting the eventual risk profile associated with the facilities under analysis. Conventional models used for consequence assessment are based on integral models, and may not be adequate to cope with the dynamic evolution of accidental scenarios and their three‐dimensional features. On the other hand, consequence assessment models based on computational fluid dynamics (CFD) approaches are promising to cope with complex scenarios and environments, but setting the simulation introduces relevant uncertainties associated with both the input data, assumptions, and with the modelling of physical effects involved. In the present study, uncertainty quantification (UQ) techniques are applied to support advanced safety studies based on CFD simulations of hazardous gas dispersion. Firstly, the accidental scenarios are characterized by defining release scenarios and conditions and quantifying source terms using integral models. At the same time, input meteorological data are gathered. This enables the development of high‐fidelity CFD simulations of gas dispersion based on different input sets and eventually the implementation of UQ techniques. The generalized polynomial chaos (gPC) expansion is employed to obtain hazardous gas concentration based on the variation of wind direction and speed. The present method is applied for the analysis of a real plant featuring a complex layout. The results show the advantages of the present approach by quantifying the influence of meteorological conditions and providing indications for supporting the development of protection systems and emergency measures.

Factor Fixing (FF) is a common method for reducing the number of model parameters to lower computational cost. FF typically starts with distinguishing the insensitive parameters from the sensitive and pursues uncertainty quantification (UQ) on the resulting reduced‐order model, fixing each insensitive parameter at a fixed value. There is a need, however, to expand such a common approach to consider the effects of decision choices in the FF‐UQ procedure on metrics of interest. Therefore, to guide the use of FF and increase confidence in the resulting dimension‐reduced model, we propose a new adaptive framework consisting of four principles: (a) re‐parameterize the model first to reduce obvious non‐identifiable parameter combinations, (b) focus on decision relevance especially with respect to errors in quantities of interest (QoI), (c) conduct adaptive evaluation and robustness assessment of errors in the QoI across FF choices as sample size increases, and (d) reconsider whether fixing is warranted. The framework is demonstrated on a spatially‐distributed water quality model. The error in estimates of QoI caused by FF can be estimated using a Polynomial Chaos Expansion (PCE) surrogate model. Built with 70 model runs, the surrogate is computationally inexpensive to evaluate and can provide global sensitivity indices for free. For the selected catchment, just two factors may provide an acceptably accurate estimate of model uncertainty in the average annual load of Total Suspended Solids (TSS), suggesting that reducing the uncertainty in these two parameters is a priority for future work before undertaking further formal uncertainty quantification.

Stochastic physics-informed neural ordinary differential equations

Solving stochastic differential equations through genetic programming and automatic differentiation