Computational building blocks in numerical solution methods, HPC impact

The solution of linear systems of algebraic equations is one of the most important building blocks scientific computing and in almost all computer simulations of various mathematical models, describing processes and systems in physics, economics, finance, biology, chemistry end more.

The so-arising linear systems are characterized by a matrix, which can be dense, sparse, square, rectangular, ill-conditioned or well-conditioned, of small or large size. All these flavors impact the
choice of the numerical solution method in order to ensure certain accuracy in the obtained solution and efficient and robust behavior of the solution procedure as a function on various problem and method parameters. In addition, thanks to the availability of HPC computing
facilities, the trend to tackle systems with large number of degrees of freedom, nowadays often hundreds of millions, entails certain reconsidering of the suitability of the algorithms, to also utilize the full potential of the HPC resources.
In this presentation we will consider some aspects of the direct and iterative solution methods for systems of linear equations, also considering the interplay between numerical and computational efficiency.