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

Lead Research Organisation:
University of Nottingham

Department Name: Sch of Mathematical Sciences

### 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

*Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients*in Computing and Visualization in Science

Description | Part One: Theoretical Framework In the first stage of project, the multilevel Monte Carlo algorithm was applied to ellipic PDEs with random coefficients. It was shown that, with MLMC method, we can get the same asymptotic cost for the stochastic problem as for the deterministic problem. Part Two: 3D Multilevel Monte Carlo Simulation Second phase of the project involved developing an efficient 3D uncertainty quantification tool. To this end, it was necessary to solve large linear systems of equations and generate realisations of Gaussian random field at reasonable cost. The algebraic multigrid (AMG) preconditioned conjugate gradient (PCG) method was used as an iterative linear solver. AMG provides a scalable algorithm which means that the work required to solve increasingly large problems grows linearly. We observed that each of permeability random fields yielded the (almost) same sparsity patterns of matrices on coarser levels. This observation led us to omit the coarse grid selection process, which is one of the expensive parts in the AMG setup phase, and use the same set of coarse grid points for all realisations. Using the cell-centred finite volume and sequential AMG algorithm in 3 dimensional space, we reduced up to almost 30% of overall MLMC simulation time by recycling the set of coarse-grid points. In order to generate realisations of large degrees of freedom especially in 3D, a fast and memory-efficient algorithm is required. The circulant embedding (CE) method of of Newsam and Dietrich and Wood and Grace has low requirements on memory and has low computational cost due to the use of Fast Fourier transform (FFT). We implemented the circulant embedding library in C++ and it was used as Gaussian random field generator in MLMC simulations. Part Three: More Realistic Model In the third phase, we extended the results from the first phase to physically more relevant quantities of interest such as the traveling time of particle, and to more realistic models. We studied the multilevel Monte Carlo technique through a case study of 2D Waste Isolation Pilot Plant (WIPP) repository in the Culebra Dolomite, New Mexico. As a quantity of interest, we used the travel time of a radioactive particle departing from the centre of repository to the cite boundary. The transmissivity measurements are available from 39 boreholes in WIPP site. In order to take into account the knowledge of transmissivity values we used the modified circulant embedding algorithm of Newsam and Dietrich, which is also implemented in our C++ library. We observed that the conditioning is more effective at reducing the variance of the estimator on the coarsest level. The conditional MLMC simulation, thus, requires to start from a finer coarsest grid, which could increase the overall complexity of this method. Nevertheless, the advantage of using the MLMC estimator over a standard MC estimator for the groundwater flow simulation, which involves direct measurements of porous medium properties, is still greater. In addition to variance reduction by conditioning, antithetic variates, which is one of widely used variance reduction techniques for the standard Monte Carlo methods, is employed for further variance reduction. We observed that almost 45% reduction in simulation times by using the antithetic variates in MLMC simulation of WIPP site. The results of the second phase are being compared to other uncertainty quantification methods such as quasi Monte Carlo method and stochastic collocation method. |

Exploitation Route | In order to increase the impact of the multilevel Monte Carlo method, we made software implementing the MLMC method which is available via the project website. The software contains procedures that are relevant to this project, including the circulant embedding method, cell-centred finite volume method, algebraic multigrid method, antithetic variates method, coarse grid variate method and the modified circulant embedding method for the conditional random field generation. The MLMC software is written in C++ with generic programming paradigm which makes easier its adaptation and extension to other quantity of interest and other applications without further effort from the user. The software has a potential in evolving from being a research tool to becoming a commercial software package. |

Sectors | Energy,Environment |

URL | https://www.maths.nottingham.ac.uk/personal/pmzmp/mlmc.html |

Description | The findings have not been used beyond academia yet. |

First Year Of Impact | 2014 |