# Multilevel Monte Carlo Methods for Elliptic Problems with Applications to Radioactive Waste Disposal

Lead Research Organisation:
University of Oxford

Department Name: Mathematical Institute

### Abstract

We propose to carry out fundamental mathematical research into efficient methods for problems with uncertain parameters and apply them to radioactive waste disposal.The UK Government's policy on nuclear power states that it is a proven low-carbon technology for generating electricity and should form part of the UK's future energy supply. Energy companies will be allowed to build new nuclear power stations provided sufficient progress is made on the radioactive waste issue. In common with other nations, geological disposal is the UK's preferred option for dealing with radioactive waste in the long term. Making a safety case for geological disposal is a major scientific undertaking. National and international research programmes have produced a good understanding of the mechanisms by which radionuclides might return to the human environment and of their consequences once there. One of the outstanding challenges is how to deal with the uncertainties inherent in geological systems and in the evolution of a repository over long time periods and this is at the heart of the proposed research.The main mechanism whereby radionuclides might return to the environment, in the event that they escape from the repository, is transport by groundwater flowing in rocks underground. The mathematical equations that model this flow are well understood, but in order to solve them and to predict the transport of radionuclides the permeability and porosity of the rocks must be specified everywhere around the repository. It is only feasible to measure these quantities at relatively few locations. The values elsewhere have to be inferred and this, inevitably, gives rise to uncertainty. In early performance assessments, relatively rudimentary approaches to treating these uncertainties were used, primarily due to the computational cost. Since then, there have been considerable advances in computer hardware and in the mathematical field of uncertainty quantification. One of the most common approaches to quantify uncertainty is to use probabilistic techniques. This means that the coefficients within the flow equations will be modelled as random fields, leading to partial differential equations with random coefficients (stochastic PDEs), and solving these is much harder and more computationally demanding than their deterministic equivalents. Many fast converging techniques for stochastic PDEs have recently emerged, which are applicable when the uncertainty can be approximated well with a small number of stochastic parameters. However, evidence from field data is such that in repository safety cases much larger numbers of stochastic parameters will be required to capture the uncertainty in the system. Only Monte Carlo (MC) sampling and averaging methods are currently feasible in this case, and the relatively slow rate of convergence of these methods is a major issue.In the work proposed here we will develop and analyse a new and exciting approach to accelerate the convergence of MC simulations for stochastic PDEs. The multilevel MC approach combines multigrid ideas for deterministic PDEs with the classical MC method. The dramatic savings in computational cost which we predict for this approach stem from the fact that most of the work can be done on computationally cheap coarse spatial grids. Only very few samples have to be computed on finer grids to obtain the necessary spatial accuracy. This method has already been applied (by one of the PIs), with great success, to stochastic ordinary differential equations in mathematical finance. In this project we will extend the technique to PDEs, developing the analysis of the method required, and apply the technique to realistic models of groundwater flow relevant to radioactive waste repository assessments. The potential impact for future work on radioactive waste disposal and also for other areas where uncertainty quantification plays a major role (e.g. carbon capture and storage) is considerable.

### Publications

Cliffe K
(2011)

*Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients*in Computing and Visualization in Science
Croci M
(2018)

*Efficient White Noise Sampling and Coupling for Multilevel Monte Carlo with Nonnested Meshes*in SIAM/ASA Journal on Uncertainty Quantification
Fang W
(2018)

*Monte Carlo and Quasi-Monte Carlo Methods*
Fang W
(2019)

*Multilevel Monte Carlo method for ergodic SDEs without contractivity*in Journal of Mathematical Analysis and Applications
Giles M
(2013)

*Monte Carlo and Quasi-Monte Carlo Methods 2012*
Giles M
(2015)

*Multilevel Monte Carlo methods*in Acta Numerica
Giles M
(2018)

*Monte Carlo and Quasi-Monte Carlo Methods*
Giles M
(2016)

*Monte Carlo and Quasi-Monte Carlo Methods*
Giles M
(2018)

*Decision-making under uncertainty: using MLMC for efficient estimation of EVPPI*in Statistics and Computing
Giles M
(2018)

*Multilevel Estimation of Expected Exit Times and Other Functionals of Stopped Diffusions*in SIAM/ASA Journal on Uncertainty Quantification
Giles M
(2014)

*Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation*in The Annals of Applied Probability
Giles M
(2015)

*Multilevel Monte Carlo Approximation of Distribution Functions and Densities*in SIAM/ASA Journal on Uncertainty Quantification
Giles M
(2019)

*Random bit multilevel algorithms for stochastic differential equations*in Journal of Complexity
Giles M
(2018)

*Random Bit Quadrature and Approximation of Distributions on Hilbert Spaces*in Foundations of Computational Mathematics
Katsiolides G
(2018)

*Multilevel Monte Carlo and improved timestepping methods in atmospheric dispersion modelling*in Journal of Computational Physics
Lester C
(2016)

*Extending the Multi-level Method for the Simulation of Stochastic Biological Systems.*in Bulletin of mathematical biology
Lester C
(2015)

*An adaptive multi-level simulation algorithm for stochastic biological systems.*in The Journal of chemical physics
Teckentrup A
(2013)

*Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients*in Numerische Mathematik
Vidal-Codina F
(2015)

*A model and variance reduction method for computing statistical outputs of stochastic elliptic partial differential equations*in Journal of Computational Physics
Vidal-Codina F
(2016)

*An Empirical Interpolation and Model-Variance Reduction Method for Computing Statistical Outputs of Parametrized Stochastic Partial Differential Equations*in SIAM/ASA Journal on Uncertainty QuantificationDescription | This project has demonstrated that the multilevel Monte Carlo method provides major improvements in the computational efficiency of Monte Carlo methods applied to the simulation of nuclear waste repositories. |

Exploitation Route | There is major potential for its use in the simulation of nuclear waste repositories, and also oil reservoir simulation. |

Sectors | Education,Energy |

URL | http://people.maths.ox.ac.uk/gilesm/mlmc.html |

Description | The mathematical approach we have developed has not yet been adopted by industry, although it is now being widely used within academia and both government and industry research labs. |

First Year Of Impact | 2011 |

Sector | Aerospace, Defence and Marine,Education,Electronics,Financial Services, and Management Consultancy |