# Compression Challenges in Large Scale Partial Differential Equation Solvers
## 2 Compression Aspects of PDE Solvers
### Data Layout Affects Compression Design Space  
One of the most important differences between PDE solvers from the compression point of view is the representation and organization of spatial data.  
General-purpose floating point compression schemes essentially ignore the underlying structure and treat the values as an arbitrary sequence of values. Examples of this approach are ISABELA, SQE, and SZ-1.1. The advantage of direct applicability to any kind of discretization comes at the cost of moderate compression factors, since position-dependent correlations in the data cannot be directly exploited.  
Structured Cartesian grids are particularly simple and allow efficient random access by index computations, but are limited to quasi-uniform resolutions and relatively simple geometries parametrized over cuboids. Cartesian structured data is particularly convenient for compression, since it allows the use of the Lorenzo predictor (fpzip) or its higher order variants (SZ-1.4), regular coarsening (AC), simple computation of multilevel decompositions (MGARD and HexaShrink), or exploiting tensor approaches such as factorized block transforms (zfp) or low-rank tensor approximations (TuckerMPI and TTHRESH).  
If complex geometries need to be discretized or if highly local solution features need to be resolved, unstructured grids are used. Their drawback is that coefficients are stored in irregular patterns, and need to be located by lookup. Fewer methods are geared towards compression of data on unstructured grids. Examples include TCUG and AT. When storing values computed on unstructured grids, the grid connectivity needs to be stored as well. In most use cases, however, e.g., iterative solvers or time stepping, many coefficient vectors have to be compressed, but few grids do. Thus, the floating point data represents the bulk of the data to be compressed. Methods developed for compression of unstructured grid geometries can in some cases been used for storing solutions or coefficient data of PDE solvers, but are mostly tailored towards computer graphics needs. We refer to the survey [33] for 3D mesh compression, as well as [34] for grid compression in PDE applications. In addition to geometry and connectivity, recent research focuses on the compression of attribute data, such as color values or texture, taking the geometry into account [35], and using progressive compression methods [36], but without rigorous error control.  
In contrast to the grid structure, the method of discretization (finite difference, finite element, or finite volume methods, or even spectral methods) is of minor importance for compression, but of course affects technical details.  
PDE solvers can also benefit from compression of further and often intermediate data that arises during the solution process and may be of completely different structure. This includes in particular preconditioners (scalar quantization, mixed precision [37; 38], hierarchical matrices [39]), discretized integral operators in boundary element methods (wavelets [40]), and boundary corrections in domain decomposition methods.