Applications of Geometric and Structure Preserving Methods
 

^Background

Numerical modelling is used to good effect in a number of applications and advances in geometric and structure preserving methods will progress this field further. These methods are a special class of numerical algorithms used to compute solutions to differential equations that preserve the underlying geometry and structure of the system. The key advantage of these methods is that they are not only computationally fast, but they also improve the accuracy of the computation since they are both quantitatively and qualitatively precise.
 
Computations of differential equations are fundamental in the mathematical models of many real-world systems, many of which are highly complex and require advanced numerical methods to solve them. This is particularly the case in areas where it is vital that model simulations are both quick and precise. In recent years there has been a shift to special classes of differential equations and purpose-built algorithms that are tailored to preserve special features of each class. This has given rise to the new fields of geometric numerical integration and of structure preserving discretization. Due to the increased speed and accuracy of geometric and structure preserving methods over other numerical algorithms to solve differential equations they have become an important tool in weather forecasting, medical imaging, defence and space and robotics.

This knowledge exchange day was part of a six month Research Programme at the Isaac Newton Institute on Geometry, compatibility and structure preservation in computational differential equations. This research programme brings together mathematicians from different communities to develop a new generation of space-time discretisation methods for differential equations needed for major scientific applications which call for advanced numerical techniques.

^{Aims and Objectives}

This workshop showcased recent applications of geometric and structure preserving methods to models of real-world systems, as well as highlight where advances in these types of numerical methods are most needed.

The programme for the day represented the breadth of application areas where geometric and structure preserving numerical methods are used and included talks from both academic research and end-used perspectives from a number of application areas. The talks highlighted recent advances in these types of numerical methods which have the potential to significantly improve simulations in areas where numerical accuracy and computational speed are vital, such as weather forecasting and medical imaging.  

The three sessions focussed on the following application areas where it has been identified that geometric and structure preserving methods have potential for significant impact:

• Weather forecasting
• Medical imaging
• Robotics
• Defence and space

 
This event brought together mathematicians and scientists working in various areas at the forefront of advances in geometric and structure preserving numerical methods, with end users from industry to further investigate opportunities for the use of these types of methods to improve the numerical solution of real-world problems. Academic talks highlighted state of the art research and techniques in this area and where they could be applied to improve the numerical solution of industrial driven models. End-user talks reflected challenges where such techniques could be used to improve the speed and accuracy of simulations.

^Posters

A poster exhibition ran alongside the workshop and during the drinks reception. 

^{Registration and Venue}

The workshop took place at the Isaac Newton Institute for Mathematical Sciences in Cambridge. Please see the Isaac Newton Institute website for further information about the venue.