ERCIM News No.22 - July 1995

In the second half of this century, Computer Science has effected a new and significant change both on science and everyday life. What is primarily astonishing is the speed at which this development has taken place. First of all a few facts:

- The computational power available today was unimaginable even just a few decades ago. We can now solve problems we could not even think about tackling before.
- The availability of powerful computational resources has opened new and exciting perspectives in mathematical modelling. It is now possible to design and perform computer simulations of complex and/or risky phenomena, for both scientific and industrial applications, at very low costs.
- Despite a widespread crisis within the high perfomance computing industry, parallel and distributed computing is changing our view of computation itself, and opening the door to new problems and new frontiers of computing.

As distinct from pure Mathematics, Computational Mathematics has a central interest in the complexity of constructive methods. In other words, embedding mathematics in the realm of computations makes one concerned with the amount of computational resources necessary and sufficient to solve the problems. Thus, the constructive aspects must be traded off against efficiency (space and time) and reliability (accuracy) requirements. From this line of argument it becomes clear that Computational Mathematics provides a sort of bridge between Computer Science and Mathematics. Its importance is also increasingly evident nowadays because new problems arising in areas such as life science and economics have to be tackled using the tools of computational mathematics. Thus, computing science has taken on the status of an autonomous scientific field, which is crucial to the practice of many other branches of science.

In my opinion Computational Mathematics will be highly influenced by parallel computing, by advances in pure mathematics and by the development of solid theoretical foundations in application domains such as life sciences. We still have not completely understood the full potential of parallelism and we still do not know if the dream of massively parallel computation is feasible; we do not even know yet which of the competing parallel machine architectures will win out in the long term. The intersection between computational mathematics and parallel computing is so considerable that any uncertainty about the future of parallel computing becomes an uncertainty as to how and where computational mathematics will evolve. This uncertainty poses severe constraints on the exploitation of parallelism itself: How cost-effective is it to develop software tools and algorithmic methodologies for classes of machines whose destiny is uncertain? Furthermore, the diversity in architectures of the current state-of-the-art parallel machines has created a new phenomenon almost absent in the case of sequential computations: parallel algorithms become machine-dependent in the sense that their efficiency can be radically different on different machines.

Let me conclude by saying that the scientists working in this field are living in very dynamic and exiting times, and are looking forward to the coming years.

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