VEC O GEN: Efficient Lorentz Vector Computation

• VEC O GEN is a framework collection that enables efficient generation and vectorized manipulation of Lorentz vectors in high-energy physics.
• It employs a flexible C++ template API supporting various coordinate systems, precisions, and backend accelerators for robust vector algebra.
• The platform leverages SYCL and USM to achieve near-native CUDA speeds, optimizing batch processing on heterogeneous computing architectures.

VEC O GEN denotes a collection of technical advances and software frameworks centered on efficient generation and manipulation of mathematical vectors, with particular focus on high-dimensional physics data and acceleration portability in modern heterogeneous computing environments. A canonical instantiation is GenVectorX, a performance-portable, SYCL-based C++ library for Lorentz vector algebra, supporting both generation ("GEN") and vectorized operation ("VEC") on large-scale datasets relevant to high energy physics. The concepts underlying VEC O GEN integrate formal algebraic representations, numerically robust transformation routines, and optimized batch processing abstraction spanning CPUs, NVIDIA/AMD/Intel GPUs, and other accelerators (Dessole et al., 2023).

1. Architecture and API Design

GenVectorX provides a generic C++ template API encapsulating Lorentz vectors in arbitrary coordinate systems and precisions, parameterized by execution backend. All types reside in the ROOT::Experimental:: namespace. The key abstraction is

$c=1$5

with coordinate system (e.g., Cartesian4D or Polar4D), floating-point precision (float or double), and backend tag (Backend::CPU, Backend::CUDA, Backend::SYCL). Compile-time traits classes, such as LVTraits<LorentzVector<...>>, expose vectorization widths, metric signatures, and dimensions. Object construction, component access (e.g., p.E(), p.px()), arithmetic operator overloads, inner products (dot(p, q)), invariant mass (p.mass()), Lorentz boosts (p.[boost](https://www.emergentmind.com/topics/boost)(β⃗)), and batch kernel invocation are all available through a concise, STL-style interface.

2. Mathematical Underpinnings and Physics Encodings

VEC O GEN frameworks encode exact physical invariants of special relativity for 4-momentum vectors:

• Four-momentum: $p^\mu = (E, p_x, p_y, p_z)$, often with $c=1$ units.
• Minkowski metric: $$\eta_{\mu\nu} = \operatorname{diag}(+1, -1, -1, -1)$$.
• Invariant mass: $m^2 = p^\mu p_\mu = (E)^2 - |\mathbf{p}|^2$.
• Lorentz-invariant dot product: $p \cdot q = \eta_{\mu\nu} p^\mu q^\nu$.
• Lorentz boosts along arbitrary $\vec{\beta}$, realized using boost matrices in the library's device-callable routines.

All mathematical routines (addition, subtraction, inner products, boosts) are implemented as device-executable methods, permitting vectorized and batched application on large event sets.

3. SYCL Implementation and Optimization

VEC O GEN leverages SYCL's device-agnostic abstraction layer to support heterogeneous acceleration using both buffer/accessor and Unified Shared Memory (USM) memory models, determined by compile-time flags. Key implementation points include:

• Kernels defined as functor structs or SYCL lambdas for batch calculations.
• Memory layouts: default Array-of-Structures (AoS), with optional Structure-of-Arrays (SoA) for optimized SIMD utilization via traits classes.
• Vectorization: Automatic vectorization of inner loops is enabled with USM+SoA layouts; SYCL subgroups expose hardware-specific shuffles and broadcasts.
• Work-group sizes tunable per architecture (optimal default: 256 threads per group on NVIDIA, smaller for AMD and Intel GPUs).
• Specialization constants encode batch or vector lengths directly into kernels, improving launch efficiency.
• USM pointer mode typically matches or outperforms buffer mode, especially for large $N$, closely approaching native CUDA performance.

4. Performance Portability and Benchmark Results

GenVectorX was benchmarked on large-scale invariant mass and Lorentz boost computations on diverse platforms:

System	GPU	Peak FP32	Speedup @ $N=10^8$ (CUDA)	Speedup (SYCL/oneAPI+USM)	Speedup (SYCL/AdaptiveCpp+USM)
OLGPU	NVIDIA A100	19.5 TFLOPS	$\sim$100×	$\sim$95×	$c=1$097×
Victus	RTX 3060	12.7 TFLOPS
OLICE	NVIDIA L4	30.3 TFLOPS

USM pointer implementations consistently achieved within 5–10% of native CUDA, while buffer-based approaches exhibited overhead for large $c=1$1. Scaling tested up to $c=1$2 vectors, with weak and strong scaling saturated only by hardware capacity. On AMD/Intel GPUs, SYCL/USM realized 80%/50% of theoretical bandwidth, respectively (Dessole et al., 2023).

5. Batch Processing, Example Usage, and Kernel Patterns

The library exposes batch kernel templates for efficient, high-throughput computation. Example workflows include:

• Construction of $c=1$3 Lorentz vectors on host, transfer to device via USM allocations.
• Batch invariant mass kernel launches, specifying workgroup (e.g., 256) and block counts, with results transferred back and postprocessed.
• Vectorized application of Lorentz boosts via batch kernel.

Typical usage involves minimal user-visible device-specific code. All device abstractions are encapsulated in headers, controlled via CMake flags to select backends (-Dcuda=ON, -Dsycl=ON). Device-side limitations include disabled I/O operators and exceptions.

6. Portability, Limitations, and Future Development

VEC O GEN is designed for source-level portability:

• The same template instantiations target CPU, CUDA, and SYCL with negligible code divergence ($c=1$4).
• SYCL allows cross-vendor GPU support (NVIDIA, AMD, Intel) and straightforward integration into event-based analytics pipelines (planned ROOT RDataFrame support).
• Planned developments include native SoA containers for optimal SIMD utilization, support for half-precision (float16) arithmetic, extended support for Poincaré transformations (compositions of boosts and rotations), and runtime autotuning of work-group/specialization constants.
• Device-side buffer limitations may require USM pointers for maximal throughput on some hardware.

7. Significance and Application in High-Performance Physics

VEC O GEN, as operationalized in GenVectorX, addresses practical challenges in the acquisition, simulation, and analysis of physics event data at modern collider facilities such as the LHC. It provides a canonical, performance-portable substrate that balances physics fidelity (via covariant vector operations and invariants), computational throughput, and code simplicity. It enables near-optimal utilization of accelerator resources without diverging source implementations, representing a key step toward unified, high-efficiency HEP data pipelines spanning heterogeneous compute ecosystems (Dessole et al., 2023).