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sp-DBA: a general framework for adaptive transform-domain computation

Jingkun Jiang, Pingchuan Deng, Yang Xia

arXiv:2609.03922v1cs.CEcs.DC

TL;DR

Existing adaptive strategies can require specialized data structures or workflow redesign, complicating reuse in mature transform-domain computations. sp-DBA adapts block execution within those workflows; across three implementations, it maintains numerical accuracy while achieving transform-domain update speedups exceeding twentyfold in the strongest configuration.

  • Problem

    Many adaptive transform-domain approaches change the field representation, requiring data reorganization or specialized structures that can limit reuse of mature workflows and efficient parallel execution.

    Source: §1 · p. 3

  • Method

    sp-DBA dynamically controls blockwise computation on the transformed field while leaving the existing transform workflow in place.

    Source: §2 · p. 5, §4 · p. 13

  • Results

    Across three implementations, numerical accuracy is maintained while transform-domain update speedups exceed twentyfold in the strongest configuration.

    Source: §1 · p. 3

  • Takeaways & Limitations

    Execution-level adaptivity leaves the transformed representation and workflow unchanged and carries across substantially different scientific and engineering problems.

    Source: §4 · p. 13

    Gains are strongest with costly updates, localized spectral activity, and low active-block ratios; activation criteria must identify solution-relevant spectral regions.

    Source: §4 · p. 13

Abstract

from arXiv · show

Transform-domain methods simplify analysis and computation, making them central to scientific computing and signal processing. However, existing adaptive strategies often introduce new data structures or require substantial workflow redesign, limiting efficient execution on massively parallel hardware. Here we present spectral dynamic block activation (sp-DBA), an adaptive acceleration framework for transform-domain workflows. sp-DBA dynamically activates transform-domain computation blocks as a calculation evolves, concentrating computation where it is needed while retaining numerical accuracy and optimized transform operations. Across representative workflows in materials science, biology, and optical communications, the demonstrated implementations achieve speedups of up to 28.1-fold for transform-domain updates and up to 8.4-fold overall acceleration; strong- and weak-scaling tests on up to eight GPUs show that the local adaptive update remains effective within distributed FFT workflows. By introducing execution-level adaptivity into mature transform-domain workflows without rebuilding existing solvers, sp-DBA extends the scale, duration, and complexity of scientific simulations and signal-processing calculations on modern parallel systems.

1 Introduction

The introduction motivates adaptivity that reuses mature transform workflows, then presents sp-DBA as blockwise execution-level adaptivity coordinated with the existing workflow.

1 Introduction

  • Existing adaptive approaches can require data reorganization or specialized structures, while irregular execution and access can reduce efficiency on parallel hardware.Direct blockwise activation also faces the mismatch between transformed modes and local physical regions, and must preserve consistency across the transform workflow.Source: §1 · p. 3
  • sp-DBA applies adaptive blockwise execution to transformed-field computation while retaining the surrounding transform workflow.The framework is demonstrated across three applications, and its computational savings can support larger domains and longer evolutions.Source: §1 · pp. 3–4, §2 · p. 5

2 Methods

  • Each block is classified as active, transition, or frozen, triggering a full update, boundary-aware weighted update, or skipped update, respectively.State preparation uses an application-specific activation measure and threshold; the threshold can be adjusted to maintain a target average active-block ratio.Source: §2 · p. 5
  • sp-DBA inserts state preparation, blockwise update, and consistency control between the forward and inverse transforms; each block maps to a CUDA thread block.The inverse transform and post-transform stages proceed as in Regular.Source: Figure 1 · p. 4, §2 · p. 5
  • The framework complements optimized transforms and preserves each solver’s equations, discretization, representation, and transform workflow.It is demonstrated across PFC, RCD, and DBP despite their differing equations and transformed-field layouts.Source: §2 · p. 5
  • About 8–16% of the corresponding Regular solver core is added for direct integration, or about 13–28% including model-adapter code.Common block operations are implemented in the shared framework while each solver reuses its existing update core.Source: §2 · pp. 5–6

3 Results

Across four workflows, sp-DBA concentrates transformed-field updates in active blocks while preserving reported solution behavior. Update-stage gains coexist with end-to-end acceleration and effective distributed execution.

3.1 Polycrystalline coarsening

  • For N ≥2048, 20%, 10%, and 5% targets yield approximately 5×, 10×, and 18–20× Fourier-space update speedups, respectively.Lower target active-block ratios produce larger speedups; on smaller grids fewer blocks can be skipped and block-state operations take a larger share of update time.Source: Figure 2 · p. 6, §3.1 · p. 7
  • At tested target ratios, sp-DBA closely tracks Regular’s normalized grain-size distribution and free-energy decline while conserving mean density.The comparisons cover late-stage grain statistics and thermodynamic quantities.Source: Figure 2 · p. 6, §3.1 · p. 7

3.2 Multicomponent pattern formation

  • With eigendecomposition, RCD update-stage speedups versus Regular are 9.6× at M = 24, 28.1× at M = 48, and 26.7× at M = 64.At smaller M, block-state overhead accounts for a larger fraction of update-stage cost.Source: Figure 3 · p. 8, §3.2 · p. 9
  • sp-DBA matches Regular’s u1 morphology and structure-factor peak positions and amplitudes, while activation concentrates in the low-wavenumber instability band.Most stored Fourier blocks remain frozen in the reported pattern-formation calculation.Source: Figure 3 · p. 8, §3.2 · p. 9

3.3 Digital backpropagation

  • At the default 10% target, DBP frequency-domain update speedup rises from 2.2× at Nt = 2^13 to 21.3× at Nt = 2^18.As Nt increases, the sampled frequency grid expands around the occupied bands, leaving more off-band blocks frozen.Source: Figure 4 · p. 10, §3.3 · p. 11
  • Across tested Nt, correlation stays near unity (lowest plotted value around 0.998) and sp-DBA NMSE remains at or below −20 dB.For this controlled input, Regular gives near-machine-precision recovery.Source: Figure 4 · p. 10, §3.3 · p. 11

3.4 Performance and scalability

  • DBP end-to-end acceleration reaches 8.4×, and RCD’s combined sp-DBA/eigendecomposition configuration gives 28.1× update-stage speedup at M = 48.The reported RCD update-stage gain is relative to Regular.Source: Figure 5 · p. 12, §3.4 · p. 12
  • Under weak scaling from one to eight GPUs, sp-DBA reduces per-GPU update time by about 2–3× while update times remain stable.The per-GPU workload stays approximately fixed and the measured active-block ratio remains near the 10% target.Source: Figure 5 · p. 12, §3.4 · p. 13

4 Discussion

sp-DBA shifts adaptivity from numerical representation changes to execution, retaining the transformed representation and workflow while controlling where computation runs. The discussion connects this approach to broader simulation reach, identifies its effective regimes and boundaries, and points to possible extensions.

Adaptivity at the execution level

  • sp-DBA adapts execution rather than the spectral set, basis, or grid, leaving the transformed representation and transform workflow unchanged across three distinct demonstrations.The authors frame this as a reusable execution strategy rather than solver-specific numerical redesign.Source: §4 · p. 13

Broader computational reach

  • Larger domains and longer grain-coarsening runs can support more reliable grain-size statistics, reduce finite-size uncertainty, and strengthen comparisons with theory and observed materials behavior.Reliable statistics require enough grains and runs long enough to distinguish transient from steady or asymptotic behavior.Source: §4 · p. 13
  • For multicomponent reaction–cross-diffusion and multimode optical transmission, saved computation can support richer models and broader parameter or operating-condition ranges.The update becomes more demanding as interacting components increase in RCD and as modes and operating conditions increase in optical transmission.Source: §4 · p. 13

Effective regimes and limitations

  • The largest gains occur when updates dominate runtime, spectral activity stays concentrated, and active-block ratios remain low; end-to-end speedup is limited by transforms, communication, and other steps.Activation criteria must identify consequential spectral regions; some optical-fiber regimes involve weak amplitude signatures and frequency transfer through four-wave mixing.Source: §4 · p. 13

Future directions

  • Phase-aware criteria, conservative activation, refresh safeguards, and distributed load balancing are proposed as extensions toward more difficult regimes and distributed execution.The authors also point toward hardware-aware block layouts and activation rules for additional transform families.Source: §4 · pp. 13–14

Supplementary Information

The supplementary information is organized around the sp-DBA framework, its solver integrations, application workflows, numerical validation, performance, distributed execution, and visualization.

S1–S3

  • The opening sections cover the sp-DBA framework, transform-domain solver integration, and XPFC polycrystalline coarsening.The listed framework topics include transition smoothing, half-spectrum neighbor activation, threshold control, and periodic refresh.Source: Supplementary Information · p. 20

S4–S5

  • Further sections address reaction–cross-diffusion pattern formation and digital backpropagation, including their governing equations and numerical configurations.Source: Supplementary Information · p. 20

S6–S9

  • The supplement also lists numerical validation, single- and multi-GPU performance, distributed FFT scaling, and visualization.Validation topics include conservation, spectral accuracy, and dealiasing compatibility.Source: Supplementary Information · p. 20

S1 sp-DBA framework

The framework manages transformed-field block states through smoothing, symmetry-aware neighbor activation, dynamic threshold control, and periodic refresh. These mechanisms retain frozen values between updates while controlling active computation and update boundaries.

S1 sp-DBA framework

  • sp-DBA manages transformed-field update boundaries through smoothing, half-spectrum neighbor activation, dynamic threshold control, and periodic refresh.Regular denotes the full-update mode with sp-DBA disabled.Source: S1 sp-DBA framework · p. 21

S1.1 Boundary-aware transition smoothing

  • Boundary-aware smoothing weights transition-block increments while frozen values persist, leaving block patterns largely unchanged and free-energy closer to Regular.The local roll-off weight is assembled from neighboring block states; smoothing applies to the increment, not stored frozen-block values.Source: S1.1 Boundary-aware transition smoothing · pp. 21–22, Figure S2 · p. 22

S1.2 Neighbor activation in half-spectrum FFT layouts

  • In half-spectrum FFT layouts, requested neighbors in the omitted half are activated through conjugate counterparts, with the symmetric partner on kx = 0 activated when required.This remapping changes block-state propagation, not stored Fourier coefficients.Source: Figure S3 · p. 23, S1.2 Neighbor activation in… · p. 23

S1.3 Dynamic threshold control and periodic refresh

  • Dynamic threshold control raises or lowers the threshold according to measured versus target active-block ratios, which include active and transition blocks.The reported PFC runs begin control after an initial 10,000-step full-activation stage, with each ratio check followed by a full-activation step.Source: S1.3 Dynamic threshold control and… · pp. 23–24
  • Shorter refresh intervals bring free-energy curves closer to Regular, while the mean density stays unchanged over tested intervals and active-block ratios return to their low baseline after refresh.Periodic refresh activates all blocks every Frefresh steps; between refresh events, selected blocks update at the base time step.Source: Figure S5 · p. 25, S1.3 Dynamic threshold control and… · p. 25

S2 sp-DBA integration with transform-domain solvers

The integration wraps each solver’s transformed-field update in a shared sp-DBA interface while retaining model-specific update routines. The supplement reports modest solver changes and contrasts this execution-level approach with adaptive methods that change representation or grid.

S2 sp-DBA integration with transform-domain solvers

S2.1 Integration procedure

  • sp-DBA calls the same model-specific update_core used by Regular, but executes it only for active and transition blocks while frozen blocks return early.Model adapters provide layout, indexing, launch configuration, and parameters for the solver-specific update.Source: S2.1 Integration procedure · p. 26

S2.2 Implementation overhead

  • 159–211 nonblank LOC add simple solver changes, or about 8–16% of each Regular solver core; including adapters brings totals to 256–356 LOC, or about 13–28%.The shared 696-LOC core is excluded from the per-solver counts.Source: Table S1 · p. 27, S2.2 Implementation overhead · p. 27

S2.3 Comparison with adaptive spectral and wavelet methods

  • Unlike adaptive spectral and wavelet workflows that alter the approximation space or grid, sp-DBA retains the transformed representation, regular array layout, transform operations, and update core.The comparison addresses implementation paths, not runtime or accuracy across methods.Source: S2.3 Comparison with adaptive spectral… · p. 27, Table S2 · p. 28

S3 XPFC polycrystalline coarsening

The XPFC application uses a pseudo-spectral model of polycrystalline coarsening and compares sp-DBA runs with Regular through grain statistics and thermodynamic quantities. Fixed-threshold tests agree with Regular, while normalized grain-size-distribution collapse becomes clearer at larger system sizes.

S3.1 Governing equations

  • The pseudo-spectral XPFC model evaluates local nonlinear terms in real space and the correlation term in Fourier space, where grain orientations broaden Bragg peaks into rings.The Fourier-amplitude regions are used to assign PFC block states.Source: S3.1 Governing equations · p. 28

S3.2 Numerical scheme and simulation configuration

  • ETD4 advances the Fourier-space linear term through an exponential propagator and evaluates the nonlinear contribution using its current value and three previous values.For small coefficient-function arguments, Taylor expansions avoid cancellation and apparent division by zero.Source: S3.2 Numerical scheme and simulation… · p. 29
  • Across four tested fixed thresholds, grain-growth trends and normalized grain-size distributions agree with Regular; dynamic threshold control keeps the active-block ratio more stable across stages and grid sizes.The production runs keep all Fourier blocks active for the first 10,000 steps to avoid premature freezing during the initial transient.Source: S3.2 Numerical scheme and simulation… · pp. 29–30

S3.3 Grain reconstruction and system-size comparison

  • The normalized grain-size-distribution collapse becomes clearer as system size increases, with the largest systems producing the smoothest late-stage distributions.Late-stage scaling plots are treated as qualitative agreement checks because grain labels contain post-processing noise.Source: S3.3 Grain reconstruction and… · pp. 31–32

S4 Reaction–cross-diffusion pattern formation

The RCD tests apply sp-DBA to a constant-coefficient multicomponent reaction–cross-diffusion model, whose Fourier modes require coupled component updates. The section also describes diagonalizing the diffusion matrix as a complementary update optimization.

S4.1 Governing equations

S4.2 Numerical scheme and simulation configuration

  • The semi-implicit Fourier update treats linear cross-diffusion implicitly and nonlinear reaction explicitly.The sp-DBA activation criterion combines component amplitudes and shares each selected block's state across all M components.Source: S4.2 Numerical scheme and simulation… · p. 33

S4.1 Governing equations

S4.2 Numerical scheme and simulation configuration

  • The RCD Fourier amplitude is localized in a compact low-wavenumber region, and the active blocks follow that region.Figure S10 visualizes a real-space component, Fourier amplitude, and block state for a 256 × 256 × 256 grid with M = 8; the average active-block ratio is about 3%.Source: S4.2 Numerical scheme and simulation… · p. 34, Figure S10 · p. 34

S4.3 Matrix eigendecomposition for GPU acceleration

  • For constant-coefficient tests, the diffusion matrix is diagonalized once, replacing a dense matrix inverse with two constant basis transforms and a k-dependent diagonal scaling.The diagonalized update holds the diagonal factor in registers and caches constant basis matrices in shared memory.Source: S4.3 Matrix eigendecomposition for GPU… · p. 34, Figure S11 · p. 35
  • Eigendecomposition reduces the cost of each coupled Fourier-mode update, while sp-DBA reduces the number of blocks where that update runs.The two optimizations therefore act on different factors of the RCD update cost.Source: S4.3 Matrix eigendecomposition for GPU… · p. 35
  • 28.1× at M = 48 compares sp-DBA + Eigen with Regular using the direct matrix update.The four configurations in Fig. 5d separate the contributions of block activation and diagonalization.Source: S4.3 Matrix eigendecomposition for GPU… · p. 35

S5 Digital backpropagation

The DBP application uses split-step Fourier propagation to recover multimode signals, applying sp-DBA to the linear half-steps. The reported spectral bands remain localized over much of the recovery distance, and activation follows their evolution.

S5.1 Governing equations

  • The multimode propagation model combines mode-wise attenuation and dispersion with fixed linear mode coupling and a time-domain Kerr phase rotation driven by total modal power.DBP approximates inverse propagation by reversing the dispersion, attenuation, and nonlinear parameters and applying the inverse mode-coupling propagator.Source: S5.1 Governing equations · p. 35

S5.2 Split-step Fourier implementation

  • Each symmetric split-step Fourier propagation step contains two frequency-domain linear half-steps and one time-domain nonlinear step.sp-DBA is applied only to the linear half-steps; FFT/IFFT calls, the nonlinear step, and split-step ordering remain unchanged.Source: S5.2 Split-step Fourier implementation · p. 36

S5.3 Simulation configuration and recovery metrics

  • sp-DBA activates a frequency block when its total modal spectral power exceeds the attenuation-rescaled threshold, applying that state to all M components in the block.Dynamic threshold control is enabled, with the default performance setting targeting a 10% average active-block ratio.Source: S5.3 Simulation configuration and… · pp. 36–37
  • The dominant spectral bands remain localized over much of the recovery distance, and the sp-DBA activation pattern follows their evolution.The figure presents spectrum versus recovery distance and frequency, alongside block-state evolution; its low, intermediate, and high values denote frozen, transition, and active regions.Source: Figure S12 · p. 37, S5.3 Simulation configuration and… · p. 38

S6 Numerical validation

The XPFC validation checks conservation and thermodynamic agreement, spectral error, and compatibility with dealiasing. The reported tests show agreement with Regular for mean density and no systematic increase in free-energy discrepancy.

S6.1 Conservation and free-energy evolution

S6.2 Spectral accuracy and numerical error

  • Outside the principal peak, the radial structure factor decays to approximately 10^-16, while the relative discrepancy after enabling sp-DBA remains about 10^-7.The reported residual is consistent with truncation-type error expected when spectrally small regions are retained from previous steps.Source: S6.2 Spectral accuracy and numerical… · p. 39, Figure S14 · p. 39

S6.3 Dealiasing compatibility

S7 Single-GPU performance evaluation

Single-GPU evaluation analyzes how sp-DBA changes update and end-to-end costs in XPFC and DBP, and compares measured acceleration with estimates based on active-block ratios.

S7 Single-GPU performance evaluation

  • Single-GPU timings used one NVIDIA A800-SXM4 GPU, GPU-synchronized wall-clock timing, and excluded file I/O and post-processing.MPI was disabled; measurements used the same CUDA version, warm-up procedure, and repeat count as the distributed tests.Source: S7 Single-GPU performance evaluation · p. 40
  • The end-to-end estimate accounts for transform, retained-model, and update costs, active-block ratio, and control overhead, with an Amdahl-type bound.The simpler active-ratio estimate omits control overhead; for the RCD comparison, measured acceleration also includes eigendecomposition.Source: S7 Single-GPU performance evaluation · pp. 40–41

S7.1 PFC cost decomposition and acceleration

  • In XPFC, most cost reduction comes from the transformed-field update, while transform and retained-model costs remain nearly unchanged.At larger grid sizes, the update workload better amortizes block-control overhead, bringing measured acceleration closer to the active-block-ratio estimate.Source: S7.1 PFC cost decomposition and… · p. 41, Figure S16 · p. 41

S7.2 DBP cost decomposition and acceleration

S8 Multi-GPU sp-DBA with distributed FFTs

Distributed tests apply sp-DBA to local transformed-field updates within a slab-decomposed FFT workflow. They assess update-stage scaling, communication, and activation stability.

S8.1 Distributed FFT workflow

S8.2 Strong and weak scaling

  • Under strong scaling, sp-DBA update time is below Regular; under weak scaling, it changes little while the active-block ratio stays near 10%.The tests use a fixed global grid for strong scaling and approximately fixed local grid area per GPU for weak scaling.Source: S8.2 Strong and weak scaling · p. 45, Figure S21 · p. 45
  • Distributed FFT and communication dominate total wall-clock cost, bounding overall acceleration by the local update’s share of the workflow.The distributed cost decomposition compares strong and weak scaling at fixed global grid size and approximately fixed local grid area, respectively.Source: S8.2 Strong and weak scaling · p. 45, Figure S21 · p. 45

S8.3 CUDA-aware MPI communication

S8.4 Activation stability under distributed execution

S9 Visualization

The visualization workflow uses separate tools for field rendering and XPFC grain reconstruction, and derives activation maps from recorded block labels.

S9 Visualization

  • ParaView renders fields and Fourier amplitudes, while OVITO handles XPFC grain reconstruction; activation maps come from recorded block labels.Scalar quantities and performance plots were prepared from exported analysis files; post-processing was excluded from solver timing unless stated otherwise.Source: S9 Visualization · p. 47
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