Roofline Model Performance Analysis

Roofline Model Performance Analysis is the visual performance modeling framework that characterizes the performance ceiling of a compute kernel as limited by either computational throughput or memory bandwidth — using arithmetic intensity (operations per byte transferred) as the key metric to identify the dominant bottleneck and guide optimization strategy.

Roofline Model Fundamentals:
- Arithmetic Intensity (AI): ratio of FLOPs to bytes transferred from/to memory — AI = total_FLOPs / total_bytes_moved; measured in FLOP/byte
- Performance Ceiling: attainable performance = min(peak_FLOPS, peak_bandwidth × AI) — the lower of compute and memory bandwidth limits determines achievable performance
- Ridge Point: the AI value where compute and memory ceilings intersect — kernels with AI below ridge point are memory-bound; above are compute-bound; ridge point = peak_FLOPS / peak_bandwidth
- Example: GPU with 100 TFLOPS peak and 2 TB/s bandwidth has ridge point at 50 FLOP/byte — matrix multiply (AI ~100+) is compute-bound; vector addition (AI = 0.25) is memory-bound

Constructing the Roofline:
- Memory Roof: diagonal line with slope = peak memory bandwidth — applies to memory-bound kernels where performance scales linearly with arithmetic intensity
- Compute Roof: horizontal line at peak computational throughput (FLOPS) — applies to compute-bound kernels where memory bandwidth is not the bottleneck
- Multiple Ceilings: additional ceilings for L1/L2 cache bandwidth, special function unit throughput, and instruction-level parallelism — each ceiling creates a lower sub-roof that may limit specific kernels
- Achievable vs. Peak: actual performance typically 50-80% of roofline ceiling — instruction overhead, pipeline stalls, and imperfect vectorization create gaps between achievable and theoretical performance

Using Roofline for Optimization:
- Memory-Bound Kernels (AI < ridge point): optimization strategies focus on reducing data movement — caching/tiling, data compression, reducing precision (FP32→FP16), and eliminating redundant loads
- Compute-Bound Kernels (AI > ridge point): optimization strategies focus on increasing computational throughput — vectorization (SIMD/tensor cores), reducing instruction count, and increasing ILP
- Increasing AI: algorithmic changes that increase FLOPs-per-byte-moved shift the kernel rightward on the roofline — tiling a matrix multiply to reuse cached data dramatically increases effective AI
- Profiling Integration: NVIDIA Nsight Compute and Intel Advisor directly plot kernel performance against the roofline — shows how far each kernel is from the ceiling and which optimization would help most

The roofline model is the essential first-step analysis tool for performance optimization — it prevents the common mistake of optimizing compute throughput for a memory-bound kernel (which yields zero improvement) or vice versa, directing engineering effort to the actual bottleneck.