While the adaptive precision scheme for mixed precision HODLR matrices offers significant advantages in terms of computational efficiency and reduced memory usage, there are several potential limitations and drawbacks to consider:
Complexity of Precision Selection: The adaptive precision selection process can introduce additional complexity in the implementation. Determining the appropriate precision for each level of the hierarchical structure requires careful analysis of the approximation error and may involve overhead in terms of computational resources.
Dependence on Matrix Properties: The effectiveness of the adaptive precision scheme is highly dependent on the properties of the matrices being processed. For matrices with highly variable rank structures or those that do not conform to the assumptions made in the analysis, the chosen precision levels may not adequately control the error, leading to potential inaccuracies in the results.
Error Accumulation: Although the scheme aims to balance errors from low-rank approximations and finite precision computations, there is still a risk of error accumulation, especially in deep hierarchical structures. As the depth of the tree increases, the potential for compounded errors may necessitate more stringent precision requirements than initially anticipated.
Limited Applicability: The adaptive precision scheme may not be universally applicable to all types of hierarchical matrices or computational scenarios. Certain applications may require stricter precision controls or may not benefit from the flexibility offered by mixed precision, limiting the generalizability of the approach.
Hardware Constraints: The performance gains from mixed precision computing are contingent upon the availability of hardware that supports various precision formats. In environments where such support is limited, the advantages of the adaptive precision scheme may not be fully realized.