ETP4HPC SRA 6 White Paper - Mathematical Methods and Algorithms
Résumé
This is a white paper released as part of the ETP4HPC’s Strategic Research Agenda 6. Importance of mathematical methods and algorithms Advancing and introducing new mathematical methods and algorithms is critical to ensure efficient use of current and future HPC architectures and technologies. Furthermore, they play an important role in enabling new ways of using HPC systems to address future societal and scientific challenges like climate change. Mathematical methods and algorithms are also needed to allow for new competitive solutions in the HPC market. Finally, mathematical methods and algorithms are crucial for exploiting future non-von-Neuman computer architectures like quantum computers. Scalable algorithms Making computer architectures more and more parallel is the key factor for reaching beyond exascale performance levels in terms of throughput of arithmetic operations. While general strategies like increasing the inherent concurrency of applications, the ability to exploit all levels of hardware-level parallelism, and avoidance of communication are well-known, their practical realisation remains highly challenging. In the section on Scalable Algorithms the following research priorities have been identified: n Communication avoiding and hiding algorithms, where new algorithms must be designed to avoid and hide communication beyond the linear solvers. n Exploiting multiple levels of parallelism: Algorithms have to become more flexible in leverage different levels of parallelism and support load balancing. n Increasing the level of parallelism: Parallel-in-Time algorithms need to be integrated with spatial parallelisation into applications. Approximate computing and correctness An attractive strategy for improving performance on current and upcoming hardware architectures is the use of approximate computing techniques. These aim to improve the performance and scalability of algorithms at the cost of a controlled loss of accuracy and robustness. For instance, today’s computing devices often provide significantly higher throughput of low-precision arithmetic operations. The section Approximate Computing and Correctness puts the focus on the following research priorities: n Low-precision arithmetics: Derive rigorous error bounds for mixed-precision algorithms and efficient implementation of such algorithms. n Lossy data compression techniques: Expand the range of applications where compression techniques can be applied and address side-effects like workloads becoming irregular and unpredictable. n Randomization: Establishment of tight error bounds for randomization techniques and better integration in numerical software libraries. Resilience and correctness Given the increasing complexity of both HPC hardware architectures as well as HPC software solutions, supporting resilience and ensuring correctness are becoming increasingly challenging. Therefore, there is a need for advances in algorithms that allow applications to maintain their performance and provide correct results despite the occurrence of hardware or software failures. Furthermore, with emerging code generation tools, the correctness of results needs to be guaranteed as software development processes evolve. Based on an analysis of the state-of-the-art and future challenges in the section Resilience and Correctness the following research topics are proposed: n Minimise or avoid check-pointing: Design algorithms that do not rely heavily on periodically saving the program's, support scaling on HPC systems, and can cope with heterogeneous architectures. n Communication-avoiding algorithms: Develop and implement communication-avoiding algorithms to enhance resiliency for HPC. n Software correctness: Research on correctness challenges that are specific to HPC and its entire software stack as well as on tools that address these challenges. Algorithms and methods for hybrid mechanistic and data-driven modelling In various areas, combining mechanistic and data-driven models has been demonstrated to be a viable strategy to model more complex systems with affordable efforts. These hybrid modelling approaches come with new challenges, where research gaps have been identified, as well as create the need for main streaming algorithms and mathematical methods developed for particular use cases. In the section Algorithms and Methods for Hybrid Mechanistic and Data-Driven Modelling the following research topic has been identified as important: n Methods and algorithms for hybrid modelling: A future is to develop and mainstream algorithms and methods that support the implementation of hybrid modelling workflows. n Dynamical-systems-based deep learning: Improve the understanding of the properties of neural network architectures and training methods. Numerical, combinatorial, and mathematical libraries With the advent of exascale HPC infrastructures, the enablement of high-fidelity and multi-query tasks has become more urgent, in particular in the context of the realisation of increasingly complex workflows as well as new paradigms like digital twins. This concerns coupling of different applications, addressing load-imbalance challenges, uncertainty quantification, sensitivity analysis etc. In the section Numerical, Combinatorial, and Mathematical Libraries the following key research challenges have been identified: n Multiscale and multi-physics simulation methods: Develop mathematical and numerical software libraries for such simulations. n Model calibration and Uncertainty Quantification: Establish algorithms and methods for assessing the modelling accuracy Mathematical methods and algorithms for HPC technologies Mathematical methods and algorithms are not only needed to advance science and engineering applications but also for creating or improving new solutions for the HPC market. In this white paper, we advocate future research and development efforts for addressing challenges related to scheduling and resource allocation as well as auto-tuning in the context of numerical libraries. After analysing the current state, in section Mathematical Methods and Algorithms for HPC Technologies the following research and development priorities are proposed: n Scheduling and resource allocation: Development of new scheduling algorithms and scheduling solutions that support increasingly important challenges including co-scheduling of complex workflows or resource allocation in the context of federated and compute continuum infrastructures. n Autotuning: Develop and implement efficient autotuning mechanisms for key numerical libraries for upcoming HPC solutions. Quantum and hybrid algorithms Quantum devices, while still in a very early state of technical readiness, start to become part of HPC infrastructures as this supports the validation of new quantum algorithms by also using powerful quantum device simulators. This development will also facilitate the development of hybrid workflows that leverage both conventional as well as quantum computing capabilities. Quantum advantage is by many considered as being demonstrated. Therefore, increased efforts in research and development efforts related to quantum and hybrid algorithms are required. In the section Quantum and Hybrid Algorithms a range of urgent research challenges are discussed: n Problem embedding, quantum-state preparation, error mitigation/correction n Algorithms on hybrid HPC/quantum architectures n Simulation of quantum circuits