Novel hybrid scalable scientific algorithms are needed with the advent of variety of novel accelerators including graphics processing units (GPUs), field-programmable gate arrays (FPGAs) as well as with the growth of the size of quantum computing devices and neuromorphic chips and various artificial intelligence (AI) specific processors. This myriad of devices requires an unified hybrid approach that allows efficient and scalable hybrid approaches combining classical and novel computing paradigms to be implemented at scale. These extreme-scale heterogeneous systems require novel scientific algorithms to hide the complexity, hide network and memory latency, have advanced communication, and have no synchronization points where possible. With the advent of AI in the past few years the need of such scalable mathematical methods and algorithms for such hybrid architectures that are able to handle data and compute intensive applications at scale becomes even more important.

Scientific algorithms for multi-petaflop and exa-flop systems also need to be fault tolerant and fault resilient, since the probability of faults increases with scale. Resilience at the system software and at the algorithmic level is needed as a crosscutting effort. Key science applications require novel mathematics and mathematical models and system software that address the scalability and resilience challenges of current- and future-generation extreme-scale heterogeneous high performance computing (HPC) systems.