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

Lead Research Organisation: University of Bath
Department Name: Mathematical Sciences


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.

Planned Impact

WHO WILL BENEFIT? Uncertainty quantification, in particular the probabilistic approach to uncertainty quantification, is a topic of growing importance for industry, business, the environment and the government. Decisions about where to build radioactive waste repositories or whether it is safe to try to mitigate the influence of CO2 on our climate via capture and storage underground, depend on an accurate quantification of the uncertainties inherent in modelling porous media flow. A number of very important areas would benefit directly from the outcomes of the research: * The radioactive waste disposal industry. Dealing with the radioactive waste is of considerable importance because, in the UK at least, further deployment of nuclear power depends on there being sufficient progress on the waste problem to convince the Government and the general population that it can be dealt with in a safe manner. * The water resources management sector, where uncertainty quantification has always played a major role. The impact of climate change means that water will be an even more valuable resource in the future and this sector will increase in importance. * The carbon capture and storage sector. This area is of extremely high current interest and is likely to become a major industrial sector in the future with important implications for tackling climate change. A number of other areas will also benefit, but the proposal does not address these applications directly: * The medical and pharmaceutical industries and other biological research sectors. There are many examples where porous media models are used to model biological systems. By their nature, these models are always subject to uncertainties. * The hydrocarbon industry. There are large economic implications for uncertainties in the hydrocarbon industry. HOW WILL THEY BENEFIT? In all the above areas the most costly computational component of a risk analysis is the solution of partial differential equations (PDEs) with uncertain coefficients, and so if the proposed research is successful, it will have enormous impact on all those fields and many more. In the safety assessments of the Sellafield site, carried out by Nirex in the mid 1990s, the treatment of uncertainty in the groundwater flow at the repository was severely limited by the computational cost. Future assessments will have to address the issue in a more comprehensive manner, and similar comments could be made regarding many of the areas mentioned above. If the proposed research is successful, it will mean that uncertainty quantification will be possible in many application areas in an affordable way where it was previously prohibitively expensive. The impact for basic and applied research and for industry is considerable. WHAT WILL BE DONE TO ENSURE THEY BENEFIT? The primary beneficiary is the radioactive waste disposal industry in the UK. It will benefit by the close involvement of the Radioactive Waste Management Directorate of the Nuclear Decommissioning Authority (NDA), the organisation with responsibility for the safe disposal or radioactive waste in the UK, and of Serco Technical and Assurance Services (TAS), a leading supplier of consultancy services and software to the radioactive waste industry worldwide. Both the NDA and Serco TAS are supporting this project with substantial in-kind contributions. They will also have representatives on an advisory board, which will provide advice on the technical direction of the project and on how to communicate the research results to the communities they represent. Other members of the board will represent the hydrology, the hydrocarbon and the carbon capture and storage sectors. Serco TAS will commercialise any software developed during the project while making it freely available to the academic community for teaching and research purposes. This will maximise the industrial and economic impact of the project.


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Cliffe K (2011) Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients in Computing and Visualization in Science

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Detommaso G (2019) Continuous Level Monte Carlo and Sample-Adaptive Model Hierarchies in SIAM/ASA Journal on Uncertainty Quantification

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Dolean V (2012) Analysis of a Two-level Schwarz Method with Coarse Spaces Based on Local Dirichlet-to-Neumann Maps in Computational Methods in Applied Mathematics

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Drzisga D (2017) Scheduling Massively Parallel Multigrid for Multilevel Monte Carlo Methods in SIAM Journal on Scientific Computing

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Graham I (2018) Analysis of Circulant Embedding Methods for Sampling Stationary Random Fields in SIAM Journal on Numerical Analysis

Description We have developed, analysed and implemented a novel uncertainty quantification tool, particularly suited for very large scale systems with a large number of uncertain parameters (in particular in the form of random fields): the multilevel Monte Carlo method. It had been applied previously only in the context of SDEs in finance. Here we studied it in the context of subsurface flow and in particular in the safety assessment of longterm radioactive waste disposal underground. The gains in the context of uncertainty quantification are significantly higher than in the SDE case.

With respect to the theoretical aspects, we significantly exceeded our research plan, giving in essence an almost complete analysis of the method in the physically relevant situation of lognormal coefficients and numerical experiments backing up all our theoretical findings. We extended the analysis to various discretisation schemes, to the case of Monte Carlo Markov chain (of huge importance to incorporate also data). And we applied the method to the WIPP test case and to some 3D model problems in radioactive waste disposal.

In parallel, we also extended the analysis of quasi-Monte Carlo methods applied to the same model problems, as well as combining the two approaches to the most efficient novel method, the multilevel QMC method - the analysis of which is essentially also completed and will be published in due course in a follow-up paper, as well as a comparison paper of all our new methods with stochastic collocation.

Another set of results that arose within the project were new multiscale and multilevel iterative techniques for solving highly heterogeneous elliptic partial differential equations efficiently and robustly, as well as their rigorous analysis. This is crucial, since the bottleneck in the novel UQ tools above ed up being the deterministic finite element solver for individual samples.
Exploitation Route We made them widely known mainly through publications in the leading international journals and through talks (many keynotes; also to engineers - oil, hydrology, aerospace, meteorology, ...).

It has spawned a whole new field of research, with academic positions on the topic being created in Germany or the US and entire minisymposia or even workshops devoted to them. The publications listed have been cited more than 450 times in the last 4 years (source: Google Scholar), which is a very big number for mathematics.

The methodology can be applied in all areas of science and engineering and people are actively applying it in a huge range of fields. In our group, we have a new EPSRC funded project together with GKN Aerospace and Mechanical Engineering in Bath where we apply the method in the context of uncertainty quantification in aerospace composites.

Through the Nuclear Decommissioning Authority (NDA) and their contractor AMEC, the specific findings in the context of radioactive waste disposal will be taken forward.
Sectors Aerospace, Defence and Marine,Energy,Environment,Manufacturing, including Industrial Biotechology

Description The project has been conducted in close collaboration with the government's Nuclear Decommissioning Authority (NDA) and their main contractor AMEC. The methods have so far only been tested on model problems and on one benchmark problem. The codes have not yet been implemented in AMEC's commercial codes or used in actual assessment of waste disposal sites by the NDA, but all the essential groundwork has been done. As a consequence of this project we have obtained EPSRC funding (both through grants and through DTG and CDT PhD studentships) as well as industry contributions from DNV GL, from GKN Aerospace, from the Met Office and from AMEC FW to investigate multilevel Monte Carlo methods in the context of other applications in aerospace composites manufacture, in risk assessment of chemical plants and oil platforms, in atmospheric dispersion and in nuclear reactor simulation. There are also huge amounts of indirect impacts of this research that are difficult to quantify, as the method is seeing such a huge uptake across disciplines.
First Year Of Impact 2012
Sector Aerospace, Defence and Marine,Energy,Environment,Manufacturing, including Industrial Biotechology
Impact Types Societal,Economic

Description EPSRC Maths for Manufacturing Grant
Amount £499,000 (GBP)
Funding ID EP/K031368/1 
Organisation Engineering and Physical Sciences Research Council (EPSRC) 
Sector Public
Country United Kingdom
Start 01/2014 
End 07/2017
Description Impact Acceleration Grant with DNV GL (risk assessment company)
Amount £56,000 (GBP)
Organisation Engineering and Physical Sciences Research Council (EPSRC) 
Sector Public
Country United Kingdom
Start 05/2014 
End 09/2014
Description Industrial CASE Studentship
Amount £97,000 (GBP)
Organisation Engineering and Physical Sciences Research Council (EPSRC) 
Sector Public
Country United Kingdom
Start 10/2013 
End 03/2017
Description Multilevel Markov Chain Monte Carlo Methods
Amount £8,600 (GBP)
Organisation Lawrence Livermore National Laboratory 
Sector Public
Country United States
Start 07/2012 
End 08/2012
Description Postdoctoral Fellowship
Amount £279,473 (GBP)
Funding ID EP/M019004/1 
Organisation Engineering and Physical Sciences Research Council (EPSRC) 
Sector Public
Country United Kingdom
Start 01/2016 
End 12/2018