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Numerical methods in uncertainty quantification

Research Project
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01.12.2014
 - 31.01.2017

In recent years it has become more and more important to model and simulate boundary value problems with random input parameters. If a statistical description of the input data is available, one can mathematically describe data and solutions as random fields and aim at the computation of corresponding deterministic statistics of the unknown random solution. Applications are, besides traditional engineering, for example biomedical or biomechanical processes. To simulate biomechanical processes one has, on the one hand, uncertain domains arising from e.g. tomographic data. On the other hand, one often has only estimates on the material parameters. Uncertainty might stem from the loading, the coefficients of the differential operator, or the domain of definition. In case of random loadings, the random solution depends linearly on the random input data. But this is not valid any more if the differential operator's coefficients or the domain of definition are random. Consequently, innovative methods must be developed in order to overcome the curse of dimension which is induced by the random process.

Collaborations & Cooperations

2035 - Participation or Organization of Collaborations on an international level
Griebel, Michael, Professor, University of Bonn and Fraunhofer Institute for Algorithms and Scientific Computing, Research cooperation

Publications

Gantner, Robert N. and Peters, Michael D. (2018) ‘Higher-Order Quasi-Monte Carlo for Bayesian Shape Inversion’, SIAM/ASA Journal on Uncertainty Quantification, 6(2), pp. 707–736. Available at: https://doi.org/10.1137/16m1096116.

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Haji-Ali, Abdul-Lateef et al. (2018) ‘Novel results for the anisotropic sparse grid quadrature’, Journal of complexity, 47, pp. 62–85. Available at: https://doi.org/10.1016/j.jco.2018.02.003.

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Harbrecht, Helmut and Peters, Michael D. (2018) ‘The second order perturbation approach for elliptic partial differential equations on random domains’, Applied Numerical Mathematics, 125, pp. 159–171. Available at: https://doi.org/10.1016/j.apnum.2017.11.002.

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Ballani, Jonas, Kressner, Daniel and Peters, Michael D. (2017) ‘Multilevel tensor approximation of PDEs with random data’, Stochastics and Partial Differential Equations, 5(3), pp. 400–427. Available at: https://doi.org/10.1007/s40072-017-0092-7.

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Dambrine, Marc et al. (2017) ‘On Bernoulli’s free boundary problem with a random boundary’, International Journal for Uncertainty Quantification, 7(4), pp. 335–353. Available at: https://doi.org/10.1615/int.j.uncertaintyquantification.2017019550.

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Harbrecht, Helmut, Peters, Michael and Siebenmorgen, Markus (2017) ‘On the quasi-Monte Carlo quadrature with Halton points for elliptic PDEs with log-normal diffusion’, Mathematics of Computation, 86, pp. 771–797. Available at: https://doi.org/10.1090/mcom/3107.

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Harbrecht, Helmut and Peters, Michael (2017) ‘Solution of free boundary problems in the presence of geometric uncertainties’, in Bergounioux, Maïtine; Oudet, Édouard; Rumpf, Martin; Carlier, Guillaume; Champion, Thierry; Santambrogio, Filippo (ed.) Topological Optimization and Optimal Transport In the Applied Sciences. Berlin-Bosten: De Gruyter (Radon Series on Computational and Applied Mathematics), pp. 20–39. Available at: https://doi.org/10.1515/9783110430417-002.

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Members (2)

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Helmut Harbrecht

Principal Investigator
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Michael Peters

Project Member