Abstract
In uncertainty quantification, an unknown quantity has to be reconstructed which depends typically on the solution of a partial differential equation. This partial differential equation itself may depend on parameters, some of them may be deterministic and some are random. To approximate the unknown quantity one therefore has to solve the partial differential equation (usually numerically) for several instances of the parameters and then reconstruct the quantity from these simulations. As the number of parameters may be large, this becomes a high-dimensional reconstruction problem.
We will address the topic of reconstructing such unknown quantities using kernel-based reconstruction methods on sparse grids. First, we will introduce into the topic, then explain the reconstruction process and finally provide new error estimates.
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Acknowledgements
Christian Rieger would like to thank the Deutsche Forschungsgemeinschaft (DFG) for financial support through the CRC 1060, The Mathematics of Emergent Effects.
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Kempf, R., Wendland, H., Rieger, C. (2019). Kernel-Based Reconstructions for Parametric PDEs. In: Griebel, M., Schweitzer, M. (eds) Meshfree Methods for Partial Differential Equations IX. IWMMPDE 2017. Lecture Notes in Computational Science and Engineering, vol 129. Springer, Cham. https://doi.org/10.1007/978-3-030-15119-5_4
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DOI: https://doi.org/10.1007/978-3-030-15119-5_4
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