# Compression Challenges in Large Scale Partial Differential Equation Solvers
## 6 Conclusions  
Data compression methods are valuable tools for accelerating PDE solvers, addressing larger problems, or archiving computed solutions. Due to floating-point data being compressed, only lossy compression can be expected to achieve reasonable compression factors; this matches perfectly with the fact that PDE solvers incur discretization and truncation errors. An important aspect is to model and predict the impact of quantization or decimation errors on the ultimate use of the computed data, in order to be able to achieve high compression factors while meeting the accuracy requirements. This requires construction and use of fast and accurate a posteriori error estimators complementing analytical estimates, and remains one of the challenging future research directions.  
Meeting such application-dependent accuracy requirements calls for problem-specific approaches to compression, e.g., in the form of transforms where the quantization of transform coefficients directly corresponds to the compression error in relevant spatially weighted Sobolev norms. Discretization-specific approaches that are able to exploit the known structure of spatial data layout, e.g., Cartesian grids or adaptively refined mesh hierarchies, are also necessary for achieving high compression factors.  
Utility and complexity of such methods are largely dictated by their position in the memory hierarchy. Sophisticated compression schemes are available and regularly used for reducing the required storage capacity when archiving solutions. On the other hand, accelerating PDE solvers by data compression is still in the active research phase, facing the challenge that computational overhead for compression can thwart performance gains due to reduced data transmission time. Thus, simpler compression schemes dominate, in particular when addressing the memory wall. Consequently, only a moderate, but nevertheless consistent, benefit of compression has been shown in the literature.  
The broad spectrum of partially contradicting requirements faced by compression schemes in PDE solvers suggests that no single compression approach will be able to cover the need, and that specialized and focused methods will increasingly be developed--a conclusion also drawn in [48].  
The trend of growing disparity between computing power and bandwidth, which could be observed during the last three decades and will persist for the foreseeable future of hardware development, means that data compression methods will only become more important over time. Thus, we can expect to see a growing need for data compression in PDE solvers in the coming years.  
**Author Contributions:** Conceptualization, methodology, software, writing-original draft preparation, writing-review and editing, project administration, visualization: S.G. and M.W.; supervision, funding acquisition: M.W.  
**Funding:** This work has been partially supported by the German Ministry for Eduction and Research (BMBF) under project grant 01 IH16005 (HighPerMeshes).  
**Acknowledgments:** We thank [PERSON] for implementing mixed-precision preconditioners, [PERSON] for implementation and testing of checkpoint/restart, and [PERSON] for many helpful discussions.  
**Conflicts of Interest:** The authors declare no conflict of interest.