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
Abstracts are not currently available in GtR for all funded research. This is normally because the abstract was not required at the time of proposal submission, but may be because it included sensitive information such as personal details.
Publications
Cliffe K
(2011)
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 |