Yvon MadayChaire Informatique et sciences numériquesCollège de FranceAnnée 2025-2026Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Karen E. Willcox : Multifidelity Proper Orthogonal DecompositionKaren E. WillcoxProfessor, Director of Oden Institute, University of Texas at Austin, USARésuméThe proper orthogonal decomposition (POD) is widely used to compute a low-dimensional basis that underpins a subsequent dimension reduction or reduced-order modeling step. POD is data-driven in the sense that it requires a training data set of high-fidelity solutions, typically referred to as snapshots. For many complex scientific applications, the computational cost of generating these snapshots is prohibitive, especially when their generation requires sampling over a high-dimensional parameter space. This talk presents a multifidelity POD (mfPOD) formulation that leverages cheaper, lower-fidelity snapshots to reduce the computational cost of computing the POD basis. MFPOD then weights high- and low-fidelity snapshot data via a control-variate formulation to guarantee an unbiased estimate of the expected high-fidelity least-squares projection error. For restrictive computational budgets, the MFPOD cost function has (under some assumptions) lower variance than the POD cost function, which makes the MFPOD subspace more robust against variations in the training data and thus less prone to overfitting. Numerical results show that mfPOD achieves an order of magnitude in computational speedup, translating into useful gains in large-scale problems. Joint work with Nicole Aretz.

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