A unified multilevel framework of upscaling and domain decomposition

We consider multiscale preconditioners for a class of mass-conservative domain-decomposition (MCDD) methods. For the application of reservoir simulation, we need to solve large linear systems, arising from finite-volume discretisations of elliptic PDEs with highly variable coefficients. We introduce an algebraic framework, based on probing, for constructing mass-conservative operators on a multiple of coarse scales. These operators may further be applied as coarse spaces for additive Schwarz preconditioners. By applying different local approximations to the Schur complement system based on a careful choice of probing vectors, we show how the MCDD preconditioners can be both efficient preconditioners for iterative methods or accurate upscaling techniques for the heterogeneous elliptic problem. Our results show that the probing technique yield better approximation properties compared with the reduced boundary condition commonly applied with multiscale methods

Mass Conservative Domain Decomposition for Porous Media Flow

publishedVersio

A unified multilevel framework of upscaling and domain decomposition

Presented at CMWR 2010 - XVIII International Conference on Computational Methods in Water Resources, June 21-24, 2010, Barcelona, SpainWe consider multiscale preconditioners for a class of mass-conservative domain-decomposition (MCDD) methods. For the application of reservoir simulation, we need to solve large linear systems, arising from finite-volume discretisations of elliptic PDEs with highly variable coefficients. We introduce an algebraic framework, based on probing, for constructing mass-conservative operators on a multiple of coarse scales. These operators may further be applied as coarse spaces for additive Schwarz preconditioners. By applying different local approximations to the Schur complement system based on a careful choice of probing vectors, we show how the MCDD preconditioners can be both efficient preconditioners for iterative methods or accurate upscaling techniques for the heterogeneous elliptic problem. Our results show that the probing technique yield better approximation properties compared with the reduced boundary condition commonly applied with multiscale methods.publishedVersio

Mathematical Analysis of Similarity Solutions for Modelling Injection of CO2 into Aquifers

Master i Anvendt matematikkMAMN-MABMAB39

Mathematical Analysis of Similarity Solutions for Modelling Injection of CO2 into Aquifers

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Auxiliary variables for 3D multiscale simulations in heterogeneous porous media

The multiscale control-volume methods for solving problems involving flow in porous media have gained much interest during the last decade. Recasting these methods in an algebraic framework allows one to consider them as preconditioners for iterative solvers. Despite intense research on the 2D formulation, few results have been shown for 3D, where indeed the performance of multiscale methods deteriorates. The interpretation of multiscale methods as vertex based domain decomposition methods, which are non-scalable for 3D domain decomposition problems, allows us to understand this loss of performance. We propose a generalized framework based on auxiliary variables on the coarse scale. These are enrichments the coarse scale, which can be selected to improve the interpolation onto the fine scale. Where the existing coarse scale basis functions are designed to capture local sub-scale heterogeneities, the auxiliary variables are aimed at better capturing non-local effects resulting from non-linear behavior of the pressure field. The auxiliary coarse nodes fits into the framework of mass-conservative domain-decomposition (MCDD) preconditioners, allowing us to construct, as special cases, both the traditional (vertex based) multiscale methods as well as their wire basket generalization

Uncertainty Analysis – 5 Challenges with Today's Practice

AbstractAs pointed out by Venkataraman and Pinto (2010), the importance of estimating project costs arises as the estimates become the benchmarks of which future costs are compared and evaluated. Although estimates become more accurate as decisions are made and uncertainties resolved, they are also chief means for assessing project feasibility, as a comparison of cost estimates with estimates of revenues and other benefits that are crucial in determining whether the project is worthwhile to carry out or not. In this paper we will discuss whether or not the uncertainty analysis is a reliable tool for supporting the cost estimation process. We present 5 challenges in connection with the way uncertainty analyses of cost estimates are done today and present findings that indicate a need to rethink the uncertainty analyses of the projects that have a high degree of uncertainty. This paper is a product of collective reflection, experience and the knowledge of the authors. It is of a qualitative nature as we do not present any quantitative or statistical evidence or methods in our approach. It is understood, due to the diverse contextual backgrounds of the projects involved, that the explanations for differences may be equally diverse. The paper is divided into five parts; The introduction – explaining the importance of the topic; part two provides a short introduction to the applied research methods; part three explain what we mean by cost estimation under uncertainty; part four presents the five identified challenges in cost estimat*ion under uncertainty; part five presents a conclusion and proposes potential areas of further research