Dimension-adaptive sparse-grid refinement for binary-black-hole initial-data models

Develop a dimension-adaptive, downward-closed Smolyak sparse-grid index set for the differentiable parametric solver of conformally flat Bowen–York binary-black-hole initial data, growing the index set greedily along parameter axes whose hierarchical interpolation surpluses remain large, in order to account for unequal parameter difficulty more efficiently than the isotropic sparse grid.

Background

The paper constructs differentiable parametric models for conformally flat Bowen–York binary-black-hole initial data using sparse interpolation followed by certified Newton refinement. Its current sparse-grid construction uses an isotropic Smolyak simplex, which assigns resolution according to the same total level across all parameter axes. The authors note that the parameter directions do not converge equally rapidly: the spin axes converge more slowly than the separation and mass-ratio axes. Consequently, an isotropic index set may expend expensive elliptic solves on parameter directions that are already well resolved while under-resolving the difficult directions.

A dimension-adaptive downward-closed index set would instead select new tensor-product contributions according to the magnitude of their hierarchical surpluses. Such a refinement could reduce the offline number of certified elliptic solves while preserving the solver’s constraint-certification and differentiability properties. The paper identifies this as a refinement not developed within the presented work.

References

Several directions remain open. First, the cheap computation of the nodal representation currently requires nested point sets. Extending it to non-nested sequences would broaden the method's applicability. Second, the weight strategies of \Cref{subsec:SelectionOfOptimalWeights} rely on knowledge of the smoothness parameters $\sigma_j$. A data-driven or adaptive selection of $$ is desirable in practice. A dimension-adaptive variant, in which the index set is built greedily from error indicators rather than prescribed a priori, could handle functions with unknown anisotropy. Finally, combining the offline phase with $\mathcal{H}$-matrix compression or Samplets would extend the method to non-local kernels at nearly linear cost.

— Anisotropic Kernel-based Multilevel Interpolation of High-Dimensional Functions on Sparse Grids  (2609.25828 - Kempf, 22 Sep 2026) in Section 6, Conclusion and Outlook

When the parameters differ in difficulty, as the slowly converging spin axes of Sec.~\ref{sec:model:peraxis} do, the isotropic simplex can be replaced by a dimension-adaptive downward-closed index set grown greedily along the axes whose surpluses remain large, a refinement we leave to future work.

— A Differentiable Parametric Model of Binary-Black-Hole Initial Data: I. Conformally flat Bowen-York punctures  (2609.10173 - Ceuster et al., 9 Sep 2026) in Section “Sparse grids for higher dimensions,” within the Parametric solver section