Abstract:
Recent convergence theory for Anderson Acceleration (AA) requires that the AA optimization norm coincide with the Hilbert space norm associated with the fixed-point operator. In practical implementations, however, the $\ell^2$ norm is among the most commonly selected norms.
Limited research has addressed this discrepancy or identified conditions under which the $\ell^2$ norm is theoretically justified. To investigate this issue, AA is applied to the Picard iteration for the Navier-Stokes Equations (NSE) with varying choices of optimization norm. A sharpened and generalized convergence estimate is first established for depth $m$ Anderson acceleration–Picard applied to the NSE when the $H^1_0$ optimization norm is employed. The analysis is problem-specific and incorporates a refined treatment of the nonlinear terms relative to prior AA-Picard convergence studies. The small-data assumption present in earlier analyses is removed. New Anderson acceleration term identities are developed to strengthen the nonlinear term estimates. A convergence result is then proved for the case in which the $L^2$ optimization norm is used. This estimate is shown to be closely aligned with the corresponding $H^1_0$ result. Although an analogous theoretical framework does not appear attainable for the $\ell^2$ norm, several numerical experiments are conducted to compare AA-Picard convergence under different optimization norm selections. The results indicate that convergence behavior is consistently similar for the $L^2$ and $H^1_0$ norms and generally, though not universally, similar for the $\ell^2$ norm. In a benchmark problem involving channel flow past a cylinder on relatively coarse meshes, AA-Picard employing the $\ell^2$ norm exhibits substantially poorer convergence than when the $L^2$ or $H^1_0$ norms are used.
Speaker’s Bio:
Dr. Leo Rebholz is a Dean’s Distinguished Professor in the School of Mathematical and Statistical Sciences at Clemson University. His scholarly work encompasses a broad spectrum of computational fluid dynamics, including the numerical analysis and development of stable, high-fidelity discretization techniques for incompressible flow models. In recent years, his research has emphasized data assimilation methodologies and the acceleration of nonlinear solvers, with particular focus on enhancing robustness, convergence properties, and computational efficiency for large-scale, multiphysics simulations.
Dr. Rebholz has authored more than 130 peer-reviewed journal articles and five scholarly monographs. He has supervised and mentored an extensive cohort of graduate and undergraduate researchers, including 13 doctoral dissertations, 14 master’s theses, and 15 undergraduate research projects, contributing significantly to the advancement of computational and applied mathematics research training.
