Yes, the low-rank approximation techniques can be further improved to achieve higher memory compression, particularly for problems characterized by complex geometries or varying material properties. Several strategies can be employed to enhance the efficiency of low-rank approximations in these contexts.
Adaptive Rank Selection: Implementing adaptive rank selection algorithms that dynamically adjust the multilinear ranks based on the local geometry and material properties can lead to more efficient representations. By analyzing the sensitivity of the solution to changes in the rank, one can optimize the rank for different regions of the computational domain, allowing for higher compression where possible.
Hierarchical Tensor Decompositions: Utilizing hierarchical tensor decompositions, such as tensor trains or hierarchical Tucker formats, can provide a more flexible framework for representing complex data structures. These methods allow for a more localized treatment of the tensor structure, which can be particularly beneficial in capturing the intricacies of complex geometries.
Geometry-Driven Approaches: Incorporating geometric information directly into the low-rank approximation process can enhance the accuracy and efficiency of the approximations. For instance, using geometric features to guide the selection of basis functions or to inform the construction of the low-rank representations can lead to better performance in terms of both accuracy and memory usage.
Material Property Adaptation: For problems with varying material properties, developing low-rank techniques that account for the spatial variability of material coefficients can improve the representation of the system matrices. This could involve creating localized low-rank approximations that adapt to the material properties in different regions of the domain.
By integrating these advanced techniques into the existing low-rank multipatch framework, it is possible to achieve significant improvements in memory compression and computational efficiency, making the method more applicable to a broader range of complex engineering problems.