Greedy Low-Rank Tensor Approximations for Elastodynamic Green’s Functions in Layered Media

Green’s functions are fundamental ingredients in elastodynamic applications, including wave propagation and dynamic soil-structure interaction. Their numerical evaluation over dense spatial and frequency grids for multiple source-receiver configurations leads to high-dimensional Green’s function tensors that impose significant computational and memory demands. To reduce these costs, the Green’s functions are recasted as a separable operator represented in compact tensor form and precomputed over the parametric domain. This is achieved using tensor decomposition algorithms, including Proper Generalized Decomposition (PGD), Greedy Tucker Approximation (GTA) and Greedy Hierarchical Tucker Approximation (GHTA), introduced here as a hierarchical extension of GTA. The low-rank approximations are built iteratively through rank-one enrichments obtained by an alternating least-squares procedure. While PGD represents the response as a sum of separable terms, GTA couples the reduced one-dimensional spaces through a global Tucker core. GHTA extends this greedy Tucker strategy by replacing the global core with hierarchical Tucker cores arranged according to a dimension tree, providing a compact hierarchical representation of the multidimensional coupling while retaining the greedy construction of the reduced spaces. The resulting reduced Green’s tensors preserve the accuracy of the reference formulation while providing compact representations that reduce storage requirements and enable efficient repeated evaluations. Green’s functions for a layered soil overlying a homogeneous halfspace are computed using PGD, GTA and GHTA, and compared with the reference direct stiffness solution. The results provide a comparative assessment of approximation accuracy, computation time, memory requirements and convergence trends.