# Compression Challenges in Large Scale Partial Differential Equation Solvers
## 5 Mass Storage
### Adjoint Solutions
#### 5.1.3 Adaptive Grid Refinement  
PDE solutions often exhibit spatially local features, e.g., corner singularities or moving fronts as in the monodomain Equation (9), which need to be resolved with small mesh width. Uniform grids with small mesh width lead to huge numbers of degrees of freedom, and therefore waste computing effort, bandwidth, and storage capacity in regions where the solution is smooth. Adaptive mesh refinement, based on error estimators and local mesh refinement, has been established as an efficient means to reduce problem size and solution time, cf. [2], and acts at the same time as a decimation/interpolation-based method for lossy data compression. The construction of adaptive meshes has been explicitly exploited by [PERSON] et al. [23] and [PERSON] [24] for scientific data compression.  
Interestingly, even though decimation and hierarchical transform coding both compete for the same spatial correlation of data, i.e., smoothness of functions to compress, their combination in PDE solvers can achieve better compression factors for a given distortion than each of the approaches alone. A simple example is shown in Figure 10 with compression factors for a fixed error tolerance given in Table 1. Using both, adaptive mesh refinement and transform coding, is below the product of individual compression factors, which indicates that there is in fact some overlap and competition for the same correlation budget. Nevertheless, it shows that even on adaptively refined grids there is a significant potential for data compression. The compression factor of adaptive mesh refinement as given in Table 1 is, however, somewhat too optimistic, since it only counts the number of coefficients to be stored. For reconstruction the mesh has to be stored as well. Fortunately, knowledge about the mesh refinement algorithm can be used for extremely efficient compression of the mesh structure [34].  
Figure 9: Goal-oriented error estimation: generated meshes with (left) and without (middle) compression (with compression factor up to 32) for Example 3b in [87]. Meshes were generated using weights according to [PERSON] [87]. Estimated errors (right) are shown for weights due [87] (ws) and to [PERSON] et al. [85] (bkr), both with and without compression. Differences between estimated errors with and without compression are barely visible, and negligible compared to the differences between the two error concepts.  
Another reason why adaptive mesh refinement and transform coding can be combined effectively for higher compression factors, is that the accuracy requirements for the PDE solution, and the solution storage can be very different, depending on the error propagation. Thus, the decimation by mesh adaptivity may need to retain information that the transform coder can safely neglect, allowing the latter to achieve additional compression on top of the former. For example, in adjoint gradient computation, the mesh resolution affects the accuracy of all future time steps and via them the reduced gradient, whereas the solution storage for backwards integration of the adjoint equation affects only one time step.