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Details of Grant 

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 Co-Investigators:
Project Partners:
Department: Mathematics
Organisation: UCL
Scheme: Standard Research
Starts: 09 July 2021 Ends: 08 July 2024 Value (£): 499,489
EPSRC Research Topic Classifications:
Mathematical Analysis
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
01 Jun 2020 EPSRC Mathematical Sciences Prioritisation Panel June 2020 Announced
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 ill-posed. 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

ill-posed 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 (a-d)

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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