On the choice of optimization norm for Anderson acceleration of the Picard iteration for Navier-Stokes equations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914051779461120 |
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| author | Hawkins, Elizabeth Rebholz, Leo |
| author_facet | Hawkins, Elizabeth Rebholz, Leo |
| contents | While the most recent Anderson acceleration (AA) convergence theory [Pollock et al, {\it IMA Num. An.}, 2021] requires that the AA optimization norm match the Hilbert space norm associated with the fixed point operator, in implementations the $\ell^2$ norm is perhaps the most common choice. Unfortunately, so far there is little research done regarding this discrepancy which might reveal when it is fine to use $\ell^2$. 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 first prove a sharpened and generalized convergence estimate for depth $m$ AA-Picard for the NSE with the $H^1_0$ AA optimization norm by using a problem-specific analysis, utilizing a sharper treatment of the nonlinear terms than previous AA-Picard convergence studies, removing a small data assumption, and developing new AA term identities in the NSE nonlinear term estimates. Next, we prove a convergence result for when $L^2$ is used as the AA optimization norm, and this estimate is found to be very similar to that of the $H^1_0$ case. While no analogous theory seems possible for the $\ell^2$ norm, several numerical tests were run to compare AA-Picard convergence with varying choices of AA optimization norm. These tests revealed that convergence behavior was always similar for $L^2$ and $H^1_0$ and {\it usually but not always} similar for $\ell^2$: on a test problem for channel flow past a cylinder with coarser meshes, convergence of AA-Picard using $\ell^2$ performs significantly worse than using $L^2$ and $H^1_0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07650 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the choice of optimization norm for Anderson acceleration of the Picard iteration for Navier-Stokes equations Hawkins, Elizabeth Rebholz, Leo Numerical Analysis While the most recent Anderson acceleration (AA) convergence theory [Pollock et al, {\it IMA Num. An.}, 2021] requires that the AA optimization norm match the Hilbert space norm associated with the fixed point operator, in implementations the $\ell^2$ norm is perhaps the most common choice. Unfortunately, so far there is little research done regarding this discrepancy which might reveal when it is fine to use $\ell^2$. 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 first prove a sharpened and generalized convergence estimate for depth $m$ AA-Picard for the NSE with the $H^1_0$ AA optimization norm by using a problem-specific analysis, utilizing a sharper treatment of the nonlinear terms than previous AA-Picard convergence studies, removing a small data assumption, and developing new AA term identities in the NSE nonlinear term estimates. Next, we prove a convergence result for when $L^2$ is used as the AA optimization norm, and this estimate is found to be very similar to that of the $H^1_0$ case. While no analogous theory seems possible for the $\ell^2$ norm, several numerical tests were run to compare AA-Picard convergence with varying choices of AA optimization norm. These tests revealed that convergence behavior was always similar for $L^2$ and $H^1_0$ and {\it usually but not always} similar for $\ell^2$: on a test problem for channel flow past a cylinder with coarser meshes, convergence of AA-Picard using $\ell^2$ performs significantly worse than using $L^2$ and $H^1_0$. |
| title | On the choice of optimization norm for Anderson acceleration of the Picard iteration for Navier-Stokes equations |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2505.07650 |