Project B07

B07 | Computational Uncertainty Quantification

Prof. Dr. Andrea Barth, University of Stuttgart
Email | Website

Prof. Dr. Oliver Deussen, University of Konstanz
Email | Website

Cedric Beschle, University of Stuttgart – Email | Website

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About this project
Results

Every scientific visualization is based on data. The data may result from direct computation, numerical simulation, or be somehow measured.

The underlying model may be subject to various uncertainties, like parameter or domain uncertainty, model uncertainty, numerical errors, or some intrinsic stochasticity of the model. Further, the uncertainty is either introduced by the measurement instrument or is the result of insufficient measurements. In all cases, however, the data is just one representative sample of an underlying distribution, i.e., a repeated numerical simulation of a stochastic model may lead to a different sample of the same underlying distribution.

Uncertainty quantification is still lacking in interpretability and the discussion on appropriate problem-dependent risk measures in general.

As described above, visualizing uncertain problems is a problem of various aspects. The focus in this project lies on a rigorous mathematical modeling and a-priori error bounds for numerical simulations.

Project B07 aims to contribute to the following research topics:

Sampling techniques for unknown distributions
Non-linear transformations and risk measures:
Inverse problems

The visualization of uncertain problems is investigated and further the influence of the visualization process on the said uncertainties.

Research Questions

How to sample efficiently for the visualization process?

How to approximate infinite-dimensional stochastic processes?

How is the visual computing pipeline influencing the uncertainties?

Fig. 1: Samples of a linear hyperbolic equation driven by a generalized-hyperbolic field and an elliptic equation with a discontinuos coefficient.

Fig. 2: Samples of a field with normal inverse Gaussian point-wise marginal distributions and Gaussian random field with different covariance operators.

Publications

A. Barth, B. Harrach, N. Hyvönen, and L. Mustonen, “Detecting Stochastic Inclusions in Electrical Impedance Tomography,” Inverse Problems, vol. 33, no. 11, Art. no. 11, 2017, doi: 10.1088/1361-6420/aa8f5c.
Abstract
This work considers the inclusion detection problem of electrical impedance tomography with stochastic conductivities. It is shown that a conductivity anomaly with a random conductivity can be identified by applying the factorization method or the monotonicity method to the mean value of the corresponding Neumann-to-Dirichlet map provided that the anomaly has high enough contrast in the sense of expectation. The theoretical results are complemented by numerical examples in two spatial dimensions.
BibTeX
@article{barth2017detecting, abstract = {This work considers the inclusion detection problem of electrical impedance tomography with stochastic conductivities. It is shown that a conductivity anomaly with a random conductivity can be identified by applying the factorization method or the monotonicity method to the mean value of the corresponding Neumann-to-Dirichlet map provided that the anomaly has high enough contrast in the sense of expectation. The theoretical results are complemented by numerical examples in two spatial dimensions.}, author = {Barth, Andrea and Harrach, Bastian and Hyvönen, Nuutti and Mustonen, Lauri}, doi = {10.1088/1361-6420/aa8f5c}, fjournal = {Inverse Problems}, journal = {Inverse Problems}, number = 11, owner = {seusdd}, pages = 115012, title = {Detecting Stochastic Inclusions in Electrical Impedance Tomography}, url = {https://doi.org/10.1088/1361-6420/aa8f5c}, volume = 33, year = 2017 }
Link
https://doi.org/10.1088/1361-6420/aa8f5c
A. Barth and A. Stein, “Approximation and simulation of infinite-dimensional Lévy processes,” Stochastics and Partial Differential Equations: Analysis and Computations, vol. 6, no. 2, Art. no. 2, 2016, doi: 10.1007/s40072-017-0109-2.
Abstract
In this paper approximation methods for infinite-dimensional Lévy processes, also called (time-dependent) Lévy fields, are introduced. For square integrable fields beyond the Gaussian case, it is no longer given that the one-dimensional distributions in the spectral representation with respect to the covariance operator are independent. When simulated via a Karhunen–Loève expansion a set of dependent but uncorrelated one-dimensional Lévy processes has to be generated. The dependence structure among the one-dimensional processes ensures that the resulting field exhibits the correct point-wise marginal distributions. To approximate the respective (one-dimensional) Lévy-measures, a numerical method, called discrete Fourier inversion, is developed. For this method, Lp-convergence rates can be obtained and, under certain regularity assumptions, mean square and Lp-convergence of the approximated field is proved. Further, a class of (time-dependent) Lévy fields is introduced, where the point-wise marginal distributions are dependent but uncorrelated subordinated Wiener processes. For this specific class one may derive point-wise marginal distributions in closed form. Numerical examples, which include hyperbolic and normal-inverse Gaussian fields, demonstrate the efficiency of the approach.
BibTeX
@article{barth2016approximation, abstract = {In this paper approximation methods for infinite-dimensional Lévy processes, also called (time-dependent) Lévy fields, are introduced. For square integrable fields beyond the Gaussian case, it is no longer given that the one-dimensional distributions in the spectral representation with respect to the covariance operator are independent. When simulated via a Karhunen–Loève expansion a set of dependent but uncorrelated one-dimensional Lévy processes has to be generated. The dependence structure among the one-dimensional processes ensures that the resulting field exhibits the correct point-wise marginal distributions. To approximate the respective (one-dimensional) Lévy-measures, a numerical method, called discrete Fourier inversion, is developed. For this method, Lp-convergence rates can be obtained and, under certain regularity assumptions, mean square and Lp-convergence of the approximated field is proved. Further, a class of (time-dependent) Lévy fields is introduced, where the point-wise marginal distributions are dependent but uncorrelated subordinated Wiener processes. For this specific class one may derive point-wise marginal distributions in closed form. Numerical examples, which include hyperbolic and normal-inverse Gaussian fields, demonstrate the efficiency of the approach.}, author = {Barth, Andrea and Stein, Andreas}, doi = {10.1007/s40072-017-0109-2}, journal = {Stochastics and Partial Differential Equations: Analysis and Computations}, note = {submitted to Stochastic and Partial Differential Equations}, number = 2, owner = {seusdd}, pages = {286-334}, title = {Approximation and simulation of infinite-dimensional {L}\'evy processes}, url = {https://doi.org/10.1007/s40072-017-0109-2}, volume = 6, year = 2016 }
Link
https://doi.org/10.1007/s40072-017-0109-2
A. Barth, R. Bürger, I. Kröker, and C. Rohde, “Computational Uncertainty Quantification for a Clarifier-thickener Model with Several Random Perturbations: A Hybrid Stochastic Galerkin Approach,” Computers & Chemical Engineering, vol. 89, pp. 11–26, 2016, doi: 10.1016/j.compchemeng.2016.02.016.
Abstract
Continuous sedimentation processes in a clarifier-thickener unit can be described by a scalar nonlinear conservation law whose flux density function is discontinuous with respect to the spatial position. In the applications of this model, which include mineral processing and wastewater treatment, the rate and composition of the feed flow cannot be given deterministically. Efficient numerical simulation is required to quantify the effect of uncertainty in these control parameters in terms of the response of the clarifier-thickener system. Thus, the problem at hand is one of uncertainty quantification for nonlinear hyperbolic problems with several random perturbations. The presented hybrid stochastic Galerkin method is devised so as to extend the polynomial chaos approximation by multiresolution discretization in the stochastic space. This approach leads to a deterministic hyperbolic system, which is partially decoupled and therefore suitable for efficient parallelisation. Stochastic adaptivity reduces the computational effort. Several numerical experiments are presented.
BibTeX
@article{journals/cce/BarthBKR16, abstract = {Continuous sedimentation processes in a clarifier-thickener unit can be described by a scalar nonlinear conservation law whose flux density function is discontinuous with respect to the spatial position. In the applications of this model, which include mineral processing and wastewater treatment, the rate and composition of the feed flow cannot be given deterministically. Efficient numerical simulation is required to quantify the effect of uncertainty in these control parameters in terms of the response of the clarifier-thickener system. Thus, the problem at hand is one of uncertainty quantification for nonlinear hyperbolic problems with several random perturbations. The presented hybrid stochastic Galerkin method is devised so as to extend the polynomial chaos approximation by multiresolution discretization in the stochastic space. This approach leads to a deterministic hyperbolic system, which is partially decoupled and therefore suitable for efficient parallelisation. Stochastic adaptivity reduces the computational effort. Several numerical experiments are presented.}, author = {Barth, Andrea and Bürger, Raimund and Kröker, Ilja and Rohde, Christian}, doi = {10.1016/j.compchemeng.2016.02.016}, ee = {https://doi.org/10.1016/j.compchemeng.2016.02.016}, journal = {Computers & Chemical Engineering}, pages = {11-26}, title = {Computational Uncertainty Quantification for a Clarifier-thickener Model with Several Random Perturbations: A Hybrid Stochastic Galerkin Approach}, url = {https://doi.org/10.1016/j.compchemeng.2016.02.016}, volume = 89, year = 2016 }
Link
https://doi.org/10.1016/j.compchemeng.2016.02.016
A. Barth and F. G. Fuchs, “Uncertainty Quantification for Hyperbolic Conservation Laws with Flux Coefficients Given by Spatiotemporal Random Fields,” SIAM Journal on Scientific Computing, vol. 38, no. 4, Art. no. 4, 2016, doi: 10.1137/15M1027723.
Abstract
In this paper hyperbolic partial differential equations (PDEs) with random coefficients are discussed. We consider the challenging problem of flux functions with coefficients modeled by spatiotemporal random fields. Those fields are given by correlated Gaussian random fields in space and Ornstein--Uhlenbeck processes in time. The resulting system of equations consists of a stochastic differential equation for each random parameter coupled to the hyperbolic conservation law. We define an appropriate solution concept in this setting and analyze errors and convergence of discretization methods. A novel discretization framework, based on Monte Carlo finite volume methods, is presented for the robust computation of moments of solutions to those random hyperbolic PDEs. We showcase the approach on two examples which appear in applications---the magnetic induction equation and linear acoustics---both with a spatiotemporal random background velocity field.
BibTeX
@article{journals/siamsc/BarthF16, abstract = {In this paper hyperbolic partial differential equations (PDEs) with random coefficients are discussed. We consider the challenging problem of flux functions with coefficients modeled by spatiotemporal random fields. Those fields are given by correlated Gaussian random fields in space and Ornstein--Uhlenbeck processes in time. The resulting system of equations consists of a stochastic differential equation for each random parameter coupled to the hyperbolic conservation law. We define an appropriate solution concept in this setting and analyze errors and convergence of discretization methods. A novel discretization framework, based on Monte Carlo finite volume methods, is presented for the robust computation of moments of solutions to those random hyperbolic PDEs. We showcase the approach on two examples which appear in applications---the magnetic induction equation and linear acoustics---both with a spatiotemporal random background velocity field.}, author = {Barth, Andrea and Fuchs, Franz G.}, doi = {10.1137/15M1027723}, ee = {http://dx.doi.org/10.1137/15M1027723}, journal = {SIAM Journal on Scientific Computing}, number = 4, pages = { A2209–A2231}, title = {Uncertainty Quantification for Hyperbolic Conservation Laws with Flux Coefficients Given by Spatiotemporal Random Fields}, url = {https://doi.org/10.1137/15M1027723}, volume = 38, year = 2016 }
Link
https://doi.org/10.1137/15M1027723