Adaptive Local Proper Orthogonal Decomposition via Graph Partitioning and Provable Error-Bounded Lossy Compression for Parametric Series in Finite-Dimensional Function Spaces

We introduce a spatially adaptive proper orthogonal decomposition (POD) method for error-bounded lossy compression of snapshot data in finite-dimensional function spaces. The method combines hierarchical graph partitioning with independent low-rank approximations on spatial subdomains, allowing the local approximation ranks to adapt to the spatial distribution of information in the data. For partitions that are orthogonal in the discrete inner product, we establish an exact decomposition of the global squared reconstruction error into local contributions and characterize the optimality of the resulting local POD approximations for fixed partitions and ranks. These results provide a rigorous mechanism for allocating a prescribed global error tolerance among subdomains and selecting the smallest admissible local ranks. We further develop a sufficient error bound for the coefficient-space normalized root-mean-square error (NRMSE). A greedy hierarchical refinement algorithm accepts a spatial subdivision only when it reduces the storage required by the local low-rank factors while preserving the prescribed reconstruction-error bound. The method is assessed using synthetic data and snapshot sequences from two- and three-dimensional compressible-flow simulations. The numerical results illustrate how spatially adaptive ranks and partitioning can improve compression relative to global POD and fixed spatial partitions, particularly when the data exhibit spatially heterogeneous dynamics.