Adaptive Local Proper Orthogonal Decomposition via Graph Partitioning and Provable Error-Bounded Lossy Compression

We introduce a spatially adaptive proper orthogonal decomposition (POD) method for error-bounded lossy compression of snapshot data. By combining hierarchical graph partitioning with independent low-rank approximations on spatial subdomains, the method adapts local approximation ranks to spatially heterogeneous information content. For partitions orthogonal in the discrete inner product, we establish an exact decomposition of the global reconstruction error into local contributions and characterize the optimality of local POD approximations. These results enable rigorous error allocation and rank selection under a prescribed global tolerance. We also derive a sufficient error bound for the coefficient-space normalized root-mean-square error (NRMSE) and develop a greedy refinement algorithm that reduces storage while maintaining the error constraint. The method is evaluated on synthetic data and two- and three-dimensional compressible-flow simulations, demonstrating improved compression over global POD and fixed spatial partitions, particularly for data with spatially heterogeneous dynamics.