Develop an effective matrix-free preconditioner for the AKV eigensolver

Construct a cheaply generated preconditioner that enables the matrix-free ARPACK-based approximate Killing vector spin algorithm to converge robustly and at lower total cost than explicitly constructing the dense matrices M and B, including for highly deformed black-hole horizons.

Background

The proposed ARPACK-based algorithm currently constructs the full dense matrices M and B and uses an LU factorization to apply the shift-inverted operator. This approach is robust but retains the costly matrix-construction step, which is the principal bottleneck at the resolutions considered.

A matrix-free alternative would apply the available operator-action functions f_M and f_B directly and use an iterative linear solver such as GMRES or BiCGSTAB. However, the iterative solver requires a good, cheaply constructed preconditioner, particularly because the shifted system can be poorly conditioned and highly deformed horizons can produce slow convergence. The authors report that they were unable to identify preconditioners that make the complete matrix-free method cheaper than full matrix construction, leaving the development of such a preconditioner unresolved.

References

We have shown that the new algorithm based on the eigensystem solver (ARPACK) and LU decomposition is close to two orders of magnitude faster than the original one based on the dense eigensystem solver (LAPACK). If we include the time required to construct the matrices, which is required by both algorithms, then the speedup is closer to one order of magnitude. We tried to use a fully matrix-free algorithm, but were unable to find preconditioners for which the overall cost of the iterations and the preconditioner construction was lower than the cost of constructing the full matrix. However, the idea of using the fixed full matrix for a number of timesteps as a preconditioner is a promising future improvement to explore.

— Finding black hole spins efficiently during a numerical binary evolution  (2608.12211 - Chaudhary et al., 12 Aug 2026) in Section 3, “Future improvements”; Appendix, Section “Matrix-free version of the AKV algorithm”; Section 4, “Conclusion”