Quantitative estimates of discretisation and modelling errors in variational data assimilation for incompressible flows

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.

Although the importance of data assimilation and the accurate solution
of associated ill-posed problems is an acknowledged fact in
computational methods for environmental sciences and weather
forecasting the problem has received undeservedly little attention
from the numerical analysis community. As a consequence the powerful
techniques developed for the design and analysis of numerical methods
for well-posed PDEs have not been made to bear on these problems. In
particular few works exist that consider quantitative error estimates
for data assimilation problems together with the question of
optimality and accuracy. Indeed an important challenge is to extract
the most information from the model and the data given measurement
errors of known maximum magnitude and fixed computational
resources. Progress on this topic could yield a strong impact in the
form of improved weather forecasting and more generally for
computations in environments where only measured data are available,
such as physiological flows and find its end users among applied
scientists and engineers working on such topics. We believe that the
detailed study on the nontrivial model problems suggested in the
present proposal can open the door to substantially improved
computations. For instance we have in recent works
designed a method for data assimilation for the heat equation for
which we prove that the convergence order for the ill-posed problem of
reconstructing the final solution over a sampling period (without
knowledge of the initial data) is the same as for the well-posed
problem where the initial data is known. The key observation is that
the method is designed to exploit numerical stability obtained by a
\emph{minimal regularisation} term together with the approximation
properties of the numerical method and the conditional stability of
the (ill-posed) physical problem. Such results can not be achieved in
the standard framework of Tikhonov type regularisation, since they
introduce an $O(1)$ perturbation of the physical system. It is
therefore our belief that the cross-over from numerical analysis to
data assimilation in environmental flows proposed in the present
project, if successful, could substantially enhance the quality of
future weather forecasting algorithms and other computations of flow
in the biosciences. The proposed project is to a very large extent
mathematical, based on numerical analysis and the development of new
tools for the optimal regularisation of ill-posed problems and their
analysis in the form of quantitative error estimates using analytical
estimates of the physical stability. It is not within the scope of the
present project to develop these techniques to a state where they are
directly applicable by
the end user, however we expect to develop the methods to a state where they can be applied in an "off the shelf fashion" by applied scientists.
In particular a database of code for different problems using the Fenics package will serve as an introduction to the practical use of the proposed methods.