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

Lead Research Organisation:
University College London

Department Name: Mathematics

### Abstract

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.

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.

### Planned Impact

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.

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.

### Organisations

### Publications

Ahmed N
(2021)

*A Pressure-Robust Discretization of Oseen's Equation Using Stabilization in the Vorticity Equation*in SIAM Journal on Numerical Analysis
Barrenechea G
(2023)

*Continuous interior penalty stabilization for divergence-free finite element methods*in IMA Journal of Numerical Analysis
Burman E
(2022)

*Eulerian time-stepping schemes for the non-stationary Stokes equations on time-dependent domains*in Numerische Mathematik
Burman E
(2022)

*Two mixed finite element formulations for the weak imposition of the Neumann boundary conditions for the Darcy flow*in Journal of Numerical Mathematics
Burman E
(2023)

*The Unique Continuation Problem for the Heat Equation Discretized with a High-Order Space-Time Nonconforming Method*in SIAM Journal on Numerical Analysis
Burman E
(2023)

*Stability estimate for scalar image velocimetry*in Journal of Inverse and Ill-posed Problems
Burman E
(2023)

*The Augmented Lagrangian Method as a Framework for Stabilised Methods in Computational Mechanics*in Archives of Computational Methods in Engineering
Burman E
(2023)

*Implicit-Explicit Time Discretization for Oseen's Equation at High Reynolds Number with Application to Fractional Step Methods*in SIAM Journal on Numerical Analysis
Burman E
(2024)

*Data assimilation finite element method for the linearized Navier-Stokes equations with higher order polynomial approximation*in ESAIM: Mathematical Modelling and Numerical Analysis
Burman E
(2021)

*Error Estimates for the Smagorinsky Turbulence Model: Enhanced Stability Through Scale Separation and Numerical Stabilization*in Journal of Mathematical Fluid Mechanics
Burman E
(2022)

*Weighted Error Estimates for Transient Transport Problems Discretized Using Continuous Finite Elements with Interior Penalty Stabilization on the Gradient Jumps*in Vietnam Journal of Mathematics
Burman E
(2022)

*A mechanically consistent model for fluid-structure interactions with contact including seepage*in Computer Methods in Applied Mechanics and Engineering
Burman E
(2023)

*Some Observations on the Interaction Between Linear and Nonlinear Stabilization for Continuous Finite Element Methods Applied to Hyperbolic Conservation Laws*in SIAM Journal on Scientific Computing
Moura R
(2022)

*Gradient jump penalty stabilisation of spectral/ h p element discretisation for under-resolved turbulence simulations*in Computer Methods in Applied Mechanics and Engineering