Multi-Resolution Processing Strategy

• Multi-resolution processing strategy is a set of methods that use variable detail levels to handle large, complex datasets in scientific computing.
• It is applied in HPC simulations, adaptive mesh refinement, and uncertainty visualization to reduce storage needs and computational overhead.
• The approach adapts error bounds with advanced post-processing to maintain high visual and numerical fidelity even under heavy compression.

A multi-resolution processing strategy refers to a set of computational methods that exploit the varied scale requirements within scientific, statistical, or engineering problems to achieve improved efficiency, accuracy, and interpretability. Such strategies are foundational in scientific data reduction, simulation, and visualization, especially in contexts involving extremely large or spatially complex datasets. They are most commonly encountered in adaptive numerical simulation (notably, Adaptive Mesh Refinement, or AMR), multi-scale data compression, and uncertainty quantification for high-performance computing (HPC) applications producing vast data volumes.

1. Scientific Motivation and Principles

Multi-resolution processing leverages the principle that many scientific phenomena do not require uniform spatial or temporal detail everywhere. Critical regions (e.g., shock fronts, turbulence, phase boundaries, or salient features) may require high spatial (or temporal) resolution, whereas more homogeneous regions can be represented with coarser fidelity without sacrificing downstream accuracy or insight. This enables dramatic improvements in storage, computational efficiency, and data transfer by focusing resources where most impactful.

In high-performance computing, these techniques are crucial for exascale simulations (cosmology, turbulence, weather), where raw data output can be petabytes per simulation, outpacing storage, I/O, and analyst interpretability.

2. Workflow Components and Algorithmic Structure

A high-quality workflow for multi-resolution scientific data reduction, as expounded in recent research, typically consists of the following stages:

1. Compression-Oriented Region of Interest (ROI) Extraction: Uniform-field simulation data are partitioned into blocks, and a quantitative importance metric (e.g., value range) is computed for each. A fraction—determined by either a percentage or scientific heuristic—of blocks with maximal significance are marked as Regions of Interest and retained at full resolution. The remainder are retained in a coarser, downsampled representation.

For each block B: r_B = max(B) – min(B) Select top x% of blocks as ROI; others downsampled.

1. Multi-Resolution Mesh Preparation: AMR simulation outputs naturally divide the domain into data blocks at varying resolutions according to dynamic refinement criteria. Uniform mesh data, once partitioned by the ROI strategy, can be transformed into a pseudo-AMR, multi-scale format, enabling compatibility with AMR-specific compression approaches.
2. Compressor Optimization & Integration:

Lossy compressors, such as SZ2, ZFP (both block-wise), and SZ3 (global predictor), are reconfigured for optimal performance on multi-resolution data: - SZ3MR (Multi-Resolution SZ3): - Implements dynamic array padding to accommodate local interpolation at all hierarchical levels in irregular, sparsely refined regions. - Utilizes an adaptive, resolution-level-dependent error bound:

$$eb_l = eb \cdot \left(\min(\alpha^{\text{maxlevel} - l}, \beta)\right)^{-1}$$

where $\alpha, \beta$ are empirically tuned. - Block-wise compressors (SZ2, ZFP): - Supplemented with a Bézier curve-based, strictly bounded post-processing step to reconcile boundary discrepancies between adjacent blocks, which often yield visible compression artifacts. The boundary update is formulated as:

$d_4' = \max(\min(B(0.5), d_4 + eb), d_4 - eb)$

where $B(t)$ is a quadratic Bézier polynomial interpolant.

1. Post-Processing for Visual and Metric Enhancement: This step enforces strict pointwise absolute error bounds while smoothing inter-block boundaries, resulting in significantly improved SSIM and PSNR, especially on block-compressed data.
2. Uncertainty Visualization: Integrated uncertainty quantification visualizes the impact of lossy compression on derived quantities (e.g., isosurfaces). Each voxel's compression error is modeled as a normal distribution and propagated, using probabilistic marching cubes, into uncertainty bands on scientific visualization products.

(Adapted: central workflow stages as described above)

3. Adaptive Mesh Refinement (AMR) and ROI Extraction

In AMR, simulation grids are dynamically refined:

• Given a mesh, each block is refined if its average value exceeds a domain-specific threshold:

$$\text{if}~\text{avg}(B) > \text{threshold}~\text{then refine block}~B$$

• The domain is represented as a union of disjoint blocks at different resolutions, which can be exploited for compression.

For uniform data, compression-oriented ROI extraction emulates AMR by identifying and preserving high-detail blocks and downsampling the remainder. Empirically, scientific feature integrity (as measured by SSIM, with $>0.999$ in case studies) is preserved even when only 15\% of blocks are kept at full resolution.

4. Compressor Adaptation and Post-Processing

Block-wise Compressors (SZ2, ZFP)

Block-based approaches optimize for speed and locality but can lose accuracy across block boundaries. The workflow implements:

• Error-bounded Bézier post-processing for boundary smoothing:

$B(t) = (1-t)^2 d_3 + 2(1-t)t d_4 + t^2 d_5$

Each midpoint is updated as $d_4' = B(0.5)$, clamped to the absolute error bound, ensuring both spatial continuity and strict error control.

Global Compressors (SZ3/SZ3MR)

For globally predicting compressors, multi-resolution blocks introduce irregular boundaries and short array dimensions. SZ3MR introduces:

• Dynamic padding: Guarantees availability of predictor neighbors.
• Level-dependent adaptive error bounds (as above) to manage sparsity and maintain global error contracts.

5. Uncertainty Quantification and Visualization

Understanding the propagation of compression-induced error is vital for scientific post-analysis:

• At compression, empirical mean and variance of per-voxel error are stored.
• During visualization, e.g., isosurface extraction, these errors are propagated using a probabilistic marching cubes algorithm, which displays "uncertainty bands"—visual regions where the compressed and original isosurface locations are uncertain due to compression.

This enables both qualitative and quantitative assessment of compression impacts, informing parameter choices and end-of-pipeline scientific validity.

6. Experimental Evaluation and Performance

Dataset Coverage

The workflow is evaluated on both uniform and AMR datasets across multiple scientific domains (e.g., Nyx, WarpX, Rayleigh-Taylor, Hurricane), with block sizes and ROI fractions calibrated to scientific need.

Key Findings

• Compression Ratios and Quality: Up to 300:1 compression achieved with SSIM near 1 and PSNR improvements of up to 10 dB over prior AMR compression baselines.
• Computational Overhead: Integration of post-processing results in 1–3% additional runtime, effectively amortized via parallelization (OpenMP).
• Visual Quality: Post-processing and compressor adaptation significantly reduce block artifacts and improve the fidelity of features in the decompressed data.
• Scientific Fidelity: Uncertainty visualization recovers true topological structures and identifies regions of interpretive risk due to heavy compression.

Performance Summary Table

Aspect	Experimental Result	Improvement
Compression Ratio	Up to 300:1	Higher than baseline AMR/ROI-only
Visual Quality	PSNR +2–10 dB, SSIM >0.999	Severe block/artifact reduction
Storage Savings	10–100x reduction vs. raw	Immediate HPC application benefit
Overhead	1–3% of total runtime for post-processing	Negligible, scalable
Usability	Uniform and AMR, in situ and offline, direct visualization	Broad domain and mode applicability

7. Scientific and HPC Significance

This workflow extends traditional multi-resolution approaches, formerly restricted to AMR-native simulation codes, to include uniform-grid data. It systematically bridges the gap between multi-scale representations and blockwise/global lossy compressors, delivers provably error-bounded and visually superior results, and incorporates uncertainty quantification into visualization. As a result, it enables both storage- and compute-efficient pipelines for massive-scale simulations, while maintaining interpretability and quantitative reliability for subsequent scientific analysis.

All algorithms, equations, workflow structure, and experimental findings are drawn directly from the content of the cited 2024 paper. For in-depth algorithmic and experimental details, see Sections 3–4 and associated figures and tables as referenced.