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RadCluster
Graph-based cluster dynamics for irradiated materials
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For the GMRES linear solver the Jacobian has the structure
\[ J = T + U V^T , \]
where \(T\) is banded, with half-bandwidth \(\max(2\,i_{mobile}, 2\,v_{mobile}) + 1\), and \(U V^T\) is a correction of rank \(r = i_{mobile} + v_{mobile}\) that comes from the coupling of the mobile species.
The Woodbury preconditioner exploits this structure exactly. It uses the LAPACK banded factorization (dgbtrf and dgbtrs) for \(T\), and a dense factorization (dgetrf and dgetrs) of the \(r \times r\) Schur complement.
| prec_type | Name | Description |
|---|---|---|
| 0 | Jacobi | diagonal scaling |
| 1 | Woodbury | bordered-banded Sherman-Morrison-Woodbury preconditioner |
The Woodbury preconditioner is the default for the full_system mode with GMRES. For the active_window mode the active system is small, and Jacobi with GMRES converges faster than the set-up cost of the Woodbury preconditioner can be recovered.
The parameters prec_bw and prec_rank set the bandwidth and the rank. They are selected automatically by default.
The preconditioner is implemented in radcluster_code/cpp_utils/core/rhs_dispatch.cpp. It probes the right-hand side through the function pointer UserData::rhs_fn, and it is therefore independent of the material model. Its derivation is in the folder docs/Formulation/reaction_admissibility_graph/supplement_S1 of the repository.
Note that the preconditioner needs LAPACK. Without LAPACK it is compiled out, and the solver uses the Jacobi preconditioner.
1.9.8