The workshop Simulation of data assimilation under a PDE constraint will be held at Sorbonne Université, 4 place Jussieu, F-75005 Paris on the 16-17th November 2023.
It will focus on the presentation of numerical methods estimating the state of a system represented by a PDE and using data assimilation.
Organizer: Guillaume Delay

The following people have been invited to give an oral presentation:

• Muriel Boulakia (Université de Versailles Saint-Quentin)
• Laurent Bourgeois (ENSTA Paris)
• Erik Burman (University College London)
• Jérémi Dardé (Université Paul Sabatier, Toulouse)
• Guillaume Delay (Sorbonne Université and INRIA, Paris)
• Deepika Garg (University College London)
• Corrie James (INRIA Paris, UVSQ-Université de Paris-Saclay)
• Damiano Lombardi (INRIA Paris)
• Philippe Moireau (INRIA and Ecole Polytechnique, Palaiseau)
• Arnaud Münch (Université Clermont Auvergne, Clermont-Ferrand)
• Lauri Oksanen (University of Helsinki)
• Mihai Nechita (Romanian Academy and Babeș-Bolyai University, Cluj-Napoca)
• Janosch Preuss (University College London)
• Maria-Luisa Rapun (Universidad Politécnica de Madrid)
• Sébastien Riffaud (INRIA Paris)
• Gaël le Ruz (Sorbonne Université, INRIA Paris and INRIA Saclay)

Program

The presentations will be 25 minutes long plus 10 minutes for questions.

Thursday 16th November 2023 (8:50-12:15)
Room 15-16-309

• 8:50 - Welcome and Opening Remarks
• 9:00 - Muriel Boulakia (Université de Versailles Saint-Quentin)
• 9:35 - Sébastien Riffaud (INRIA Paris)
• 10:10 - Coffee Break
• 10:30 - Damiano Lombardi (INRIA Paris)
• 11:05 - Guillaume Delay (Sorbonne Université and INRIA, Paris)
• 11:40 - Maria-Luisa Rapun (Universidad Politécnica de Madrid)
• 12:15 - Lunch

Thursday 16th November 2023 (14:00-17:55)
Room 16-26-113

• 14:00 - Janosch Preuss (University College London)
• 14:35 - Arnaud Münch (Université Clermont Auvergne, Clermont-Ferrand)
• 15:10 - Lauri Oksanen (University of Helsinki)
• 15:45 - Mihai Nechita (Romanian Academy and Babeș-Bolyai University, Cluj-Napoca)
• 16:20 - Coffee Break
• 16:45 - Corrie James (INRIA Paris, UVSQ-Université de Paris-Saclay)
• 17:20 - Erik Burman (University College London)

Friday 17th November 2023 (9:00-12:15)
Room 15-16-309

• 9:00 - Jérémi Dardé (Université Paul Sabatier, Toulouse)
• 9:35 - Deepika Garg (University College London)
• 10:10 - Coffee Break
• 10:30 - Philippe Moireau (INRIA and Ecole Polytechnique, Palaiseau)
• 11:05 - Gaël le Ruz (Sorbonne Université, INRIA Paris and INRIA Saclay)
• 11:40 - Laurent Bourgeois (ENSTA Paris)
• 12:15 - End of the workshop and Lunch

Abstracts

• Muriel Boulakia (Université de Versailles Saint-Quentin): Numerical methods for data assimilation in fluid models.
  
  We will consider different PDE models in fluid mechanics and present a numerical method for the reconstruction of the velocity and the pressure from local measurements of the velocity. This method which has consistency properties is based on the stabilization of the discretized Finite Element formulation of the equation. We will present results on the analysis of the reconstruction error which rely on the quantification of the unique continuation property and illustrate this method for the reconstruction of the blood flow in a vessel.


• Laurent Bourgeois (ENSTA Paris): Data assimilation problems: the Morozov principle revisited.
  
  In this talk we consider a very simple data assimilation problem which can be formulated as "trying to invert a non surjective operator". We address such problem with the help of a mixed formulation of the Tikhonov regularization and the Morozov principle to choose the regularization parameter.
  
  It happens that the operator at stake does not have a dense range, which brings us to revisit the classical formulation of the Morozov principle. The Morozov solution is then obtained by using duality in optimization. We eventually propose an extension of our approach to the case when we impose some constaints to the solution.
  This is a joint work with Jérémi Dardé, from IMT Toulouse.


• Erik Burman (University College London): What is the best convergence we can hope for when approximating ill-posed problems with conditional stability?
  
  Recently there has been some interest in weakly consistent methods for the approximation of the unique continuation of solutions to second order elliptic pdes. Such methods can typically be shown to satisfy error bounds with a convergence order that depends on the approximation properties of the finite element space and the (conditional) stability of the unique continuation problem. In this talk we will recall the ideas behind such methods in the framework of continuous finite element methods. Then we will discuss if, and in what sense, the resulting approximations can be considered optimal. Finally, we will consider two situations in which the accuracy can be improved and how this is reflected in the computational method.


• Jérémi Dardé (Université Paul Sabatier, Toulouse): Iterated quasi-reveribility method for data assimilation problems.
  
  In this talk, we consider a variant of the quasi-reversibility [QR] method, the iterated QR-method, and its application to a very simple data assimilation problem.
  
  The iterated QR method has previously been studied in the context of data completion problems, where in a sense one try to invert a non-surjective linear operator with dense range. The usual QR method presents a parameter of regularisation, which has to be chosen "small" (to obtain a good reconstruction), but not "too small" (to be able to invert the system), and ideally function of the level of noise, which is possible but usually numerically demanding. In the iterated QR method, one solved iteratively M standard QR problems, for a fixed parameter of regularisation, which can surprisingly be chosen arbitrarily large: the problems are then easily invertible, leading to a fast and accurate method, and choosing when to stop the iterations in function of the level of noise on the data is simple and direct.
  
  The situation is slightly modified in the context of data assimilation, as the range of the operator we would like to invert is not dense anymore. As data are always contaminated by noise, this can lead to very bad reconstructions. The iterated QR method easily adapts to this new context: the case of exact data is similar, whereas the case of noisy data needs some modifications. We present several numerical examples, in the case of data completion for elliptic and parabolic operators and data assimilation problems for elliptic operators, to illustrate its feasibility and efficiency.


• Guillaume Delay (Sorbonne Université and INRIA, Paris): The unique continuation problem for the heat equation discretized with a high-order space-time method.
  
  We discretize a unique continuation problem subject to the heat equation and present the associated numerical analysis. This unique continuation problem consists in reconstructing the solution of the heat equation in a target space-time subdomain given its (noised) value in a subset of the computational domain. Both initial and boundary data can be unknown.
  
  We discretize a space-time variational formulation using discontinuous Galerkin in time and hybridized discontinuous Galerkin in space. The numerical solution is searched as the saddle point of a Lagrangian functional where stabilization terms are considered, including a Tikhonov-like regularization. Inf-sup stability and a priori error bounds are established for a (weak) residual norm. Error bounds in energy norm are proved using the stability of the continuous problem. Some numerical tests are presented to validate the established convergence rate. This is joint work with Erik Burman and Alexandre Ern.


• Deepika Garg (University College London): Data assimilation nonconforming finite element methods for transient Stokes problem.
  
  In this talk, we will consider the unique continuation problem for reconstructing the final state of the transient Stokes problem when the initial data is unknown, but additional data is given in a subdomain in space-time. The backward differentiation method is used to discretize the time derivative and utilize standard nonconforming affine finite element approximation for the discretization in space. The discrete system is regularized by adding a penalty of the $H^1$-semi-norm of the initial data, scaled with the mesh parameter. An optimal error estimate in $L^2(T_1,T;H^1(\Omega))$, $T_1>0$, is derived using the Lipschitz stability of the reconstruction problem. The theory is validated on some numerical examples.
  This is a joint work with Erik Burman and Janosch Preuss.


• Corrie James (INRIA Paris, UVSQ-Université de Paris-Saclay): Data Assimilation for the Laplacian with Population Data.
  
  In the context of a unique continuation problem, we study how the inclusion of population data improves results. Firstly, starting with the Laplacian equation and measurements taken over a spatially coarse resolution, we integrate a database of measurements via the Proper Orthogonal Decomposition (POD) basis, and solve using a stabilized finite element method. From here, we obtain a theoretical error estimation as well as numerical results.


• Damiano Lombardi (INRIA Paris): Optimal recovery for state estimation in haemodynamics.
  
  The problem of state estimation in haemodynamics consists in reconstructing the 3d blood flow in a portion of the vasculature given some data. In view of setting up fast reconstruction methods, we consider an optimal recovery formulation of the state estimation problem. The starting point of the present work is the so-called Parametrised Background Data Weak (PBDW) method. In this, we consider a linear subspace approximation of the set of solutions of the model describing the system (in the present context, the parametric Navier-Stokes equations). The optimal recovery consists in finding, among the elements of the space which have vanishing discrepancy, the one which is the closest to the linear subspace. We describe two contributions: in the first one, instead of considering a single subspace approximation, we consider the case in which we can construct multiple subspaces approximating the solutions set.
  
  The optimal recovery is classically formulated as a time independent state estimation. This means that for time dependent problems it results in a sequence of unrelated estimations.
  
  The second contribution consists in proposing a possible extension of optimal recovery for time dependent systems. This relies on the approximation of the solutions set by using tensor methods. In particular, we will use the Tensor Train format. The counterpart of the Kalman observability criterion can be proved. Several numerical experiments will be presented.


• Philippe Moireau (INRIA and Ecole Polytechnique, Palaiseau): Optimal filtering for PDEs and with PDEs.
  
  Optimal filtering approaches in data assimilation are an old theory that is theoretically attractive but computationally prohibitive because of the curse of dimensionality in numerical implementation. In this talk, we propose to completely rethink this theory for two different problems. First, using parabolic PDEs, we show that additional regularity results imply that the Riccati operator belongs to the class of Hilbert-Schmidt operators and hence associated with kernels. This regularity allows us to perform the numerical analysis of the space-time discretization of the Kalman estimator and justifies the implementation of a numerically effective Kalman algorithm thanks to the use of H-matrices originally developed for the discretization of integral equations. The second problem concerns nonlinear finite dimensional problems, where this time the optimal filter can be computed from a solution of a Hamilton-Jacobi-Bellman (HJB) equation defined in the state space. After choosing a particular splitting time-scheme for this HJB equation, we recognize inf-convolutions and proximal operators, which are now popular in optimal transport and optimization. Through the use of the softmax approximation, this allows us to limit the burden of the resulting algorithm and paves the way for the use of optimal filters for nonlinear PDEs when combined with model reduction strategies.


• Arnaud Münch (Université Clermont Auvergne, Clermont-Ferrand): Constructive proof of exact controllability for semi-linear wave equation.
  
  We briefly discuss constructive proofs of controllability for semi-linear wave equations of the form $\partial_tt y -\Delta y + f(y)=0$, assuming that the function $f$ is $C^1$ and does not grow too fast at infinity. Existing proofs of controllability are usually based on non constructive fixed point arguments. We show how we can define, within an appropriate Carleman functional space setting, contracting fixed point operators yielding to strongly convergent approximations of the nonlinear problem. Numerical illustrations will be given together with possible perspectives for the corresponding dual inverse problems. Joint works with Jérôme Lemoine (Clermont-Ferrand).


• Lauri Oksanen (University of Helsinki): Spacetime finite element methods for inverse and control problems subject to the wave equation.
  
  There is a well-known duality between inverse initial source problems and control problems for the wave equation, and analysis of both these boils down to the so-called observability estimates. I will present recent results on numerical analysis of these problems. The inverse initial source problem gives a model for the acoustic step of photoacoustic tomography, a biomedical imaging modality based on the photoacoustic effect. The talk is based on joint work with Erik Burman (University College London), Ali Feizmohammadi (University of Toronto) and Arnaud Münch (Université Clermont Auvergne).


• Mihai Nechita (Romanian Academy and Babeș-Bolyai University, Cluj-Napoca): Unique continuation for the Helmholtz equation using conforming FEM and physics-informed neural networks.
  
  We consider the unique continuation problem for the Helmholtz equation and study its numerical approximation with two methods: conforming FEM and physics-informed neural networks (PINNs). For conforming FEM, regularization is added on the discrete level using gradient jump penalty, Galerkin least squares and a scaled Tikhonov term. The method is shown to converge in terms of the stability of the problem and the polynomial degree of approximation. For PINNs, one can use the conditional stability of the problem to bound the generalization error. We present numerical experiments in 2d for different frequencies and for geometric configurations with different stability bounds.


• Janosch Preuss (University College London): Unique continuation for the wave equation using a discontinuous Galerkin time discretization.
  
  We revisit the unique continuation problem for the wave equation in the time domain which has been solved previously in [1]. Instead of using a full space time discretization like in this reference, we propose a more standard approach based on a discontinuous Galerkin time discretization combined with continuous finite elements in space. The resulting method leads to the same error estimates as in [1] and allows for a fairly efficient solution of the linear systems by exploiting the natural decomposition of the space-time domain induced by the discontinuous Galerkin discretization in time. We illustrate our method with numerical experiments and report on current investigations to combine it with a multi-level strategy to further improve its efficiency.
  This is joint work with Erik Burman.
  
  References:
  [1] E. Burman, A. Feizmohammadi, A. Münch, L. Oksanen (2021). Space time stabilized finite element methods for a unique continuation problem subject to the wave equation. ESAIM: M2AN, 55, S969–S991.


• Maria-Luisa Rapun (Universidad Politécnica de Madrid): Combination of mode decompositions and topological derivatives for active thermographic inspection.
  
  In this work we propose a new numerical method for processing a sequence of thermograms obtained in experimental non-destructive tests of metallic plates. The method is based on the combination of two data processing tools: the higher order dynamic mode decomposition and the topological derivative. The first one is used to clean experimental noise and to identify the thermographic modes. Then, the topological derivative of a cost functional defined in terms of such modes is used to diagnose defects inside the plate. The performance of the method will be evaluated for a real sequence of experimental thermograms.
  
  This is a joint work with J.M. Perales and J.M. Vega.


• Sébastien Riffaud (INRIA Paris): A low-rank solver for parameter estimation and uncertainty quantification in nonlinear systems of parametric PDEs.
  
  This work has been motivated by data assimilation in blood flow models. In many realistic applications, the knowledge of the model parameters as well as the boundary conditions is not perfect. We focus here on the estimation of the input parameters from observations of the output solution. The resulting estimate is then exploited to perform uncertainty quantification tasks. However, there are two major difficulties when trying to solve parameter estimation problems. First, some parameters may not be identifiable. For instance, if only partial noisy measurements of the solution are available, several different parameter values may be associated with the same observation. Second, if the parameters depend nonlinearly on the solution, the parameter estimation problem results in a nonlinear non-convex optimization problem that can be difficult to solve. For these reasons, we employ a sequential Markov Chain Monte Carlo (MCMC) method using the Metropolis-Hasting algorithm to estimate the parameters. This bayesian approach consists in sampling the posterior probability distribution of the parameters and allows to identify eventual correlations between the parameter values. However, updating this sampling is computationally expensive since the solution must be evaluated repeatedly for each queried parameter. For this reason, we developed a low-rank solver to significantly reduce the time and storage requirements associated with the MCMC procedure. The performance of the resulting method is demonstrated on three different applications.


• Gaël le Ruz (Sorbonne Université, INRIA Paris and INRIA Saclay): Optimal filtering on manifolds.
  
  This work was motivated by data assimilation for wildfire propagation, where the state and the observations of the system are naturally modeled in the manifold of contours. Typically, one can use an estimate-then-project method to address this problem. However, this is purely empirical and, in addition, an embedding in the Euclidean space need to be accessed, which is clearly artificial in the case of contours. Writing and solving the filtering problem directly on the manifold (without using the embedding in the ambient space) is a novel promising research direction, as some recent results in optimization and optimal control suggest. In this talk, using the example of a first-order dynamics on the two-sphere Riemannian manifold, we propose to develop a framework for computing optimal filters in a general manifold from the solution of a Hamilton-Jacobi-Bellman equation in the state space. We then reduce the cost of the resulting algorithm by using a quadratic approximation of the value function solution of the Hamilton-Jacobi-Bellman equation.