Forschung
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MATH+ projects
- Uncertainty Quantification and Design of Experiments for Data-Driven Control
- Machine Learning Enhanced Filtering Methods for Inverse Problems
- Wasserstein Gradient Flows for Generalised Transport in Bayesian Inversion
Uncertainty quantification for PDEs using quasi-Monte Carlo methods
Modern quasi-Monte Carlo (QMC) methods are based on tailoring specially designed cubature rules for high-dimensional integration problems. By leveraging the smoothness and anisotropy of an integrand, it is possible to achieve faster-than-Monte Carlo convergence rates. QMC methods have become a popular tool for solving partial differential equations (PDEs) involving random coefficients, a central topic within the field of uncertainty quantification.
Problems in uncertainty quantification can be roughly divided into two categories. Forward uncertainty quantification is the study of how uncertainties in mathematical or computational models and their inputs affect the system response. Conversely, inverse uncertainty quantification is concerned with the inverse problem of estimating the system inputs based on observations of the response of the system. In both cases, state-of-the-art cubature rules are essential to make the quantification of uncertainty tractable in practice.
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Publications
Peer-reviewed papers
- P. A. Guth and V. Kaarnioja. Application of dimension truncation error analysis to high-dimensional function approximation. To appear in: A. Hinrichs, P. Kritzer, F. Pillichshammer (eds.). Monte Carlo and Quasi-Monte Carlo Methods 2022. Springer Verlag (preprint).
- V. Kaarnioja, F. Y. Kuo, and I. H. Sloan. Lattice-based kernel approximation and serendipitous weights for parametric PDEs in very high dimensions. To appear in: A. Hinrichs, P. Kritzer, F. Pillichshammer (eds.). Monte Carlo and Quasi-Monte Carlo Methods 2022. Springer Verlag (preprint).
Submitted manuscripts
- I. H. Sloan and V. Kaarnioja. Doubling the rate – improved error bounds for orthogonal projection in Hilbert spaces. Preprint 2023. arXiv:2308.06052 [math.NA]
- H. Hakula, H. Harbrecht, V. Kaarnioja, F. Y. Kuo, and I. H. Sloan. Uncertainty quantification for random domains using periodic random variables. Preprint 2022. arXiv:2210.17329 [math.NA]
- P. A. Guth and V. Kaarnioja. Generalized dimension truncation error analysis for high-dimensional numerical integration: lognormal setting and beyond. Preprint 2022. arXiv:2209.06176 [math.NA]
- V. Kaarnioja and A. Rupp. Quasi-Monte Carlo and discontinuous Galerkin. Preprint 2022. arXiv:2207.07698 [math.NA]
Miscellaneous
- H. Hakula, H. Harbrecht, V. Kaarnioja, F. Y. Kuo, and I. H. Sloan. Domain uncertainty quantification using periodic random variables with application to elliptic PDEs. Poster presented at Bayes Comp 2023, Levi, Finland, March 15, 2023.
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PDE-constrained optimal control problems under uncertainty
In PDE-constrained optimal control problems, the target function is the solution of a PDE, steered by a control function. However, if the material properties of a mathematical model are poorly known or even completely unknown, it may be necessary to model these uncertainties using random fields.
To account for the presence of uncertainty in optimal control problems, one may compose the objective with a risk measure. Risk measures such as the expected value or the entropic risk measure involve high-dimensional integrals over the stochastic variables. This leads to challenging high-dimensional integration problems, which may be tackled using, e.g., QMC methods.
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Publications
Submitted manuscripts
- P. A. Guth, V. Kaarnioja, F. Y. Kuo, C. Schillings, and I. H. Sloan. Parabolic PDE-constrained optimal control under uncertainty with entropic risk measure using quasi-Monte Carlo integration. Preprint 2022. arXiv:2208.02767 [math.NA]
Miscellaneous
- P. A. Guth, V. Kaarnioja, F. Y. Kuo, C. Schillings, and I. H. Sloan. Quasi-Monte Carlo methods for optimal control problems subject to parabolic PDE constraints under uncertainty. Poster presented at Symposium on Inverse Problems: From experimental data to models and back, University of Potsdam, September 19, 2022.
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