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This research investigates the effects of different optimization norms in Anderson acceleration applied to Picard iterations for Navier-Stokes equations.

June 26, 2026Open Access

On the Choice of Optimization Norm for Anderson Acceleration of the Picard Iteration for Navier–Stokes Equations

Key Points

This research investigates the effects of different optimization norms in Anderson acceleration applied to Picard iterations for Navier-Stokes equations.
Examined varying AA optimization norms in numerical implementations of Picard iterations for Navier-Stokes equations.
Proved generalized convergence estimates for multiple optimization norms.
Conducted numerical tests comparing convergence iterations needed across different optimization norms.
Observed comparable convergence for different norms under shared conditions in numerical tests.
Noted significant performance differences in convergence iterations for specific flows, particularly when using certain norms.

Abstract

ABSTRACT While recent Anderson acceleration (AA) convergence theory Pollock et al., IMA Num. An. , 2021 requires that the AA optimization norm match the Hilbert space norm associated with the fixed point operator, in implementations the norm is the most common choice. So far there is little research done regarding this discrepancy. To address this issue, we consider AA applied to the Picard iteration for the Navier–Stokes equations (NSE) with varying choices of the AA optimization norm. We prove a sharpened and generalized convergence estimate with the AA optimization norm, and an essentially equivalent (in terms of associated Lipschitz constants) result for when is used. While no analogous theory seems possible for , numerical tests were run to compare varying choices of AA optimization norms. These tests revealed similar convergence for and in terms of iterations needed for convergence and usually but not always similar for : for flow past a cylinder, convergence using performs significantly worse.

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