### Background and Motivation

The mathematical sciences play a vital part in all aspects of modern society. Without research and training in mathematics, there would be no engineering, economics or computer science; no smart phones, MRI scanners, bank accounts or PIN numbers. Mathematics is playing a key role in tackling the modern-day challenge of cyber security and in predicting the consequences of climate change, whereas in manufacturing sectors such as automotive and aerospace industries benefit from superior virtual design processes. Likewise the life sciences sector, with significant potential for economic growth, would not be in such a strong position without mathematics research and training, providing the expertise integral to the development of areas such as personalised healthcare and pharmaceuticals, and underpinning development of many medical technologies. The

emergence of truly massive datasets across most fields of science and engineering increases the need for new tools from the mathematical sciences.

One of the classic ways in which mathematical science research plays a role in the economy is through collecting data towards understanding it, by using tools and techniques, enabling the discovery of new relationships or models. Modelling of physical phenomena already dates back several centuries, and well-known systems of equations with the names of Maxwell, Navier-Stokes, Korteweg-de Vries and more recently the Schrödinger equation plus many others are now well established. But it was not until the advent of computers in the middle of the previous century, and the development of sophisticated computational methods (like iterative solution methods for large sparse linear systems) that this could be taken to a higher level, by performing computations using these models. Software tools with advanced computational mathematical techniques for the solution of the aforementioned systems of equations have become common place, and are heavily used by engineers and scientists.

Mirroring this activity is increased awareness by society and industry that mathematical simulation is ubiquitous to address the challenging problems of our times. Industrial processes, economic models and critical events like floods, power failures or epidemics have become so complicated that its realistic description does not require the simulation of a single model, but rather the cosimulation of various models. Better scientific understanding of the factors governing these will provide routes to greater innovation power and economic well-being across an increasingly complex, networked world with its competitive and strongly interacting agents. Industry, but also science, is highly dependent on the development of virtual environments that can handle the complex problems that we face today, and in the future.

For example, if the origins of life are to be explained, biologists and mathematicians need to work together, and most of the time spent will be on evaluating and simulating the mathematical models. Using the mathematics of evolutionary dynamics, the change from no life to life (referring to the self-replicating molecules dominating early Earth) can be explained. Another example is the electronics industry, which all of us rely on for new developments in virtually every aspect of our everyday life. Innovations in this branch of industry are impossible without the use of virtual design environments that enable engineers to develop and test their complex designs behind a screen, without ever having to go into the time-consuming (several months) process of prototyping.

Principles of computational science and engineering rooted in modern applied mathematics are at the core of these developments, subjects that are set to undergo a renaissance in the 21st century. Indeed, no less a figure than Stephen Hawking is on record as saying that the 21st century will be the century of complexity. Another great figure, yet young, is Fields medallist Terence Tao, who was a major contributor to the recently published document entitled “The mathematical sciences in 2025”, stating: “Mathematical sciences work is becoming an increasingly integral and essential component of a growing array of areas of investigation in biology, medicine, social sciences, business, advanced design, climate, finance, advanced materials, and many more – crucial to economic growth and societal well- being”.

Growing computing power, nowadays including multicore architectures and GPU’s, does not provide the solution to the ever growing demand for more complex and more realistic simulations. In fact, it has been demonstrated that Moore’s Law, describing the advances in computing power over the last 40 years, equally holds for mathematical algorithms. Hence, it is important to develop both faster computers and faster algorithms, at the same time. This is essential if we wish to keep up with the growing demands by science and technology for more complex simulations. Traditionally, algorithmic speed-ups have come from developments in the area of linear system solution, in which iterative algorithms developed since the 1970’s have been prominent and very effective. Since the start of the new century, however, another powerful development is seen in mathematics, as well as in the systems and control area. This field, which we term ‘model reduction’ for simplicity (we will detail this further in the next section), aims at capturing essential features of models, thereby drastically reducing the size of the problem to be solved. As it holds many promises, this will be the basis for the challenge addressed in this COST Action.