{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/reduced-order-modeling-for-elliptic-problems","title":"Reduced order modeling for elliptic problems with high contrast diffusion coefficients","arxiv_id":"2304.10971","date":"2023-04-21","proceeding":null,"authors":["Albert Cohen","Matthieu Dolbeault","Agustin Somacal","Wolfgang Dahmen"],"abstract":"We consider the parametric elliptic PDE $-{\\rm div} (a(y)\\nabla u)=f$ on a spatial domain $\\Omega$, with $a(y)$ a scalar piecewise constant diffusion coefficient taking any positive values $y=(y_1, \\dots, y_d)\\in ]0,\\infty[^d$ on fixed subdomains $\\Omega_1,\\dots,\\Omega_d$. This problem is not uniformly elliptic as the contrast $\\kappa(y)=\\frac{\\max y_j}{\\min y_j}$ can be arbitrarily high, contrarily to the Uniform Ellipticity Assumption (UEA) that is commonly made on parametric elliptic PDEs. Based on local polynomial approximations in the $y$ variable, we construct local and global reduced model spaces $V_n$ of moderate dimension $n$ that approximate uniformly well all solutions $u(y)$. Since the solution $u(y)$ blows as $y\\to 0$, the solution manifold is not a compact set and does not have finite $n$-width. Therefore, our results for approximation by such spaces are formulated in terms of relative $H^1_0$-projection error, that is, after normalization by $\\|u(y)\\|_{H^1_0}$. We prove that this relative error decays exponentially with $n$, yet exhibiting the curse of dimensionality as the number $d$ of subdomains grows. We also show similar rates for the Galerkin projection despite the fact that high contrast is well-known to deteriorate the multiplicative constant when applying Cea's lemma. We finally establish uniform estimates in relative error for the state estimation and parameter estimation inverse problems, when $y$ is unknown and a limited number of linear measurements $\\ell_i(u)$ are observed. A key ingredient in our construction and analysis is the study of the convergence of $u(y)$ to limit solutions when some of the parameters $y_j$ tend to infinity.","url_abs":"https://arxiv.org/abs/2304.10971v1","url_pdf":"https://arxiv.org/pdf/2304.10971v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"reduced-order-modeling-for-elliptic-problems","repo_url":"https://github.com/agussomacal/romhighcontrast","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}