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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

Submitted manuscripts

• V. Kaarnioja, F. Y. Kuo, and I. H. Sloan. Lattice-based kernel approximation and serendipitous weights for parametric PDEs in very high dimensions. Preprint 2023. arXiv:2303.17755 [math.NA]
• P. A. Guth and V. Kaarnioja. Application of dimension truncation error analysis to high-dimensional function approximation. Preprint 2023. arXiv:2301.13693 [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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