# Group actions in function approximation spaces

Lead Research Organisation:
University of Kent

Department Name: Sch of Maths Statistics & Actuarial Scie

### Abstract

In 1987, Britain experienced a hurricane that was famously not predicted by the UK Met Office.The PI heard that mathematicians working at the Met Office decided, after due deliberation, that the reason was that the numerical code used to predict extreme weather had a particular subtle flaw: that a conservation law, known as potential vorticity, was not built into the computer code. This kind of conservation law is guaranteed by a famous theorem, proved by Emmy Noether in the early 1900's, which links the symmetries of the physical model directly to the conservation laws.The long term aim of the line of research of the project is to build such laws, exactly, intonumerical code that models physical systems. This involves first understanding how the physical symmetries, which are understood in terms of smooth actions on smooth spaces,are mapped into actions in discrete or digital spaces; actually, onto the approximate functions on such spaces used to model the smooth functions. The second part involves investigatinghow Noether's theorem transfers to the discrete case in a concrete and practical way.

### Planned Impact

Variational problems are endemic throughout physics and engineering. Principles of optimisation such as minimum time, minimum flow resistance and minimum power expenditure have been invoked through the history of science and common speech... The deterministic success of these principles has become so routine that success is taken for granted. (from A. Bejan, Shape and Structure, from Engineering to Nature, Cambridge University Press, Cambridge, 2000.) Variational problems with a continuous symmetry group have guaranteed conservation laws. The PI has been concerned about automatically embedding conservation laws into numerical code ever since speaking to a Met Office mathematician. She was told that the reason the Met Office failed to predict the 1987 hurricane was decided, after due investigation, to be that the relevant numerical models did not preserve potential vorticity. This is a conservation law arising via Noether's Theorem from the particle relabelling symmetry; a rather subtle and seemingly non physical pseudogroup action. As computers become faster and more efficient, it becomes feasible to incorporate properties of the original physical models into their associated numerical models, and indeed it is philosophically attractive to do so. This is the premise of the `hot' field of geometric integration. The possible impact of this proposal is high because 1) the range of physical phenomena that are modelled by a variational principle and have a continuous symmetry group is broad, and 2) the approach is algorithmic, that is, is centered on general methods with proven properties of outputs. While incorporating potential vorticity exactly into numerical code is a ``holy grail , along the way there should be a wide range of applications whose impact is assured although the specifics can only be conjectured at present.

## People |
## ORCID iD |

Elizabeth Louise Mansfield (Principal Investigator) |

### Publications

Giesselmann J
(2015)

*Reduced relative entropy techniques for a priori analysis of multiphase problems in elastodynamics*in BIT Numerical Mathematics
Giesselmann J
(2015)

*Energy consistent discontinuous Galerkin methods for a quasi-incompressible diffuse two phase flow model*in ESAIM: Mathematical Modelling and Numerical Analysis
Giesselmann J
(2015)

*A Posteriori Analysis of Discontinuous Galerkin Schemes for Systems of Hyperbolic Conservation Laws*in SIAM Journal on Numerical Analysis
Goncalves TMN
(2010)

*Noether's Conservation Laws: smooth and discrete cases*
Goncalves TMN
(2016)

*Moving Frames and Noether's Conservation Laws - the General Case*in Forum in Mathematics, Sigma
Hydon P
(2011)

*Extensions of Noether's Second Theorem: from continuous to discrete systems*in Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Lakkis O
(2013)

*A Finite Element Method for Nonlinear Elliptic Problems*in SIAM Journal on Scientific Computing
Mansfield E
(2013)

*Discrete Moving Frames and Discrete Integrable Systems*in Foundations of Computational Mathematics
Mansfield EL

*Discrete Moving Frames on Lattices*Description | (1) we have shown that conservation of energy, linear and angular momentum can be achieved in numerical (Finite Element) schemes for variational problems, a priori, that is, without being forced. We did this by proving a weak form of the law that was achieved for any mesh, and that this could be made as close as desired to the smooth law by an adaptive scheme. (2) we have shown that for smooth variational problems, moving frames can be used to elucidate the structure of the conservation laws, in the general case (arbitrary dependent and independent variables, general Lie group actions) (3) We have accumulated evidence for two conjectures that overthrow the conventional wisdom, namely, that potential vorticity can indeed be conserved in a numerical scheme, and secondly, that one can indeed conserve both energy and a symplectic form. |

Exploitation Route | Meteorologists are keen to have conservation of potential vorticity. This was thought to be impossible in a numerical scheme. The conservation of energy and momenta are endemic in the landscape of mathematical models for physical and engineering problems. To achieve this a priori in a numerical scheme will have wide impact. |

Sectors | Aerospace, Defence and Marine,Construction,Digital/Communication/Information Technologies (including Software),Other |

Description | (1) The meteorologists are keen for our methods conservation of potential velocity to be finalised. (2) We have used the resulting numerical schemes in consultancy projects with fish innovation ltd. |

First Year Of Impact | 2014 |

Sector | Aerospace, Defence and Marine,Construction,Other |

Description | LMS research in pairs |

Amount | £1,000 (GBP) |

Funding ID | 41214 |

Organisation | London Mathematical Society |

Sector | Learned Society |

Country | United Kingdom |

Start | 10/2013 |

End | 12/2013 |

Description | ELM-Brazil |

Organisation | Federal University of Sao Carlos |

Country | Brazil |

Sector | Academic/University |

PI Contribution | Intellectual |

Collaborator Contribution | Intellectual |

Impact | On going series of papers with TMN Goncalves, now in Brazil |

Start Year | 2012 |

Description | ELM-Surrey |

Organisation | University of Surrey |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | Intellectual input, training of PhD student |

Collaborator Contribution | Intellectual input |

Impact | Journal papers co-authored by Elizabeth Mansfield and Peter Hydon. One paper close to submission also co-authored by Linyu Peng. |

Description | TP-Crete |

Organisation | University of Crete |

Country | Greece |

Sector | Academic/University |

PI Contribution | During this time TP was part of the Archimedes Center for Modeling, Analysis and Computation. TP undertook research on the topic of energy conservative numerical methods for dispersive fluid flow PDEs. |

Collaborator Contribution | Collaborative research |

Impact | This collaboration yielded a pair of research partners with whom I still collaborate |

Start Year | 2012 |