On the Compressibility of Integral Operators in Anisotropic Wavelet Coordinates

The present article is concerned with the s*-compressibility of classical boundary integral operators in anisotropic wavelet coordinates. Having the s*-compressibility at hand, one can design adaptive wavelet algorithms which are asymptotically optimal, meaning that any target accuracy can be achieved at a computational expense that stays proportional to the number of degrees of freedom (within the setting determined by an underlying wavelet basis) that would ideally be necessary for realising that target accuracy if full knowledge about the unknown solution were given. As we consider anisotropic wavelet bases, we can achieve higher convergence rates compared to the standard, isotropic setting. Especially, edge singularities of anisotropic nature can be resolved.

💡 Research Summary

The paper investigates the s*-compressibility of classical boundary integral operators when discretised with anisotropic tensor‑product wavelet bases. The authors begin by recalling that solving operator equations L u = f with non‑local operators such as boundary integral operators leads to dense system matrices, making standard linear‑complexity solvers infeasible. Adaptive algorithms based on nonlinear approximation, in particular best N‑term approximations, can achieve optimal rates provided the underlying infinite matrix is sufficiently compressible.

The authors construct anisotropic wavelet bases on the unit interval and extend them to two‑dimensional tensor‑product bases on the unit square □ =