EPSRC Reference: 
EP/T033126/1 
Title: 
Quantitative estimates of discretisation and modelling errors in variational data assimilation for incompressible flows 
Principal Investigator: 
Burman, Professor EN 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
UCL 
Scheme: 
Standard Research 
Starts: 
09 July 2021 
Ends: 
08 July 2024 
Value (£): 
499,489

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
The assimilation of data in computational models is a very important
task in predictive science in the natural environment. In particular
for weather forcasting and biological flow problems such as
cardiovascular flows, measured data must be used to complete the
model. More often than not the available data is not compatible with
the partial differential equations modelling the physical
phenomenon. The problem is illposed. Under certain mild assumption on
the model and measurement errors one can nevertheless use the model
together with the data to obtain computational predictions, typically
using Tikhonov regularisation to control instabilities due to the
illposed character. Two important tools for this are 3DVAR and
4DVAR. These are variational data assimilation methods that, by and
large, look for a solution minimising some norm of the difference
between the solution to the measurements, or to a so called background
state in case it exists, under the constraint of the physical pde model, in our case represented by a partial differential equation. The difference between 3DVAR and 4DVAR is that in 3DVAR data assimilation time evolution is not accounted for. It is therefore applicable only to stationary problem or to repeated assimilation of data ``snapshots'' followed by evolution. In 4DVAR data is expected to be distributed in space time and all space time data is used to produce the assimilated solution.
 In spite of the important literature on the topic of data
assimilation using 3DVAR/4DVAR there appears to be no rigorous numerical
analysis for two or three dimensional problems (for an exception in
one space dimension see [JBFS15]) combining the effect on the solution of
(a) modelling errors;
(b) discretisation of the partial differential equations;
(c) perturbation due to regularisation;
(d) perturbations of the measured data.
 The aim of the present project is to provide sharp rigorous
estimates for the effect on the approximate solution of points (ad)
above in the challenging case of incompressible flow problems.
The derivation of such estimates will give a clear indication on what
type of regularisations are optimal and also what kind of quantities can reasonably be approximated given a set of measured data. Typically the tendency in computational methods
is to evolve from low order approaches to high resolution methods. The
ambition is to design and analyse such high resolution methods for
variational data assimilation problems.

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