📝 SummarXiv

Abstract

We introduce a low-rank framework for adaptive isogeometric analysis with truncated hierarchical B-splines (THB-splines) that targets the main bottleneck of local refinement: memory- and time-intensive matrix assembly once the global tensor-product structure is lost. The method interpolates geometry-induced weight and source terms in separable spline spaces and computes their tensor-train (TT) representations via the alternating minimal energy (AMEn) solver, enabling level-wise assembly of system operators using univariate quadrature. To recover separability in the adaptive setting, we reduce the active basis to tensor-product domains and partition active/non-active cells into a small number of Cartesian cuboids, so each contributes a Kronecker factor that is accumulated and rounded in TT. We realize the two-scale relation with truncation in low rank and assemble the global hierarchical operators in a block TT format suitable for iterative solvers. A prototype MATLAB implementation built on the GeoPDEs package and the TT-Toolbox demonstrates that, for model problems with moderately complex refinement regions, the approach reduces memory footprint and assembly time while maintaining accuracy; we also discuss limitations when ranks grow with geometric or refinement complexity. This framework advances scalable adaptive IgA with THB-splines, particularly in three dimensions.