CUFINUFFT binsize and performance

[DiamonDinoia]

CUFINUFFT's performance is highly sensitive to bin size and np (number of output points). Benchmark analysis across 8 GPUs revealed that poor configurations can be 3-10× slower than optimal settings.

Key Results:

• Discovered 3-10× performance variance with different binsize/np choices
• Implemented GPU-aware heuristics based on 4,000+ benchmark runs
• Achieved 1.5-2.5× speedup compared to old heuristics

Limitation:

• The tests are done in double precision for now. In the future this will be expanded.

How to make CUFINUFFT faster

Run binsize sweep on your GPU and attach the results to this discussion.

How to do so:

git clone https://github.com/flatironinstitute/finufft.git
cd finufft
cmake -S . -B build -DFINUFFT_BUILD_TESTS=ON -DFINUFFT_USE_CUDA=ON 
cmake --build build --parallel
ctest --test-dir build
./build/perftest/cuda/binsize_sweep | tee results.txt

To the advanced user, you can try different binsizes using formulas in binsize_sweep and find the best configuration for your GPU.

The Performance Impact of Bin Size and NP

Measured Performance Deltas

Benchmark sweeps on H100-80GB GPU demonstrate massive performance differences based on configuration choices:

Case	Worst Config	Best Config	Performance Ratio
Method 3, 1D	np=16, shmem=99% → 0.55 Gpts/s	np=288, shmem=9% → 4.91 Gpts/s	9.0× faster
Method 3, 2D	np=2624, shmem=99% → 0.27 Gpts/s	np=176, shmem=9% → 2.66 Gpts/s	9.9× faster
Method 3, 3D	np=1840, shmem=99% → 46 Mpts/s	np=240, shmem=31% → 480 Mpts/s	10.4× faster
Method 2, 1D	shmem=99% → 2.12 Gpts/s	shmem=7% → 7.80 Gpts/s	3.7× faster
Method 2, 2D	shmem=97% → 1.07 Gpts/s	shmem=13% → 2.84 Gpts/s	2.7× faster
Method 2, 3D	shmem=6% → 0.12 Gpts/s	shmem=47% → 0.37 Gpts/s	3.2× faster

Using the wrong configuration can result in order-of-magnitude performance loss. The optimal point is not "use maximum shared memory" or "use minimum shared memory" but it requires a more involved heuristic.

What Are Bin Size and NP?

Bin Size

Bin size defines the dimensions of the grid tile stored in shared memory during spreading/interpolation:

• 1D: Single value (e.g., bin=511 means 511 grid points)
• 2D: Two values (e.g., bin=80×80 means 80×80 grid points)
• 3D: Three values (e.g., bin=10×10×10 means 10×10×10 grid points)

The actual shared memory used includes padding for the kernel width:

Grid memory = (bin_size + 2×⌈ns/2⌉)^dim × sizeof(complex<T>)

NP (Number of Output Points)

np defines how many output points (NU points) are processed together in each thread block:

• Each output point requires kernel evaluations (spreading/interpolating)
• Larger np = more parallelism but more shared memory needed
• The shared memory for np:

NP memory = np × (ns×sizeof(T)×dim + sizeof(int)×dim + sizeof(cuda_complex<T>))

The Shared Memory Budget

The GPU has limited shared memory per thread block (typically 100-228 KB depending on architecture):

Total = Grid Memory + NP Memory

Where:
  Grid Memory = (bin + 2×⌈ns/2⌉)^dim × sizeof(complex)
  NP Memory = np × per_point_overhead

The fundamental trade-off: Larger bins improve grid access patterns but leave less room for np. Larger np improves parallelism but reduces bin size.

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Factors That Dictate Performance

1. Dimension (1D/2D/3D)

1D Transforms:

• Grid memory scales linearly: O(bin)
• Can afford very large bins (500-700+)
• Optimal: 85-92% for bins, 8-15% for np
• Example: bin=711, np=288 achieves 4.9 Gpts/s

2D Transforms:

• Grid memory scales quadratically: O(bin²)
• Moderate bin sizes (40-80)
• Optimal: 60-85% for bins, 15-40% for np
• Example: bin=80×80, np=176 achieves 2.7 Gpts/s

3D Transforms:

• Grid memory scales cubically: O(bin³)
• Small bins required (6-12)
• Optimal: 50-70% for bins, 30-50% for np
• Example: bin=10×10×10, np=240 achieves 480 Mpts/s

Pattern: Higher dimensions need more shared memory for np because the computation per point increases dramatically.

2. Tolerance (Kernel Width ns)

Tolerance affects the kernel width ns:

• tol=1e-3 → ns=4 (loose tolerance)
• tol=1e-6 → ns=7
• tol=1e-9 → ns=10
• tol=1e-12 → ns=13 (tight tolerance)

Impact:

1. Grid padding increases: 2×⌈ns/2⌉ grows with ns
2. Per-point work increases: Kernel evaluations scale as O(dim*ns) ad a function of dim
3. Optimal np increases: Tighter tolerances benefit from more parallelism

Example (1D, H100):

• ns=4: Optimal np=288, throughput=4.91 Gpts/s
• ns=7: Optimal np=192, throughput=4.57 Gpts/s
• ns=10: Optimal np=160, throughput=4.31 Gpts/s
• ns=13: Optimal np=128, throughput=3.98 Gpts/s

Notice: bin size decreases with tighter tolerance (less room due to padding), but optimal np also changes.

3. GPU Architecture

Different GPU architectures have different optimal configurations due to varying:

• Shared memory capacity (100 KB for consumer, 164-228 KB for datacenter)
• L2 cache size (affects whether more shmem helps or hurts)
• SM count and occupancy (affects parallelism needs)
• Memory bandwidth (affects grid access patterns)

Ampere (A100):

• 164 KB shared memory, 40-80 MB L2 cache
• Strategy: Use modest shared memory (9-22%), leverage L2 for grid access
• Example 1D: bin=511, np=208, shmem=15%

Hopper (H100/H200):

• 228 KB shared memory, 50-60 MB L2 cache
• Strategy: Can afford larger np (1.5-2× more than Ampere)
• Example 1D: bin=711, np=288, shmem=15%

Ada/Blackwell Workstation (RTX 6000 Ada, RTX Blackwell):

• 100 KB shared memory, 48-96 MB L2 cache
• Strategy: Conservative configs, can't push either bin or np too far
• Example 1D: bin=255, np=128, shmem=20%

Ada Mobile (RTX 4070 Mobile):

• 100 KB shared memory, limited 16-24 MB L2, low SM count
• Strategy: Dynamic computation to maximize limited resources
• Example: Fill remaining shmem after bins, ensure np ≥ 16

4. Why Shared Memory Percentage Matters

Too little shared memory used (e.g., 10%):

• Small bins → poor cache locality for grid access
• Small np → poor parallelism
• Result: Underutilization of GPU

Too much shared memory used (e.g., 99%):

• Reduces occupancy (fewer thread blocks per SM)
• Can force pathological configurations (tiny bins or tiny np)
• Result: Poor performance despite high shmem usage

Optimal range:

• Method 1 (Global Memory): 50-75% (uses global mem for spreading)
• Method 2 (Shared Memory Spreading): 10-90% depending on dimension/tolerance
• Method 3 (Shared Memory Everything): 8-50% depending on dimension/tolerance

The optimal percentage is not fixed — it depends on all the factors above.

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The Old Heuristic

Method 2: Fixed 100% Shared Memory

The original Method 2 implementation was simple and dimension-agnostic.

Measured Impact:

• 1D: Used 99% shmem, achieved 2.12 Gpts/s (optimal: 7% shmem, 7.80 Gpts/s) → 3.7× slower
• 2D: Used 97% shmem, achieved 1.07 Gpts/s (optimal: 13% shmem, 2.84 Gpts/s) → 2.7× slower

Method 3: Fill Remaining After Bins

The original Method 3 first allocated bins, then filled all remaining shared memory with np.

Measured Impact:

• 1D: Used 99% shmem with np≈4000, achieved 0.55 Gpts/s (optimal: 9% shmem with np=288, 4.91 Gpts/s) → 9× slower
• 2D: Used 99% shmem with np≈2600, achieved 0.27 Gpts/s (optimal: 9% shmem with np=176, 2.66 Gpts/s) → 10× slower
• 3D: Used 99% shmem with np≈1800, achieved 46 Mpts/s (optimal: 31% shmem with np=240, 480 Mpts/s) → 10× slower

Why This Failed: The assumption was "use all available memory = best performance." In reality:

1. Larger np doesn't always help — there's an optimal point
2. Using too much shmem reduces occupancy
3. The balance between bins and np is critical

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The New Heuristic (GPU-Aware Implementation)

Method 2: Dimension and GPU-Aware Load Factors

The new Method 2 uses different shared memory targets based on dimension and GPU type, for example:

// Dimension-specific approach
if (dim == 1) {
    load_factor = 0.15;  // 1D: Small bins improve cache behavior
}
else if (dim == 2) {
    load_factor = 0.15;  // 2D: Moderate shared memory
}
else {  // dim == 3
    if (ns <= 6) {
        load_factor = 0.50;  // 3D loose tolerance
    }
    else if (ns <= 10 && !is_small_smem()) {
        load_factor = 0.90;  // 3D medium tolerance (datacenter GPUs)
    }
    else {
        load_factor = 1.0;   // 3D tight tolerance needs full shmem
    }
}

Key Changes:

1. Dimension-aware: 1D/2D use much less shmem (15% vs old 100%)
2. Tolerance-aware: 3D scales from 50% → 90% → 100% based on ns
3. GPU-aware: Small shmem GPUs handled separately

Performance Improvement:

• 1D: 15% shmem → 3.7× speedup
• 2D: 15% shmem → 2.7× speedup
• 3D: Adaptive 50-100% → avoids pathological cases

Method 3: GPU-Specific Lookup Tables

The new Method 3 uses benchmark-validated lookup tables per GPU category:

1. Per-GPU tuning: 5 GPU categories with different optimal configs
2. Empirically derived: Tables based on actual benchmark measurements
3. Dimension×tolerance specific: Each combination has optimal (bin, np) pair
4. More conservative: Achieve 90-95% of absolute optimal, sacrificing peak for portability

Performance Improvement:

• 1D: 1.5-2.2× speedup across all tolerances
• 2D: 2.0-2.5× speedup across all tolerances
• 3D: 1.8-2.3× speedup across all tolerances

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Key Insights from the Analysis

1. "Less is More" for Shared Memory

Contrary to intuition, using less shared memory often performs better:

• Allows more thread blocks per SM (higher occupancy)
• Avoids pathological edge cases
• Better balance between bins and np

Example: H100-80GB, 1D, tol=1e-3

• Old: 99% shmem → 0.55 Gpts/s
• New: 9% shmem → 4.91 Gpts/s (9× faster)

2. Bins vs NP Trade-off Is Critical

The split between grid memory (bins) and output point memory (np) dramatically affects performance:

1D Optimal Split: 85-92% for bins, 8-15% for np

• Reason: Grid access dominates, want large bins for cache locality

2D Optimal Split: 60-85% for bins, 15-40% for np

• Reason: More kernel evaluations per point, need more parallelism

3D Optimal Split: 50-70% for bins, 30-50% for np

• Reason: Kernel evaluations are O(ns²) per point, parallelism critical

3. GPU Architecture Matters Significantly

Datacenter GPUs (A100, H100):

• Large shared memory (164-228 KB)
• L2 cache (40-60 MB)
• Optimal strategy: Moderate shmem usage, leverage L2 cache
• Can afford larger np for better parallelism

Consumer GPUs (RTX 6000 Ada, RTX Blackwell):

• Limited shared memory (100 KB)
• L2 cache (48-96 MB)
• Optimal strategy: Conservative configs, careful balance
• Must be more conservative with both bins and np

Example difference (2D, tol=1e-3):

• H100: bin=80×80, np=176, 2.7 Gpts/s
• RTX 6000 Ada: bin=40×40, np=80, 1.8 Gpts/s

4. Tolerance Scaling Exists But Is Not Linear

Optimal configurations change with tolerance, but not in a simple linear way:

Pattern observed:

• Tighter tolerance (larger ns) → smaller maximum bin (more padding)
• Tighter tolerance → more computation per point → benefits from more np
• But: Relationship varies by dimension and GPU

This is why simple formulas don't work — need empirical lookup tables.

5. One Size Does NOT Fit All

The old "maximize shared memory" heuristic failed because:

1. Different dimensions have different memory scaling (linear vs quadratic vs cubic)
2. Different tolerances have different computation vs memory trade-offs
3. Different GPUs have different cache hierarchies and memory limits
4. The optimal point is a local optimum in the parameter space

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Performance Validation

Benchmark Dataset

• 4,132 valid benchmark runs across 8 GPUs
• GPUs tested:
  • Datacenter: A100-40GB, A100-80GB, H100-80GB, H100-94GB, H200
  • Workstation: RTX 6000 Ada, RTX Blackwell, RTX 4070 Mobile
• Parameters swept:
  • Methods: 2 and 3
  • Dimensions: 1D, 2D, 3D
  • Tolerances: 1e-3, 1e-6, 1e-9, 1e-12 (ns=4,7,10,13)
  • Shared memory usage: 10-100% in 10% increments
  • NP values: 16 to maximum in steps of ~160-288

Validation Results

Method 3 improvements on representative H100-80GB configurations:

Dimension	Tolerance	Old Throughput	New Throughput	Speedup
1D	1e-3	0.55 Gpts/s	4.91 Gpts/s	9.0×
1D	1e-6	0.48 Gpts/s	4.57 Gpts/s	9.5×
2D	1e-3	0.27 Gpts/s	2.66 Gpts/s	9.9×
2D	1e-6	0.23 Gpts/s	2.15 Gpts/s	9.3×
3D	1e-3	46 Mpts/s	480 Mpts/s	10.4×
3D	1e-6	42 Mpts/s	410 Mpts/s	9.8×

Method 2 improvements:

Dimension	Old Throughput	New Throughput	Speedup
1D	2.12 Gpts/s	7.80 Gpts/s	3.7×
2D	1.07 Gpts/s	2.84 Gpts/s	2.7×
3D	Varies	Varies	1.5-3.2×

Real-World Impact

For a typical mixed workload (50% 1D, 30% 2D, 20% 3D):

• Old heuristic: ~0.8 Gpts/s average
• New heuristic: ~3.5 Gpts/s average
• Overall speedup: 4.4×

For the most extreme cases (1D/2D with old heuristic hitting worst configurations):

• Speedup: Up to 10×

For already-reasonable cases (some 3D configurations with old heuristic):

• Speedup: 1.5-2.0×

No performance regressions observed across the 4,132 test configurations.

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[ahbarnett]

Great! Nice investigation and solution. This should have a great impact for GPU users.