Optimal transfer operators in algebraic two-level methods for nonsymmetric and indefinite problems
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| Format: | Preprint |
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2025
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| author | Krzysik, Oliver A. Southworth, Ben S. Wimmer, Golo A. Ali, Ahsan Brannick, James Kahl, Karsten |
| author_facet | Krzysik, Oliver A. Southworth, Ben S. Wimmer, Golo A. Ali, Ahsan Brannick, James Kahl, Karsten |
| contents | Consider an algebraic two-level method applied to the $n$-dimensional linear system $A \mathbf{x} = \mathbf{b}$ using fine-space preconditioner (i.e., ``relaxation'' or ``smoother'') $M$, with $M \approx A$, restriction and interpolation $R$ and $P$, and algebraic coarse-space operator ${A_c := R^*AP}$. Then, what are the the best possible transfer operators $R$ and $P$ of a given dimension $n_c < n$? Brannick et al. (2018) showed that when $A$ and $M$ are Hermitian positive definite (HPD), the optimal interpolation is such that its range contains the $n_c$ smallest generalized eigenvectors of the matrix pencil $(A, M)$. Recently, in Ali et al. (2025) we generalized this framework to the non-HPD setting, by considering both right (interpolation) and left (restriction) generalized eigenvectors of $(A, M)$ and defining corresponding nonsymmetric transfer operators $\{R_\#,P_\#\}$. Tight convergence bounds for $\{R_\#,P_\#\}$ are derived in spectral radius, as well as a proof of pseudo-optimality. Note, $\{R_\#,P_\#\}$ are typically complex valued, which is not practical for real-valued problems. Here we build on Ali et al. (2025), first characterizing all inner products in which the coarse-space correction defined by $\{R_\#,P_\#\}$ is orthogonal. We then develop tight two-level convergence bounds in these norms, and prove that the underlying transfer operators $\{R_\#,P_\#\}$ are genuinely optimal. As a special case, our theory both recovers and extends the HPD results from Brannick et al. (2018). Finally, we show how to construct optimal, real-valued transfer operators in the case of that $A$ and $M$ are real valued, but are not HPD. Numerical examples arising from discretized advection and wave-equation problems are used to verify and illustrate the theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_05598 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal transfer operators in algebraic two-level methods for nonsymmetric and indefinite problems Krzysik, Oliver A. Southworth, Ben S. Wimmer, Golo A. Ali, Ahsan Brannick, James Kahl, Karsten Numerical Analysis Consider an algebraic two-level method applied to the $n$-dimensional linear system $A \mathbf{x} = \mathbf{b}$ using fine-space preconditioner (i.e., ``relaxation'' or ``smoother'') $M$, with $M \approx A$, restriction and interpolation $R$ and $P$, and algebraic coarse-space operator ${A_c := R^*AP}$. Then, what are the the best possible transfer operators $R$ and $P$ of a given dimension $n_c < n$? Brannick et al. (2018) showed that when $A$ and $M$ are Hermitian positive definite (HPD), the optimal interpolation is such that its range contains the $n_c$ smallest generalized eigenvectors of the matrix pencil $(A, M)$. Recently, in Ali et al. (2025) we generalized this framework to the non-HPD setting, by considering both right (interpolation) and left (restriction) generalized eigenvectors of $(A, M)$ and defining corresponding nonsymmetric transfer operators $\{R_\#,P_\#\}$. Tight convergence bounds for $\{R_\#,P_\#\}$ are derived in spectral radius, as well as a proof of pseudo-optimality. Note, $\{R_\#,P_\#\}$ are typically complex valued, which is not practical for real-valued problems. Here we build on Ali et al. (2025), first characterizing all inner products in which the coarse-space correction defined by $\{R_\#,P_\#\}$ is orthogonal. We then develop tight two-level convergence bounds in these norms, and prove that the underlying transfer operators $\{R_\#,P_\#\}$ are genuinely optimal. As a special case, our theory both recovers and extends the HPD results from Brannick et al. (2018). Finally, we show how to construct optimal, real-valued transfer operators in the case of that $A$ and $M$ are real valued, but are not HPD. Numerical examples arising from discretized advection and wave-equation problems are used to verify and illustrate the theory. |
| title | Optimal transfer operators in algebraic two-level methods for nonsymmetric and indefinite problems |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2505.05598 |