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

Roberto Nuca, Matteo Parsani, Lisandro Dalcin

arXiv:2610.31375v1math.NAcs.NA

TL;DR

Large simulation snapshot sequences create substantial storage demands, motivating compression with a prescribed reconstruction-error bound. The paper combines hierarchical graph partitioning with independent local POD approximations and adaptive local ranks. Experiments on synthetic data and compressible-flow snapshots illustrate compression advantages over global POD and fixed partitions, especially for spatially heterogeneous dynamics.

  • Problem

    Large-scale simulations produce snapshot data that can be inconvenient to store, transfer, or analyze, motivating lossy compression under a prescribed reconstruction-error tolerance.

    Source: §1 · pp. 1–2

  • Method

    The method uses hierarchical graph partitioning, independent local POD approximations, and a greedy storage-aware refinement rule that preserves the global error certificate.

    Source: §1 · p. 3, §6 · pp. 22–23

  • Results

    Experiments on synthetic data and two- and three-dimensional compressible-flow snapshots illustrate potential compression advantages over global POD and fixed spatial partitions, with errors within prescribed tolerances.

    Source: §1 · pp. 3–4, §6 · p. 23

  • Takeaways & Limitations

    Spatially adaptive ranks can reflect differing approximation requirements across regions, particularly for data with spatially localized or heterogeneous dynamics.

    Source: §6 · p. 23

    The greedy search is restricted to its hierarchical partition space and does not guarantee a globally minimum-storage representation; reported compression ratios exclude partition metadata and encoding overhead.

    Source: §4 · p. 14, §6 · p. 23

Abstract

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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.

1 Introduction

Large simulations generate snapshot data whose storage burden affects downstream scientific workflows, motivating compression under a prescribed error tolerance. This paper combines spatially local POD with graph-based partitioning and error-budgeted rank selection, then evaluates the approach against global POD and fixed partitions.

1 Introduction

  • The introduction frames snapshot compression as a response to storage burdens affecting checkpointing, visualization, post-processing, data transfer, uncertainty quantification, and surrogate-model construction.The motivating goal is to reduce storage while keeping reconstruction error below a prescribed tolerance.Source: §1 · pp. 1–2
  • A single global POD basis can require many modes for spatially localized or multiscale structures, whereas this representation retains separate local bases across spatial regions.Other hierarchical approaches may use partitions to construct one global reduced space rather than retaining local bases in the compressed representation.Source: §1 · p. 2
  • The method combines graph-based spatial partitioning with independent local POD approximations, selecting the smallest admissible rank in each subdomain under a shared global error budget.A split is retained only when the child factors reduce algebraic storage while preserving the error certificate.Source: §1 · p. 3
  • For partitions orthogonal in the discrete inner product, global squared error decomposes into local contributions, enabling local error budgets and smallest admissible ranks.For fixed partitions and ranks, local POD is optimal within the block-separable approximation class; the coefficient-space NRMSE bound is sufficient but can be conservative.Source: §1 · p. 3
  • The partition search is restricted to the graph-generated hierarchy and its greedy form is not claimed to find the minimum-storage partition over all possibilities.The numerical assessment compares the approach with classical POD and adaptive local POD on fixed spatial partitions.Source: §1 · pp. 3–4

2 Mathematical Setting

The mathematical setting represents snapshots in a fixed finite-dimensional function space on a fixed mesh, with a fixed basis and Gram matrix across snapshots. Mesh cells are grouped into spatial subdomains and mapped to disjoint coefficient-index blocks, which organize the snapshot matrix for subsequent local approximation.

2 Mathematical Setting

  • The analysis assumes snapshots share a fixed finite-dimensional function space, basis, and Gram matrix on a fixed computational mesh.A deforming-mesh problem is included only after mapping snapshots to a common reference representation with fixed basis and Gram matrix.Source: §2 · p. 4

2.1 Mesh Description, Mesh Partitioning and Degrees of Freedom

  • Cells are grouped into nonempty, connected subdomains, then associated with disjoint coefficient-index sets that define spatial row blocks of the snapshot matrix.Graph partitioning may use weights such as degrees of freedom per cell; the illustrated weighted partition balances degrees of freedom rather than cell counts.Source: §2.1 · pp. 5–6, Figure 1 · p. 6

2.2 Spaces of Functions

  • The function space is represented through coefficient vectors and a Gram matrix encoding the chosen inner product; the Gram matrix is symmetric positive definite.A general Gram matrix need not be diagonal, and the text identifies diagonal matrices as ideal for computation.Source: §2.2 · p. 7

3 Local Surrogates

The section constructs local snapshot approximations by partitioning spatially associated rows of the snapshot matrix. It relates local SVD and POD errors to global reconstruction error and coefficient-space NRMSE.

3 Local Surrogates

  • The snapshot matrix is organized with spatially associated degrees of freedom in rows and times, parameter values, or realizations in columns.Each local data matrix selects the rows assigned to its index set; the row mapping is retained when assembling local approximations.Source: §3 · p. 8

3.1 Local singular value decomposition

  • For a fixed partition and local ranks, local truncated SVDs minimize Frobenius error among approximations with those row-block rank limits.The global error aggregates the errors of the disjoint row blocks.Source: §3.1 · p. 10

3.2 Local proper orthogonal decomposition

  • Local POD modes are orthonormal in their local Gram matrices, and their truncation errors yield a global error estimate when summed across subdomains.The local POD construction uses a symmetric positive-definite local Gram matrix.Source: §3.2 · pp. 10–11

3.3 Connection between Frobenius norm and NRMSE

  • A sufficient coefficient-space NRMSE bound is obtained by allocating the global error budget among subdomains and choosing ranks to satisfy the local conditions.The proposed allocation is size-proportional, with each subdomain's weight equal to its number of rows divided by the total number of rows.Source: §3.3 · pp. 11–12

4 Graph Partitioning, Adaptivity and Data Compression

The method explores spatial partitions through hierarchical graph subdivision and compares the storage of local low-rank factors. It retains storage-reducing splits while preserving the assigned error budget, but the greedy search is not globally optimal.

Storage and compression

  • The stated storage counts factor matrices and singular values, but excludes partition data, index maps, preprocessing parameters, file headers, and numerical precision.The passage describes these excluded items as minimal computational overhead in practice, even for small data matrices.Source: §4 · p. 12

Greedy hierarchical refinement

  • Hierarchical search begins with N0 subdomains and tentatively splits selected leaves into N+ children, inducing the index partition from the current leaves.The method recursively applies the procedure to accepted child subdomains.Source: §4 · p. 13
  • Accepted splits preserve the region's squared-error budget and global NRMSE guarantee while strictly reducing algebraic factor storage.Under the size-proportional allocation, storage decreases monotonically along accepted refinements.Source: §4 · p. 14
  • The greedy refinement does not generally find the minimum-storage partition in the complete hierarchical search space.A split that immediately increases storage may lead to a deeper refinement that would reduce it.Source: §4 · p. 14

Storage and compression

  • If local ranks remain equal to the global rank, partitioning strictly increases factor storage because each subdomain repeats temporal factors.Localization helps only when reduced local ranks compensate for that repeated storage.Source: §4 · pp. 12, 14

5 Numerical experiments

The experiments test adaptive POD on controlled synthetic data and two- and three-dimensional compressible-flow snapshots. Across the synthetic and flow tests, adaptive compression handles spatially heterogeneous data, including cases where global POD cannot compress at very strict error bounds.

5 Numerical experiments

  • The numerical tests combine synthetic datasets designed to stress the algorithm with compressible-flow simulations of a NACA 0012 airfoil and an inclined cube.The flow cases use density snapshots.Source: §5 · p. 14

5.1 Synthetic test with solitons

  • The soliton test sums two traveling waves with spline-controlled centers, amplitudes, and support radii; their nonoverlapping supports leave many data rows at the zero initialization.The waves are positioned in the bottom-left and top-right parts of a nonuniform square mesh.Source: §5.1 · pp. 14–15
  • Adaptive compression outperforms regular POD on the soliton test and selectively truncates local ranks, while regular POD can fail to compress at strict requested NRMSE.Refinement localizes in the bottom-left region, where the signal has more points and variation.Source: §5.1 · p. 16, Figure 5 · p. 17

5.2.1 Subsonic flow past a NACA 0012 airfoil

  • The airfoil test uses two partitioning trajectories, 10-4 and 20-2; the latter explores more partitions because each subdomain is split in two.Initial and subsequent subdivision counts are user-defined inputs.Source: §5.2.1 · p. 18
  • On the airfoil snapshots, adaptive POD retains compression at very low requested NRMSE while regular POD does not, with a posteriori error close to the requested bound.Local rank distributions distinguish regions with little information from regions with stronger temporal fluctuations.Source: §5.2.1 · p. 18, Figure 9 · p. 20, Figure 10 · p. 20

5.2.2 Supersonic flow past an inclined cube

  • Compression performance remains consistent across the cube’s initial and fully developed 100-snapshot datasets, despite markedly different spatial and temporal features.The tests compress density snapshots from both datasets.Source: §5.2.2 · p. 21, Figure 14 · p. 22

6 Conclusions

The method combines graph-based spatial partitioning with local low-rank approximations, supported by error guarantees for orthogonal partitions and a storage-aware refinement rule. Experiments illustrate compression advantages over global POD and fixed partitions, especially for heterogeneous dynamics.

6 Conclusions

  • Graph-based partitioning with independent local low-rank approximations allows ranks to vary across subdomains and adapt to spatially varying data complexity.This is the proposed spatially adaptive local POD method for error-bounded lossy compression.Source: §6 · pp. 22–23
  • For partitions orthogonal in the discrete inner product, global squared reconstruction error decomposes exactly into local contributions, and fixed-rank local POD minimizes error in its block-separable class.These properties support allocating a global error budget and selecting the smallest admissible local ranks; a sufficient bound also controls coefficient-space NRMSE.Source: §6 · p. 23
  • Hierarchical refinement is accepted only when it reduces low-rank-factor storage while preserving the global error guarantee.Source: §6 · p. 23
  • Experiments on synthetic data and 2D and 3D compressible-flow snapshots illustrate compression advantages over global POD and fixed spatial partitions, especially for heterogeneous dynamics.Reconstruction errors remain within prescribed tolerances, consistently with the theoretical bounds.Source: §6 · p. 23
  • The greedy partitioning strategy does not guarantee globally minimum storage, and the reported compression ratio excludes partition metadata and encoding overhead.Source: §6 · p. 23

CRediT authorship contribution statement

Roberto Nuca, Matteo Parsani, and Lisandro Dalcin contributed to the work across conceptual, methodological, computational, and writing roles.

CRediT authorship contribution statement

AI usage statement

The authors report no AI use in developing the ideas, mathematical arguments, code, or tests, and limited AI assistance to manuscript language.

AI usage statement

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