Applied Mathematics Colloquium
Monday,
February 24, 2020
4:00 PM - 5:00 PM
Speaker:
Lorenzo Tamellini, CNR - Istituto di Matematica Applicata e Tecnologie Informatiche “E. Magenes”, Pavia, Italy
Title: Sparse-grids-based Uncertainty Quantification of geochemical compaction of sedimentary basins
Abstract: In this work we propose a methodology based on sparse grids for the Uncertainty Quantification (UQ) of sedimentary basins undergoing mechanical and geochemical compaction processes, which we model as a coupled, time-dependent, non-linear, monodimensional (depth-only) system of PDEs with uncertain parameters.
We discuss both forward and inverse UQ for this problem, whose quantities of interest (QoI) are the in-depth profiles of porosity, temperature and pressure at T=today. The methodology proposed is based on a sparse-grid approximation of the QoI, and in particular we will discuss an efficient methodology for the computation of the Sobol indices, to evaluate the impact of each random parameter on the total variability of the QoI. The inverse problem will be tackled with a Maximum Likelihood approach, sped up by replacing the full model evaluation with its sparse-grid approximate counterpart.
We then consider the case of multi-layered basins, in which each layer is characterized by a different material. The multi-layered structure gives rise to discontinuities in the map from the uncertain parameters to the QoI. Because of these discontinuities, an appropriate treatment is needed to apply sparse grids quadrature and interpolation for UQ purposes. To this end, we propose a two-steps methodology which relies on a change of coordinates to align the interfaces among layers of different materials; note that the map from the physical to the reference domain is random because the location of the interfaces also depends on the values of the random parameters. Once this alignment has been computed (again by means of a sparse grid), a standard sparse-grid-based UQ analysis of the QoI can be performed within each layer. This procedure can then be seen as a composition of sparse grids, or "deep sparse grid approximation". The effectiveness of this procedure is due to the fact that the physical locations of the interfaces among layers feature a smooth dependence on the random parameters and are therefore themselves amenable to sparse grid polynomial approximations.
We showcase the capabilities of our numerical methodologies through some synthetic test cases.
References:
[1] Ivo Colombo, Fabio Nobile, Giovanni Porta, Anna Scotti, Lorenzo Tamellini, Uncertainty Quantification of geochemical and mechanical compaction in layered sedimentary basins, Computer Methods in Applied Mechanics and Engineering, 2018. https://doi.org/10.1016/j.cma.2017.08.049
[2] Giovanni Porta, Lorenzo Tamellini, Valentina Lever, Monica Riva, Inverse modeling of geochemical and mechanical compaction in sedimentary basins through Polynomial Chaos Expansion, 2014. https://doi.org/10.1002/2014WR015838
[3] Luca Formaggia, Alberto Guadagnini, Ilaria Imperiali, Valentina Lever, Giovanni Porta, Monica Riva, Anna Scotti, Global sensitivity analysis through polynomial chaos expansion of a basin-scale geochemical compaction model, Computational Geosciences, 2013. https://doi.org/10.1007/s10596-012-9311-5
Lorenzo Tamellini, CNR - Istituto di Matematica Applicata e Tecnologie Informatiche “E. Magenes”, Pavia, Italy
Title: Sparse-grids-based Uncertainty Quantification of geochemical compaction of sedimentary basins
Abstract: In this work we propose a methodology based on sparse grids for the Uncertainty Quantification (UQ) of sedimentary basins undergoing mechanical and geochemical compaction processes, which we model as a coupled, time-dependent, non-linear, monodimensional (depth-only) system of PDEs with uncertain parameters.
We discuss both forward and inverse UQ for this problem, whose quantities of interest (QoI) are the in-depth profiles of porosity, temperature and pressure at T=today. The methodology proposed is based on a sparse-grid approximation of the QoI, and in particular we will discuss an efficient methodology for the computation of the Sobol indices, to evaluate the impact of each random parameter on the total variability of the QoI. The inverse problem will be tackled with a Maximum Likelihood approach, sped up by replacing the full model evaluation with its sparse-grid approximate counterpart.
We then consider the case of multi-layered basins, in which each layer is characterized by a different material. The multi-layered structure gives rise to discontinuities in the map from the uncertain parameters to the QoI. Because of these discontinuities, an appropriate treatment is needed to apply sparse grids quadrature and interpolation for UQ purposes. To this end, we propose a two-steps methodology which relies on a change of coordinates to align the interfaces among layers of different materials; note that the map from the physical to the reference domain is random because the location of the interfaces also depends on the values of the random parameters. Once this alignment has been computed (again by means of a sparse grid), a standard sparse-grid-based UQ analysis of the QoI can be performed within each layer. This procedure can then be seen as a composition of sparse grids, or "deep sparse grid approximation". The effectiveness of this procedure is due to the fact that the physical locations of the interfaces among layers feature a smooth dependence on the random parameters and are therefore themselves amenable to sparse grid polynomial approximations.
We showcase the capabilities of our numerical methodologies through some synthetic test cases.
References:
[1] Ivo Colombo, Fabio Nobile, Giovanni Porta, Anna Scotti, Lorenzo Tamellini, Uncertainty Quantification of geochemical and mechanical compaction in layered sedimentary basins, Computer Methods in Applied Mechanics and Engineering, 2018. https://doi.org/10.1016/j.cma.2017.08.049
[2] Giovanni Porta, Lorenzo Tamellini, Valentina Lever, Monica Riva, Inverse modeling of geochemical and mechanical compaction in sedimentary basins through Polynomial Chaos Expansion, 2014. https://doi.org/10.1002/2014WR015838
[3] Luca Formaggia, Alberto Guadagnini, Ilaria Imperiali, Valentina Lever, Giovanni Porta, Monica Riva, Anna Scotti, Global sensitivity analysis through polynomial chaos expansion of a basin-scale geochemical compaction model, Computational Geosciences, 2013. https://doi.org/10.1007/s10596-012-9311-5
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