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- K. A. Cliffe, Michael B. Giles, Robert Scheichl, Aretha L. Teckentrup
- Computat. and Visualiz. in Science
- 2011

We consider the numerical solution of elliptic partial differential equations with random coefficients. Such problems arise, for example, in uncertainty quantification for groundwater flow. We describe a novel variance reduction technique for the standard Monte Carlo method, called the multilevel Monte Carlo method. The main result is that in certain… (More)

- Ivan G. Graham, P. O. Lechner, Robert Scheichl
- Numerische Mathematik
- 2007

We consider additive Schwarz domain decomposition preconditioners for piecewise linear finite element approximations of elliptic PDEs with highly variable coefficients. In contrast to standard analyses, we do not assume that the coefficients can be resolved by a coarse mesh. This situation arises often in practice, for example in the computation of flows in… (More)

- Julia Charrier, Robert Scheichl, Aretha L. Teckentrup
- SIAM J. Numerical Analysis
- 2013

We consider a finite element approximation of elliptic partial differential equations with random coefficients. Such equations arise, for example, in uncertainty quantification in subsurface flow modelling. Models for random coefficients frequently used in these applications, such as log-normal random fields with exponential covariance, have only very… (More)

In this paper we describe a new class of domain deomposition preconditioners suitable for solving elliptic PDEs in highly fractured or heterogeneous media, such as arise in groundwater flow or oil recovery applications. Our methods employ novel coarsening operators which are adapted to the heterogeneity of the media. In contrast to standard methods (based… (More)

In this paper we present a new preconditioner suitable for solving linear systems arising from finite element approximations of elliptic PDEs with high-contrast coefficients. The construction of the preconditioner consists of two phases. The first phase is an algebraic one which partitions the degrees of freedom into “high” and “low” permeability regions… (More)

We study two–level overlapping domain decomposition preconditioners with coarse spaces obtained by smoothed aggregation in iterative solvers for finite element discretisations of second-order elliptic problems. We are particularly interested in the situation where the diffusion coefficient (or the permeability) α is highly variable throughout the domain.… (More)

- Aretha L. Teckentrup, Robert Scheichl, Michael B. Giles, Elisabeth Ullmann
- Numerische Mathematik
- 2013

We consider the application of multilevel Monte Carlo methods to elliptic PDEs with random coefficients. We focus on models of the random coefficient that lack uniform ellipticity and boundedness with respect to the random parameter, and that only have limited spatial regularity. We extend the finite element error analysis for this type of equation, carried… (More)

- Robert Scheichl, Panayot S. Vassilevski, Ludmil Zikatanov
- SIAM J. Numerical Analysis
- 2012

In this paper we generalize the analysis of classical multigrid and two-level overlapping Schwarz methods for 2nd order elliptic boundary value problems to problems with large discontinuities in the coefficients that are not resolved by the coarse grids or the subdomain partition. The theoretical results provide a recipe for designing hierarchies of… (More)

In this paper we discuss new domain decomposition preconditioners for piecewise linear finite element discretisations of boundary-value problems for the model elliptic problem −∇ · (A∇u) = f , (1) in a bounded polygonal or polyhedral domain Ω ⊂ R, d = 2 or 3 with suitable boundary data on the boundary ∂Ω. The tensor A(x) is assumed isotropic and symmetric… (More)

- Ivan G. Graham, Frances Y. Kuo, Dirk Nuyens, Robert Scheichl, Ian H. Sloan
- J. Comput. Physics
- 2011