# Investigating Supercomputer Performance with Sustainability in the Era of Artificial Intelligence
## 2 Fundamentals
### Performance  
Measuring the performance of supercomputers is a complex task, in view of the fact that there is no universally accepted metric that accounts for every aspect of the operations. Instead, the capabilities and behavior of the supercomputers are typically measured using multiple metrics. The number of operations performed and the computational time taken under prescribed conditions within a specific context are the two fundamental parameters for evaluating the performance of supercomputers. These metrics can be used either in isolation or in combination, depending on the context. The most widely used metric for measuring supercomputer performance is FLOPS. A floating-point operation is the multiplication or addition of real numbers formatted in a machine-understandable format, allowing for flexible manipulation. However, the true performance of a supercomputer lies in its ability to solve real-world problems and produce useful results with societal impact, such as simulating physical phenomena. To facilitate performance comparisons across supercomputers, the HPC community adopted standardized benchmarks for measuring the performance [18]. Supercomputers typically consist of hundreds of thousands to millions of processor cores. Unlike cores in conventional small-scale computers (e.g., laptops), the cores in a supercomputer are either physical CPU cores or physical GPU cores with interconnected communication systems. However, CPUs and GPUs are often combined into a single supercomputing system, so both types of cores or other accelerators can be found in one machine (e.g., the El Capitan supercomputer has 11,039,616 combined CPU and GPU cores, and Frontier has a combined 8,699,904 CPU and GPU cores). These cores work collaboratively to solve complex problems. The number of cores significantly influences the computational power, performance, and power consumption of a supercomputer. Currently, the most powerful and fastest supercomputer is El Capitan with 1742.00 Pflops and over 11 million cores.  
The performance of supercomputers is measured using benchmarks. Benchmarking involves testing and measuring a supercomputer's performance by evaluating the time taken to run a standard program. Two key metrics, Rmax (flops) and Rpeak (flops), are typically used to compare performance. Rpeak represents the theoretical peak performance of a supercomputer, calculated by multiplying the peak performance of its cores by the total number of cores. This metric is based on the hardware specifications and assumes ideal conditions for achieving Max efficiency. Rmax refers to the performance achieved at the Max level during the execution of LINPACK. LINPACK is a benchmark used for mathematical computations on large-scale square matrices (Nmax X Nmax). Each matrix is a floating-point number matrix table with the flexibility to choose a big Nmax. Rmax and the Rpeak are used by TOP500 for ranking the performance of supercomputers. Rmax is the actual performance of a supercomputer, which is typically lower than Rpeak due to inefficiencies or unforeseen issues encountered during the execution of the benchmark in real-world environments [19].