New Strategies Based on Hierarchical Matrices for Matrix Polynomial Evaluation in Exascale Computing Era

Advancements in computing platform deployment have acted as both push and pull elements for the advancement of engineering design and scientific knowledge. Historically, improvements in computing platforms were mostly dependent on simultaneous developments in hardware, software, architecture, and algorithms (a process known as co-design), which raised the performance of computational models. But, there are many obstacles to using the Exascale Computing Era sophisticated computing platforms effectively. These include but are not limited to massive parallelism, effective exploitation, and high complexity in programming, such as heterogeneous computing facilities. So, now is the time to create new algorithms that are more resilient, energy-aware, and able to address the demands of increasing data locality and achieve much higher concurrency through high levels of scalability and granularity. In this context, some methods, such as those based on hierarchical matrices (HMs), have been declared among the most promising in the use of new computing resources precisely because of their strongly hierarchical nature. This work aims to start to assess the advantages, and limits, of the use of HMs in operations such as the evaluation of matrix polynomials, which are crucial, for example, in a Graph Convolutional Deep Neural Network (GC-DNN) context. A case study from the GCNN context provides some insights into the effectiveness, in terms of accuracy, of the employment of HMs.