%0 Journal Article %1 WOS:000453873700016 %A Barth, Andrea %A Stein, Andreas %C 3600 UNIV CITY SCIENCE CENTER, PHILADELPHIA, PA 19104-2688 USA %D 2018 %I SIAM PUBLICATIONS %J SIAM-ASA JOURNAL ON UNCERTAINTY QUANTIFICATION %K imported from:brittalenz ians %N 4 %P 1707-1743 %R 10.1137/17M1148888 %T A Study of Elliptic Partial Differential Equations with Jump Diffusion Coefficients %V 6 %X As a simplified model for subsurface flows, elliptic equations may be utilized. Insufficient measurements or uncertainty in those are commonly modeled by a random coefficient, which then accounts for the uncertain permeability of a given medium. As an extension of this methodology to flows in heterogeneous/fractured/porous media, we incorporate jumps in the diffusion coefficient. These discontinuities then represent transitions in the media. More precisely, we consider a second order elliptic problem where the random coefficient is given by the sum of a (continuous) Gaussian random field and a (discontinuous) jump part. To estimate moments of the solution to the resulting random partial differential equation, we use a pathwise numerical approximation combined with multilevel Monte Carlo sampling. In order to account for the discontinuities and improve the convergence of the pathwise approximation, the spatial domain is decomposed with respect to the jump positions in each sample, leading to path-dependent grids. Hence, it is not possible to create a nested sequence of grids which is suitable for each sample path a priori. We address this issue by an adaptive multilevel algorithm, where the discretization on each level is sample-dependent and fulfills given refinement conditions.

@article{WOS:000453873700016, abstract = {As a simplified model for subsurface flows, elliptic equations may be utilized. Insufficient measurements or uncertainty in those are commonly modeled by a random coefficient, which then accounts for the uncertain permeability of a given medium. As an extension of this methodology to flows in heterogeneous/fractured/porous media, we incorporate jumps in the diffusion coefficient. These discontinuities then represent transitions in the media. More precisely, we consider a second order elliptic problem where the random coefficient is given by the sum of a (continuous) Gaussian random field and a (discontinuous) jump part. To estimate moments of the solution to the resulting random partial differential equation, we use a pathwise numerical approximation combined with multilevel Monte Carlo sampling. In order to account for the discontinuities and improve the convergence of the pathwise approximation, the spatial domain is decomposed with respect to the jump positions in each sample, leading to path-dependent grids. Hence, it is not possible to create a nested sequence of grids which is suitable for each sample path a priori. We address this issue by an adaptive multilevel algorithm, where the discretization on each level is sample-dependent and fulfills given refinement conditions.}, added-at = {2021-09-13T10:24:35.000+0200}, address = {3600 UNIV CITY SCIENCE CENTER, PHILADELPHIA, PA 19104-2688 USA}, affiliation = {Barth, A (Corresponding Author), Univ Stuttgart, IANS SimTech, Stuttgart, Germany. Barth, Andrea; Stein, Andreas, Univ Stuttgart, IANS SimTech, Stuttgart, Germany.}, author = {Barth, Andrea and Stein, Andreas}, author-email = {andrea.barth@mathematik.uni-stuttgart.de andreas.stein@mathematik.uni-stuttgart.de}, biburl = {https://puma.ub.uni-stuttgart.de/bibtex/28cbdb621961cdba1ad33e1ae663468e8/mathematik}, da = {2021-08-10}, doc-delivery-number = {HF0SR}, doi = {10.1137/17M1148888}, funding-acknowledgement = {German Research Foundation (DFG) as part of the Cluster of Excellence in Simulation Technology at the University of Stuttgart {[}EXC 310/2]; junior professorship program of Baden-Wurttemberg}, funding-text = {This research was supported by the German Research Foundation (DFG) as part of the Cluster of Excellence in Simulation Technology (EXC 310/2) at the University of Stuttgart and by the junior professorship program of Baden-Wurttemberg.}, interhash = {4d5bdeac74652866e67e029befd67cc0}, intrahash = {8cbdb621961cdba1ad33e1ae663468e8}, issn = {2166-2525}, journal = {SIAM-ASA JOURNAL ON UNCERTAINTY QUANTIFICATION}, journal-iso = {SIAM-ASA J. Uncertain. Quantif.}, keywords = {imported from:brittalenz ians}, language = {English}, number = 4, number-of-cited-references = {40}, oa = {Green Submitted}, pages = {1707-1743}, publisher = {SIAM PUBLICATIONS}, research-areas = {Mathematics; Physics}, times-cited = {0}, timestamp = {2023-12-07T16:38:56.000+0100}, title = {A Study of Elliptic Partial Differential Equations with Jump Diffusion Coefficients}, type = {Article}, unique-id = {WOS:000453873700016}, usage-count-last-180-days = {0}, usage-count-since-2013 = {0}, volume = 6, web-of-science-categories = {Mathematics, Interdisciplinary Applications; Physics, Mathematical}, year = 2018 }