A Tree-Based Localized Reduced Basis Method for Parametrized Parabolic PDEs
Abstract
We propose a localized reduced basis method for parametrized parabolic partial differential equations based on an adaptive partitioning of the parameter domain using hierarchical binary trees. A POD-Greedy procedure is employed to construct reduced spaces associated with each parameter subdomain. A certified \emph{a posteriori} error estimator is derived from an inf-sup stability analysis and shown to be equivalent to the underlying approximation error. This equivalence, together with suitable regularity assumptions on the parameter-to-solution map, enables a rigorous convergence analysis and is used to derive a strategy for selecting the number of POD modes to retain in each local reduced space. Numerical experiments for a parametrized convection--diffusion problem demonstrate the effectiveness of the proposed approach and confirm the predicted convergence behavior.
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BibTeX
@misc{barakat2026treebased,
title = {{A Tree-Based Localized Reduced Basis Method for Parametrized Parabolic PDEs}},
author = {Mohamed Barakat and Diane Guignard},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36141},
eprint = {2609.36141},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36141},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Tree-Based Localized Reduced Basis Method for Parametrized Parabolic PDEs
AU - Mohamed Barakat
AU - Diane Guignard
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.36141
UR - https://arxiv.org/abs/2609.36141
AB - We propose a localized reduced basis method for parametrized parabolic partial differential equations based on an adaptive partitioning of the parameter domain using hierarchical binary trees. A POD-Greedy procedure is employed to construct reduced spaces associated with each parameter subdomain. A certified \emph{a posteriori} error estimator is derived from an inf-sup stability analysis and shown to be equivalent to the underlying approximation error. This equivalence, together with suitable regularity assumptions on the parameter-to-solution map, enables a rigorous convergence analysis and is used to derive a strategy for selecting the number of POD modes to retain in each local reduced space. Numerical experiments for a parametrized convection--diffusion problem demonstrate the effectiveness of the proposed approach and confirm the predicted convergence behavior.
ER -