## Contact

Swiss National Science Foundation (SNSF)

Wildhainweg 3P.O. Box

CH-3001 Bern

Phone +41 31 308 22 22

English title | Efficient numerical methods for flow and transport phenomena in heterogeneous random porous media. |
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Applicant | Nobile Fabio |

Number | 140574 |

Funding scheme | Project funding (Div. I-III) |

Research institution | EPFL - SB - SMA-GE |

Institution of higher education | EPF Lausanne - EPFL |

Main discipline | Mathematics |

Start/End | 01.04.2012 - 30.04.2015 |

Approved amount | 164'480.00 |

multivariate polynomial approximation; Flow and transport in porous media; High Performance Computing; Monte Carlo Sampling; Stochastic Galerkin and collocation methods; Equations with random coefficients

Lead |
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Lay summary |

This project focuses in the development of efficient numerical methods for groundwater flow and solute transport phenomena in heterogeneous aquifers. To account for the lack of measurements and the strong level of uncertainty in the characterization of the properties of subsurface media, a common practice in hydrology is to describe the porosity and permeability of the materials by means of spatially correlated log-normally distributed random fields. The project builds around the idea of approximating the solution of the flow and transport problems by multivariate polynomials of a finite (or countably infinite) number of random variables used to parametrize the log-normal permeability field. Such polynomial approximations can be constructed by either projecting the equations on a suitable polynomial subspace (Stochastic Galerkin) or by interpolating the solution on a suitable set of points in the parameter space (Stochastic Collocation). Specific tasks of the project include: 1) Design good (nearly optimal) polynomial spaces targeted to treat the case of log-normal permeability, and potentially able to deliver effective approximations also in the limit case of an infinite (countable) number of random variables; 2) Combine polynomial approximations with Monte Carlo techniques to treat the case of a non-smooth covariance kernel (which implies non smooth realizations of the permeability field); 3) Specifically address the case of permeability random fields conditioned to available measurements from observation wells; 4) Develop a Stochastic Domain Decomposition approach to treat aquifers with multiple facies with independent randomness. We will also focus on efficient methods for the probabilistic delineation of well catchments and time-related capture zones. For this we will investigate two alternative approaches, either by simulating particle trajectories to see which ones reach the well, or by solving a backward transport equation that retropropagates a given solute concentration injected from the well. Both Monte Carlo and polynomial approximations will be investigated and compared. |

Direct link to Lay Summary | Last update: 21.02.2013 |

Name | Institute |
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Name | Institute |
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Publication |
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An adaptive sparse grid algorithm for elliptic {PDE}s with lognormal diffusion coefficient |

Convergence of quasi-optimal sparse grid approximation of Hilbert-space-valued functions: application to random elliptic {PDE}s |

Multi index Monte Carlo: when sparsity meets sampling |

A multi level Monte Carlo method with control variate for elliptic {PDE}s with log-normal coefficients |

Comparison of Clenshaw--Curtis and Leja quasi-optimal sparse grids for the approximation of random {PDE}s |

Group / person | Country |
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Types of collaboration |
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MCSE, King Abdullah University of Science and Techology | Saudi Arabia (Asia) |

- in-depth/constructive exchanges on approaches, methods or results - Publication |

Title | Type of contribution | Title of article or contribution | Date | Place | Persons involved |
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Algorithms and Applications Workshop | Poster | Comparison of quasi-optimal and adaptive sparse-grids for groundwater ow problems Advances in Uncertainty Quantification Methods | 06.01.2015 | KAUST, Thuwal, Saudi Arabia | Tesei Francesco; |

Workshop - Sparse grids and Applications | Individual talk | Quasi-optimal sparse grids for PDEs with random coefficients | 01.09.2014 | University of Stuttgart, Germany, Germany | Nobile Fabio; |

ENUMATH conference 2013 | Talk given at a conference | Multi Level Monte Carlo methods with Control Variate for elliptic SPDEs | 30.08.2014 | EPFL, Lausanne, Switzerland | Tesei Francesco; |

SIMAI 2014 | Individual talk | Sampling and collocation methods for PDEs with random data | 07.07.2014 | Taormina, Italy | Nobile Fabio; |

Swiss Numerics Colloquium 2014 | Talk given at a conference | Analysis of the Multi Level Monte Carlo method with ControlVariate applied to elliptic SPDEs | 25.04.2014 | I-Math Institute für Mathematik, Zürich, Switzerland | Tesei Francesco; |

MASCOT NUM 2014 meeting | Poster | Multi Level Monte Carlo methods with Control Variate for elliptic Stochastic Partial Differential Equations | 23.04.2014 | ETH Zürich, Switzerland | Tesei Francesco; |

MATHMET 2014 | Individual talk | Sampling based polynomial chaos approaches for uncertainty propagation | 24.03.2014 | PTB Berlin, Germany | Nobile Fabio; |

SIAM UQ 2014 | Talk given at a conference | Multilevel Monte Carlo Methods with Control Variate for elliptic SPDEs | 01.03.2014 | Hyatt Regency Savannah, United States of America | Tesei Francesco; |

Algorithms and Applications Workshop | Poster | Multi Level Monte Carlo methods with Control Variate for elliptic SPDEs, Advances in Uncertainty Quantification Methods | 06.01.2014 | KAUST, Thuwal, Saudi Arabia | Tesei Francesco; |

Workshop - Advances in Uncertainty Quantification Methods, Algorithms and Applications | Individual talk | Collocation methods for uncertainty quantification in PDE models with random data | 06.01.2014 | KAUST, Saudi Arabia | Nobile Fabio; |

Workshop - Partial Differential Equations with Random Coefficients | Individual talk | Stochastic collocation and MLMC methods for elliptic PDEs with random coefficients | 13.11.2013 | WIAS, Berlin, Germany | Nobile Fabio; |

workshop - Multiscale and High-Dimensional Problems | Individual talk | Analysis of collocation methods for elliptic PDEs with stochastic coefficients | 28.07.2013 | Oberwolfach, Germany | Nobile Fabio; |

Ninth IMACS Seminar on Monte Carlo Methods | Talk given at a conference | Multi Level Monte Carlo methods with Control Variate for elliptic SPDEs | 16.07.2013 | Universite de Savoie, Annecy-le-Vieux, France | Tesei Francesco; |

25th Biannual Numerical Analysis Conference | Talk given at a conference | Multi Level Monte Carlo methods with Control Variate | 26.06.2013 | University of Strathclyde, Glasgow, Great Britain and Northern Ireland | Tesei Francesco; |

Workshop on Numerical Methods for Uncertainty Quantification | Poster | Multi Level Monte Carlo methods with Control Variate for elliptic SPDEs, Hausdorff Center for Mathematics | 13.05.2013 | Mathematik-Zentrum, Bonn, Germany | Tesei Francesco; |

Swiss Numerics Colloquium 2013 | Poster | Multilevel Monte Carlo methods with Control Variate for elliptic SPDEs | 05.04.2013 | EPFL, Lausanne, Switzerland | Tesei Francesco; |

Workshop - Numerical Methods for PDE Constrained Optimization with Uncertain Data | Individual talk | Collocation approaches for forward uncertainty propagation in PDE models with random input data | 28.01.2013 | Oberwolfach, Germany | Nobile Fabio; |

29th GAMM-Seminar Leipzig on Numerical Methods for UQ | Individual talk | Collocation methods for uncertainty quantification in PDE models with random data | 21.01.2013 | MPI, Leipzig, Germany | Nobile Fabio; |

International Workshop on Numerical Methods of SDE | Individual talk | Stochastic polynomial and MLMC methods for elliptic PDEs with random coefficients, | 15.10.2012 | Chinese Academy of Sciences, Beijing, China | Nobile Fabio; |

Workshop - Stochastic Analysis and Applications | Individual talk | Numerical methods for PDEs with random coefficients | 04.06.2012 | Centre Interfacultaire Bernoulli, EPFL, Switzerland | Nobile Fabio; |

Seminar | Individual talk | Collocation methods for PDE models with random data | 31.05.2012 | MPI, Magdeburg, Germany | Nobile Fabio; |

Title | Date | Place |
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85th GAMM annual meeting | 10.03.2014 | Erlangen, Germany |

Workshop Numerical Methods for Uncertainty Quantification | 07.05.2013 | Mathematik-Zentrum, Bonn , Germany |

Number | Title | Start | Funding scheme |
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182236 | Model order reduction based on functional rational approximants for parametric PDEs with meromorphic structure | 01.01.2019 | Project funding (Div. I-III) |

146360 | Dynamical low rank approximation of evolution equations with random parameters | 01.05.2013 | Project funding (Div. I-III) |

This project focuses in the development of efficient numerical methods for groundwater flow and solute transport phenomena in heterogeneous aquifers. To account for the lack of measurements and the strong level of uncertainty in the characterization of the properties of subsurface media, a common practice in hydrology is to describe the porosity and permeability of the materials by means of spatially correlated log-normally distributed random fields. Although this approach can describe, in principle, heterogeneity at several scales, we focus specifically on macroscopic heterogeneity, i.e. long-range variations of the physical properties within an aquifer, with a characteristic correlation length of the same order of the aquifer size. We address, therefore, situations that are well far from the homogenization limit.The project builds around the idea of approximating the solution of the flow and transport problems by multivariate polynomials of a finite (or countably infinite) number of random variables used to parametrize the log-normal permeability field. Such polynomial approximations can be constructed by either projecting the equations on a suitable polynomial subspace (Stochastic Galerkin) or by interpolating the solution on a suitable set of points in the parameter space (Stochastic Collocation).Building on our previous experience on the solution of elliptic equations with random coefficients, we will explore and develop several new ideas to solve efficiently the flow problem in heterogeneous random porous media: 1) Design good (nearly optimal) polynomial spaces targeted to treat the case of log-normal permeability, and potentially able to deliver effective approximations also in the limit case of an infinite (countable) number of random variables; 2) Combine polynomial approximations with Monte Carlo techniques to treat the case of a non-smooth covariance kernel (which implies non smooth realizations of the permeability field); 3) Specifically address the case of permeability random fields conditioned to available measurements from observation wells; 4) Develop a Stochastic Domain Decomposition approach to treat aquifers with multiple facies with independent randomness.One of the goals of the project is to provide efficient methods for the probabilistic delineation of well catchments and time-related capture zones. For this we will investigate two alternative approaches, either by simulating particle trajectories to see which ones reach the well, or by solving a backward transport equation that retropropagates a given solute concentration injected from the well. Both Monte Carlo and polynomial approximations will be investigated and compared. Reduced order modeling for the backward transport equation will be analyzed as well.This project has a strong component in methodological development and theoretical analysis. At the same time, the most successful techniques developed will be implemented in a parallel finite element code and adapted to High Performance Computing to tackle realistic applications of practical interest in hydrology.

Swiss National Science Foundation (SNSF)

Wildhainweg 3P.O. Box

CH-3001 Bern

Phone +41 31 308 22 22

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