# Mixed finite element analysis of lognormal diffusion and multilevel Monte Carlo methods

@article{Graham2013MixedFE, title={Mixed finite element analysis of lognormal diffusion and multilevel Monte Carlo methods}, author={Ivan G. Graham and Robert Scheichl and Elisabeth Ullmann}, journal={Stochastics and Partial Differential Equations Analysis and Computations}, year={2013}, volume={4}, pages={41-75} }

This work is motivated by the need to develop efficient tools for uncertainty quantification in subsurface flows associated with radioactive waste disposal studies. We consider single phase flow problems in random porous media described by correlated lognormal distributions. We are interested in the error introduced by a finite element discretisation of these problems. In contrast to several recent works on the analysis of standard nodal finite element discretisations, we consider here mass…

## 23 Citations

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An abstract, problem-dependent theorem is given on the cost of the new multilevel estimator based on a set of simple, verifiable assumptions for a typical model problem in subsurface flow and shows significant gains over the standard Metropolis--Hastings estimator.

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In this work, we present a numerical analysis of a probabilistic approach to quantify the migration of a contaminant, under the presence of uncertainty on the permeability of the porous medium. More…

### Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients

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- 2013

It is proved that convergence of the multilevel Monte Carlo algorithm for estimating any bounded, linear functional and any continuously Fréchet differentiable non-linear functional of the solution is convergence.

### Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients

- Mathematics, Computer ScienceNumerische Mathematik
- 2013

It is proved that convergence of the multilevel Monte Carlo algorithm for estimating any bounded, linear functional and any continuously Fréchet differentiable non-linear functional of the solution is convergence.

### Computable Error Estimates for Finite Element Approximations of Elliptic Partial Differential Equations with Rough Stochastic Data

- Mathematics, Computer ScienceSIAM J. Sci. Comput.
- 2016

This work proposes goal-oriented estimates, based on local error indicators, for the pathwise Galerkin and expected quadrature errors committed in standard, continuous, piecewise linear finite element approximations of linear elliptic partial differential equations with rough stochastic coefficients.

### Multilevel Monte Carlo and improved timestepping methods in atmospheric dispersion modelling

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- 2018

### Multi-index Stochastic Collocation Convergence Rates for Random PDEs with Parametric Regularity

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- 2016

We analyze the recent Multi-index Stochastic Collocation (MISC) method for computing statistics of the solution of a partial differential equation (PDE) with random data, where the random coefficient…

### Multi-index Stochastic Collocation Convergence Rates for Random PDEs with Parametric Regularity

- Mathematics, Computer ScienceFound. Comput. Math.
- 2016

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The scalability of the sampling method with nonmatching mesh embedding, coupled with a parallel forward model problem solver, for large‐scale 3D MLMC simulations with up to 1.9·109 unknowns is demonstrated.

### Circulant embedding with QMC: analysis for elliptic PDE with lognormal coefficients

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A convergence analysis for the quasi-Monte Carlo method in the case when the QMC method is a specially designed randomly shifted lattice rule is provided, which can be independent of the number of stochastic variables under certain assumptions.

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