Nora Lüthen


Last Name

Lüthen

First Name

Nora

ORCID

0000-0002-3765-4222

Organisational unit

03962 - Sudret, Bruno / Sudret, Bruno

Filters

2018 - 201952020 - 202526
Reset filters

Search Results

Publications 1 - 10 of 31
  • An improved a priori error analysis of Nitsche’s method for Robin boundary conditions
    Item type: Journal Article
    Lüthen, Nora; Juntunen, Mika; Stenberg, Rolf (2018)
    Numerische Mathematik
  • Surrogate modelling for UQ in Engineering
    Item type: Presentation
    Lüthen, Nora (2024)
    Uncertainty analyses typically require numerous function evaluations, which can become prohibitively costly for complex engineering models. To reduce computational effort, surrogate models can be used, which approximate the original model and are cheap to evaluate. We consider black-box methods, i.e., methods which do not need knowledge of the inner workings of the original model. In our lectures, we introduce the two popular surrogate modeling techniques polynomial chaos expansions and Kriging, and discuss their use in sensitivity and reliability analysis. We also explore extensions, such as the case of non-scalar output. The theory is illustrated with case studies and accompanied by practical exercises.
  • Simulationsbasierte Risikoanalyse
    Item type: Presentation
    Lüthen, Nora (2024)
  • Simulationsbasierte Risikoanalyse
    Item type: Presentation
    Lüthen, Nora (2021)
  • Adaptive designs for multi-output polynomial chaos expansions and sensitivity analysis
    Item type: Other Conference Item
    Lüthen, Nora; Marelli, Stefano; Sudret, Bruno (2024)
    SIAM Conference on Uncertainty Quantification (UQ 2024). Searchable Abstract Document
    Adaptive design of experiments has been demonstrated very effective in reducing the computational costs of complex uncertainty quantification tasks, such as reliability and sensitivity analysis. While traditionally associated with local- and kernel- based surrogate models like Gaussian process modelling and support vector machines, recent research has demonstrated that adaptive design of experiments can also benefit regression- based surrogate models, such as polynomial chaos expansions (PCEs). Nevertheless, one core limitation of most adaptive design strategies is that, with few exceptions, they are designed for scalar-output models only. When dealing with multiple output models, optimality conditions become much more difficult to define, and the literature on the subject is still sparse. In this contribution, we extend a recently proposed semi-supervised sequential design approach for sparse PCE, to the case of vector-output computational models. Thanks to the well-known synergy between PCE and Sobol' indices- based sensitivity analysis, this design of experiments strategy is also well suited for sensitivity analysis applications. We demonstrate the performance of this sequential design strategy on a number of well-known benchmarks from the surrogate modelling and sensitivity analysis literature.
  • Sparse spectral surrogate models for deterministic and stochastic computer simulations
    Item type: Doctoral Thesis
    Lüthen, Nora (2022)
    Computer simulations are an invaluable tool for modeling and investigating real-world phenomena and processes. However, as any model, simulations are affected by uncertainty caused by imperfect knowledge or natural variability of their parameters, initial conditions, or input values. This leads to uncertainty in the model response, which needs to be quantified to make subsequent conclusions and decisions trustworthy. To alleviate the considerable cost of uncertainty analyses for expensive computational models, the latter are often replaced by surrogates, i.e., by approximations with an explicit functional form that can be created based on a rather small number of model evaluations, and can be evaluated at low cost. Some computer models are affected by uncertainty only through their input parameters: for fixed values of the inputs, they always produce the same response. These models are called deterministic simulators. In contrast, models that feature inherent stochasticity are called stochastic simulators. The latter generate a different result each time they are run even if their input parameters are held at fixed values. In other words, they behave like random fields whose index set is the space of input parameters. In this thesis, we investigate spectral surrogate models, which are a class of global non-intrusive methods that expand the computational model onto an orthonormal basis of a suitable function space. We focus on sparse expansions, i.e., representations that only include a small finite subset of the basis elements. Sparse representations are typically computed by regression with sparsity-encouraging constraints, often using ideas originating from the field of compressed sensing. In particular, for deterministic simulators we explore the popular sparse polynomial chaos expansions (PCE) method, which utilizes a polynomial basis that is orthonormal with respect to the distribution of the input variables. We conduct an extensive literature survey as well as a benchmark of several promising methods on multiple models of varying dimensionality and complexity. The benchmark results are aggregated and visualized in a novel way to extract reliable recommendations about which methods should be used in practice. We also investigate the recently proposed Poincaré chaos expansions, which rely on a generally non-polynomial basis consisting of eigenfunctions of a specific differential operator connected to the Poincaré inequality. By construction, this basis is well suited for derivative-based global sensitivity analysis, which we explore both analytically and numerically. Furthermore, we propose a new surrogate model for stochastic simulators. Taking the random function view of a stochastic simulator, we approximate its trajectories by sparse PCE and perform Karhunen-Loève expansion on them. The latter is a well-known spectral representation for a random field which separately characterizes its spatial and stochastic variation. The joint distribution of the random coefficients is inferred using the marginal-copula framework. The resulting surrogate model is able to approximate marginal distributions, mean, and covariance function of the stochastic simulator, and can generate new trajectories.
  • Surrogating stochastic simulators from sample trajectories using a non-Gaussian random field approach
    Item type: Other Conference Item
    Lüthen, Nora; Marelli, Stefano; Sudret, Bruno (2022)
    SIAM Conference on Uncertainty Quantification (UQ22). Searchable Abstracts Document
    Stochastic simulators are a class of computational models that give a different response each time they are run, even if the same input parameters are used. Such a simulator can be viewed as a random field, indexed by the space of its input parameters. We focus on a class of stochastic simulators for which it is possible to generate trajectories, i.e., evaluations of the simulator throughout the space of input parameters for which the latent variables that induce the stochasticity of the simulator are held fixed (e.g., by fixing the random seed). Stochastic simulators are typically highly complex and expensive to run, which makes uncertainty analysis and optimization costly. These costs can be alleviated by replacing the simulator with a suitable surrogate model, which captures the essential characteristics of the original model while being much cheaper to evaluate. We propose a surrogate model that combines sparse polynomial chaos expansion, extended Karhunen-Loève expansion, and parametric inference of joint distributions in the marginal-copula framework to represent the stochastic simulator based on a number of model evaluations. The resulting surrogate model has an analytical form that can easily be used to compute moments and to sample new trajectories. In this talk, we demonstrate the performance of our surrogate model on a real-world engineering application, and show how it can be utilized to perform conditional prediction.
  • Surrogating stochastic simulators using Karhunen-Loève expansion, sparse PCE and advanced statistical modelling
    Item type: Other Conference Item
    Lüthen, Nora; Marelli, Stefano; Sudret, Bruno (2021)
    The classical uncertainty quantification approach models all uncertainty about a physical process in the form of input uncertainty to a deterministic computational model. However, this is not always possible: sometimes part of the uncertainty such as high-dimensional environmental variables cannot be easily modelled (e.g., earthquakes, wind fields), or there is intrinsic randomness in the model (e.g., epidemiological SIR models). Then the model is a so-called stochastic simulator: even when holding all input parameters at a fixed value, the model response is still a random variable. A stochastic simulator can also be seen as a random field, where the input space acts as its index set. To simulate random fields, a widely used method is Karhunen-Loève expansion (KLE), which represents the random field as an infinite series involving orthonormal deterministic basis functions and a countable number of uncorrelated random variables. However, for inferring a random field from a small set of model evaluations, two challenges have often limited the applicability of KLE [1,2]: the covariance function, which is needed to compute the KLE, is usually not known; and the joint distribution of KL random variables is in general complicated, non-Gaussian and possibly highly dependent, and therefore difficult to model. Our approach addresses these challenges. Building on the success of sparse polynomial chaos expansions (PCE) as surrogate models for deterministic engineering models, we propose to use them to approximate trajectories from the stochastic simulator of interest, which results in a continuous covariance function. After computing the KLE and the KL random variables associated with the trajectories, we infer a parametric form of their joint distribution by using state-of-the-art probabilistic modelling techniques such as vine copulas [3] and generalized lambda distributions [4]. We demonstrate that our approach results in a stochastic emulator with accurate marginals and covariance function, which furthermore can be sampled to obtain new realizations. [1] Poirion, F., & Zentner, I. (2014). Stochastic model construction of observed random phenomena. Probabilistic Engineering Mechanics, 36, 63-71. [2] Azzi, S., Huang, Y., Sudret, B., & Wiart, J. (2019). Surrogate modeling of stochastic functions - application to computational electromagnetic dosimetry. International Journal for Uncertainty Quantification, 9(4). [3] Torre, E., Marelli, S., Embrechts, P., & Sudret, B. (2019). A general framework for data-driven uncertainty quantification under complex input dependencies using vine copulas. Probabilistic Engineering Mechanics, 55, 1-16. [4] Karian, Z. A., & Dudewicz, E. J. (2000). Fitting statistical distributions: the generalized lambda distribution and generalized bootstrap methods. CRC press.
  • Sparse Polynomial Chaos Expansions: Literature Survey and Benchmark
    Item type: Journal Article
    Lüthen, Nora; Marelli, Stefano; Sudret, Bruno (2021)
    SIAM/ASA Journal on Uncertainty Quantification
    Sparse polynomial chaos expansions (PCE) are a popular surrogate modelling method that takes advantage of the properties of PCE, the sparsity-of-effects principle, and powerful sparse regression solvers to approximate computer models with many input parameters, relying on only a few model evaluations. Within the last decade, a large number of algorithms for the computation of sparse PCE have been published in the applied math and engineering literature. We present an extensive review of the existing methods and develop a framework for classifying the algorithms. Furthermore, we conduct a unique benchmark on a selection of methods to identify which approaches work best in practical applications. Comparing their accuracy on several benchmark models of varying dimensionality and complexity, we find that the choice of sparse regression solver and sampling scheme for the computation of a sparse PCE surrogate can make a significant difference of up to several orders of magnitude in the resulting mean-squared error. Different methods seem to be superior in different regimes of model dimensionality and experimental design size.
  • Spectral decomposition of H¹(µ) and Poincaré inequality on a compact interval — Application to kernel quadrature
    Item type: Journal Article
    Roustant, Olivier; Lüthen, Nora; Gamboa, Fabrice (2024)
    Journal of Approximation Theory
Publications 1 - 10 of 31