Creating tomorrow’s computational methods

Fredrik Fryklund will receive his doctoral degree in mathematics from KTH Royal Institute of Technology in 2021. Thanks to a grant from Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position with Professor Leslie Greengard at the Courant Institute of Mathematical Sciences, New York University, USA.

Mathematics has an astounding ability to describe our world. For example, mathematical equations can be used to predict how natural phenomena, like the weather, will develop over a period of time. However, equations rarely provide a solution that can be written as a formula; a computer must usually provide an approximate answer. The objective of this project is to develop numerical methods for better computations.

Improvements in computer power have allowed the study of more complex problems, demanding increasingly sophisticated computational methods. Important examples are new integral equation methods that have proved successful. They are thorough and can easily provide solutions that are very accurate (correct to a large number of decimal places), complex geometries are simple to incorporate, they are flexible, and with rapid algorithms, they are also efficient. A disadvantage is that integral methods are limited to specific problems in mathematical physics.

There are already methods for expanding the areas of use for integral equations, but some important issues are still unsolved. One challenge is to find the optimal way of using computational capacity, so that more computer-resources are used when high resolution is required and fewer when a lower resolution is adequate. Another challenge is to increase the efficiency of the new methods by combining them with existing methods for fast computations. These methods can in some cases reduce the time for computer calculations from a couple of years to a couple of minutes. Fredrik Fryklund’s plan is to try to broaden the integral models and create new computational mathematics with applications in other fields of research and industry.