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Efficient numerical methods for flow and transport phenomena in heterogeneous random porous media.

English title Efficient numerical methods for flow and transport phenomena in heterogeneous random porous media.
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
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Keywords (6)

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

Lay Summary (English)

Lead
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

Responsible applicant and co-applicants

Employees

Name Institute

Publications

Publication
An adaptive sparse grid algorithm for elliptic {PDE}s with lognormal diffusion coefficient
Nobile F., Tamellini L., Tesei F., Tempone R. (2016), An adaptive sparse grid algorithm for elliptic {PDE}s with lognormal diffusion coefficient, in Garcke J. (ed.), Springer, berlin, germany, 191-220.
Convergence of quasi-optimal sparse grid approximation of Hilbert-space-valued functions: application to random elliptic {PDE}s
Nobile F., Tamellini L., Tempone R. (2016), Convergence of quasi-optimal sparse grid approximation of Hilbert-space-valued functions: application to random elliptic {PDE}s, in Numer. Math., 134(2), 343-388.
Multi index Monte Carlo: when sparsity meets sampling
Haji-Ali A-L., Nobile F., Tempone R. (2016), Multi index Monte Carlo: when sparsity meets sampling, in Numer. Math., 132(4), 767-806.
A multi level Monte Carlo method with control variate for elliptic {PDE}s with log-normal coefficients
Nobile F., Tesei F. (2015), A multi level Monte Carlo method with control variate for elliptic {PDE}s with log-normal coefficients, in Stoch. PDEs: Anal. & Comp., 3(3), 398-444.
Comparison of Clenshaw--Curtis and Leja quasi-optimal sparse grids for the approximation of random {PDE}s
Nobile F., Tamellini L., Tempone R. (2015), Comparison of Clenshaw--Curtis and Leja quasi-optimal sparse grids for the approximation of random {PDE}s, in Kirby R. M. (ed.), Springer, Heidelberg/Berlin, Germany, 475-482.

Collaboration

Group / person Country
Types of collaboration
MCSE, King Abdullah University of Science and Techology Saudi Arabia (Asia)
- in-depth/constructive exchanges on approaches, methods or results
- Publication

Scientific events

Active participation

Title Type of contribution Title of article or contribution Date Place Persons involved
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;


Self-organised

Title Date Place
85th GAMM annual meeting 10.03.2014 Erlangen, Germany
Workshop Numerical Methods for Uncertainty Quantification 07.05.2013 Mathematik-Zentrum, Bonn , Germany

Associated projects

Number Title Start Funding scheme
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)

Abstract

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.
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