### Workshop A6 - Mathematical Foundations of Data Assimilation and Inverse Problems

**Organizers:** Jean-Frédéric Gerbeau (INRIA, France) - Sebastian Reich (Universitât Potsdam, Germany) - Karen Willcox (Massachusetts Institute of Technology, USA)

## Talks

July 10, 14:30 ~ 15:00

## Unlocking datasets by calibrating populations of models to data density: a study in atrial electrophysiology

### Kevin Burrrage

### Queensland University of Technology, the University of Oxford, Australia, UK - kevin.burrage@gmail.com

The understanding of complex physical or biological systems nearly always requires a characterisation of the variability that underpins these processes. In addition, the data used to calibrate such models may also often exhibit considerable variability. A recent approach to deal with these issues has been to calibrate populations of models (POMs), that is multiple copies of a single mathematical model but with different parameter values. To date this calibration has been limited to selecting models that produce outputs that fall within the ranges of the dataset, ignoring any trends that might be present in the data. We present here a novel and general methodology for calibrating POMs to the distributions of a set of measured values in a dataset. We demonstrate the benefits of our technique using a dataset from a cardiac atrial electrophysiology study based on the differences in atrial action potential readings between patients exhibiting sinus rhythm (SR) or chronic atrial fibrillation (cAF) and the Courtemanche-Ramirez-Nattel model for human atrial action potentials. Our approach accurately captures the variability inherent in the experimental population, and allows us to identify the differences underlying stratified data as well as the effects of drug block.

Joint work with Brodie A. J. Lawson (Queensland University of Technology, Brisbane, Australia), Christopher C. Drovandi (Queensland University of Technology, Australia), Nicole Cusimano (Basque Center for Applied Mathematics, Bilbao, Spain), Pamela Burrage (Queensland University of Technology, Australia) and Blanca Rodriguez (University of Oxford, Oxford, United Kingdom).

July 10, 15:00 ~ 15:30

## Reduced Basis' Acquisition by a Learning Process for Rapid On-line Approximation of Solution to PDE's : laminar flow past a backstep

### Yvon Maday

### Université Pierre et Marie Curie, Laboratoire J.-L. Lions, France - maday@ann.jussieu.fr

Reduced Basis Methods for the approximation to parameter dependent Partial Differential Equations are now well developed and start to be used in industrial framework. The classical implementation of the Reduced Basis Method goes through two stages : in the the first one, offline and time consuming, from standard approximation methods a reduced basis is constructed, then in a second stage, online and very cheap, a small problem, of the size of the reduced basis, is solved.

The offline stage is a learning one from which the online stage can proceed efficiently. In this presentation we propose to complement the offline stage with some statistical learning ingredients in order to build more knowledge in the process so as to tackle either different classes of problems or even speed more the online approximation. The method is presented on a simple flow problem governed by the Navier Stokes equation and illustrated on the test case of a flow pas a backward step.

Joint work with P. Gallinari and O. Schwander (Sorbonne Universités, UPMC Univ. Paris 06 and CNRS, UMR 7606, LIP6),, Y. Maday (Sorbonne Universités, UPMC Univ. Paris 06 and CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, Institut Universtaire de France, Division of Applied Mathematics, Brown University),, M. Sangnier (Sorbonne Universités, UPMC Univ Paris 06, F-75005, Paris, France LSTA ), and T. Taddei (Sorbonne Universités, UPMC Univ. Paris 06 and CNRS, UMR 7598, Laboratoire Jacques-Louis Lions)..

July 10, 15:30 ~ 16:00

## EnKF-based interacting particle filter formulations

### Jana de Wiljes

### University Potsdam, Germany - wiljes@uni-potsdam.de

In a nonlinear setting the filtering distribution can be approximated via the empirical measure provided that an ensemble of samples is available. A computationally feasible option to generate these particles for each time instance is to define an appropriate modified evolution equation that describes the dynamics of the particles with respect to the incoming data. The most famous example of such interacting particle filter formulations is the ensemble Kalman filter (EnKF). Although it works remarkably well in practice its success is not well understood from a mathematical point of view. In a recent study we were able to derive stability and accuracy results for a specific variant of the EnKF in a continuous setting with a small number of particles. Inspired by the EnKF we explored more general interacting particle filter formulations that allow to overcome weaknesses of the EnKF as well as drawbacks of classical sequential resampling schemes. More precisely, we consider the recently proposed ensemble transform particle filter (ETPF) which is an adaption of the standard particle filter where the resampling step is replaced by a linear transformation. However, the transformation step is computationally expensive and leads to an underestimation of the ensemble spread for small and moderate ensemble sizes.

These shortcomings have recently been addressed by developing second-order accurate extensions of the ETPF. It is also demonstrated that the nonlinear ensemble transform filter (NETF) arises as a special case of our general framework. Numerical results for the Lorenz-63 and Lorenz-96 models demonstrate the effectiveness of the proposed modified particle filters.

Joint work with Wilhelm Stannat (Technical University Berlin), Sebastian Reich (University Potsdam and University of Reading) and Walter Acevedo (University Potsdam).

July 10, 16:00 ~ 16:30

## Data-driven operator inference for non-intrusive projection-based model reduction

### Karen Willcox

### MIT, USA - kwillcox@mit.edu

This talk presents a non-intrusive projection-based model reduction approach for full models based on time-dependent partial differential equations. Projection-based model reduction constructs the operators of a reduced model by projecting the equations of the full model onto a reduced space. Traditionally, this projection is intrusive, which means that the full-model operators are required either explicitly in an assembled form or implicitly through a routine that returns the action of the operators on a given vector; however, in many situations the full model is given as a black box that computes trajectories of the full-model states and outputs for given initial conditions and inputs, but does not provide the full-model operators. Our non-intrusive operator inference approach solves an optimization problem to infer approximations of the reduced operators from the initial conditions, inputs, trajectories of the states, and outputs of the full model, without requiring the full-model operators. The inferred operators are the solution of a least-squares problem and converge, with sufficient state trajectory data, in the Frobenius norm to the reduced operators that would be obtained via an intrusive projection of the full-model operators.

Reference: Peherstorfer, B. and Willcox, K., Data-driven operator inference for nonintrusive projection-based model reduction, Computer Methods in Applied Mechanics and Engineering, Vol. 306, pp. 196-215, 2016.

Joint work with Benjamin Peherstorfer (University of Wisconsin - Madison).

July 10, 17:00 ~ 18:00

## Waves and imaging in random media

### Josselin Garnier

### Ecole Polytechnique, France - josselin.garnier@polytechnique.edu

In sensor array imaging an unknown medium is probed by waves emitted by an array of sources and recorded by an array of receivers. Sensor array imaging in a randomly scattering medium is usually limited because coherent signals recorded by the receiver array and coming from a reflector to be imaged are weak and dominated by incoherent signals coming from multiple scattering by the medium. Stochastic and multiscale analysis allows to understand the direct problem and helps solving the inverse problem. We will see in this talk how correlation-based imaging techniques can mitigate or even sometimes benefit from the multiple scattering of waves. Applications to seismic interferometry, non-destructive testing, and intensity correlation imaging in optics will be discussed.

July 11, 14:30 ~ 15:00 - Room T1

## Observer strategies for inverse problems associated with wave-like equations

### Philippe Moireau

### Inria - LMS, Ecole Polytechnique, CNRS - Université Paris-Saclay, France - philippe.moireau@inria.fr

We present a novel strategy to perform estimation for wave-like systems and more general evolution PDEs with uncertain initial conditions and parameters. We adopt a filtering approach on the dynamical system formulation to construct a joint state-parameter estimator that uses some measurements available in standard operating conditions. Namely, the aim is to obtain a modified dynamical system converging to the reference by incorporating correction terms using the data. First, in the case of known parameters, state estimation is performed using a Luenberger observer inspired from feedback control theory. This type of state estimator is chosen for its particular effectiveness and robustness. In particular, unlike the classical Kalman approach, this filter is computationally tractable for numerical systems arising from the discretization of PDEs and – although optimality is lost – the exponential stability of the corresponding error system gives exponential convergence of the estimator/observer. With uncertain parameters we extend the estimator by incorporating the parameters in an augmented dynamical system. The effect of the first stage state filter then consists in essence in circumscribing the uncertainty to the parameter space – which is usually much “smaller” than the parameter space – and allows for an $H^2$ type filter in the resulting low rank space. This second step is related to "reduced rank filtering" procedures in data assimilation. The convergence of the resulting joint state-parameter estimator can be mathematically established , and we demonstrate its effectiveness by identifying localized parameters. We propose to illustrate every aspect of this strategy from the observer formulation, analysis, robustness to noise, discretization up do the numerical implementation with various examples based on wave-like models in bounded but also unbounded domains.

July 11, 15:00 ~ 15:30 - Room T1

## Optimal experimental design for large-scale PDE-constrained Bayesian inverse problems

### Omar Ghattas

### The University of Texas at Austin, USA - omar@ices.utexas.edu

We address the problem of optimal experimental design (OED) for infinite-dimensional nonlinear Bayesian inverse problems. We seek an A-optimal design, i.e., we aim to minimize the average variance of a Gaussian approximation to the inversion parameters at the MAP point, i.e. the Laplace approximation. The OED problem includes as constraints the optimality condition PDEs defining the MAP point as well as the PDEs describing the action of the posterior covariance. A randomized trace estimator along with low rank approximations of the Hessian of the data misfit lead to efficient OED cost function computation, and an adjoint approach leads to efficient gradient computation for the OED problem. We provide numerical results for the optimal sensor locations for inference of the permeability field in a porous medium flow problem. The results indicate that the cost of the proposed method (measured by the number of forward PDE solves) is independent of the parameter and data dimensions.

Joint work with Alen Alexanderian (North Carolina State University), Noemi Petra (University of California, Merced) and George Stadler (New York University).

July 11, 15:30 ~ 16:00 - Room T1

## Inverse problem of electrocardiography: a multiphysics data assimilation approach

### Jean-Frédéric Gerbeau

### Inria, France - jean-frederic.gerbeau@inria.fr

We address the inverse problem of electrocardiography by combining electrical and mechanical measurements. Our strategy, proposed in [1], relies on a model of the electromechanical contraction which is registered on ECG data [2,3] but also on measured mechanical displacements of the heart tissue typically extracted from medical images. In this respect, we establish the convergence of a sequential estimator which combines for such coupled problems various state of the art sequential data assimilation methods in an efficient framework [4]. We aggregate a Luenberger observer for the mechanical state and a Reduced Order Unscented Kalman Filter applied on the parameters to be identified, and a POD projection of the electrical state. Then using synthetic data we show the benefits of our approach for the estimation of the electrical state of the ventricles along the heart beat compared with more classical strategies which only consider an electrophysiological model with ECG measurements. Our numerical results actually show that the mechanical measurements improve the identifiability of the electrical problem allowing to reconstruct the electrical state of the coupled system more precisely. Therefore, this work is intended to be a first proof of concept, with theoretical justifications and numerical investigations, of the advantage of using available multi-modal observations for the estimation and identification of an electromechanical model of the heart.

References:

[1] C. Corrado, J-F. Gerbeau, P. Moireau. Identification of weakly coupled multiphysics problems. Application to the inverse problem of electrocardiography. Journal of Computational Physics, 283, pp. 271-298, 2015.

[2] M. Boulakia, S. Cazeau, M. Fern\'andez, J-F. Gerbeau, N. Zemzemi. Mathematical modeling of electrocardiograms: a numerical study, Annals of biomedical engineering, 38 (3), 1071-1097, 2010.

[3] E. Schenone, A. Collin, J-F. Gerbeau. Numerical simulation of electrocardiograms for full cardiac cycles in healthy and pathological conditions. International journal for numerical methods in biomedical engineering, 32(5), 2016

[4] P. Moireau, D. Chapelle, P. Le Tallec. Joint state and parameter estimation for distributed mechanical systems. Computer methods in applied mechanics and engineering, 197 (6), 659-677, 2008.

Joint work with Philippe Moireau (Inria, France) and Cesare Corrado (King's College London, UK)..

July 11, 17:00 ~ 18:00 - Room T1

## Bayesian inference via low-dimensional couplings

### Youssef Marzouk

### Massachusetts Institute of Technology, USA - ymarz@mit.edu

Integration against an intractable probability measure is among the fundamental challenges of statistical inference, particularly in the Bayesian setting. A principled approach to this problem seeks a deterministic coupling of the measure of interest with a tractable ``reference'' measure (e.g., a standard Gaussian). This coupling is induced by a transport map, and enables direct simulation from the desired measure simply by evaluating the transport map at samples from the reference. Yet characterizing such a map---e.g., representing, constructing, and evaluating it---grows challenging in high dimensions.

We will present links between the conditional independence structure of the target measure and the existence of certain low-dimensional couplings, induced by transport maps that are sparse or decomposable. We also describe conditions, common in Bayesian inverse problems, under which transport maps have a particular low-rank structure. Our analysis not only facilitates the construction of couplings in high-dimensional settings, but also suggests new inference methodologies. For instance, in the context of nonlinear and non-Gaussian state space models, we describe new variational algorithms for online filtering, smoothing, and parameter estimation. These algorithms implicitly characterize---via a transport map---the full posterior distribution of the sequential inference problem using only local operations while avoiding importance sampling or resampling.

Joint work with Alessio Spantini (Massachusetts Institute of Technology, USA) and Daniele Bigoni (Massachusetts Institute of Technology, USA).

July 12, 14:30 ~ 15:00 - Room T1

## Estimation for front propagation models with front level-set data using observers. Applications in medicine and in fire propagation.

### Annabelle Collin

### Inria, Université Bordeaux, Bordeaux INP, France - annabelle.collin@inria.fr

In this presentation, we present a sequential — observer based — data assimilation strategy for front propagation models (as reaction-diffusion, eikonal or reaction-transport equations), and for data corresponding to propagation isochrones. First, an original similarity measure between the computed front and the observed front is introduced. Then an efficient feedback adapted to this similarity measure is presented based on shape derivatives leading in fine to a so-called Luenberger observer. This shape observer can be completed with the introduction of the topological derivative of the measure in order to take into account the breakthrough of new fronts. Mathematical justifications of the stabilization property brought by the feedback are detailed. We also discuss the extension to joint state-parameter estimation by using a Kalman based strategy such as RO-UKF. Finally, numerical illustrations are presented on synthetic data and on real data in three applications domains (cardiac electrophysiology, tumor growth and fire propagation) revealing the potential of the approach.

Joint work with Dominique Chapelle (Inria, Université Paris Saclay) and Philippe Moireau (Inria, Université Paris Saclay).

July 12, 15:00 ~ 15:30 - Room T1

## Localized Model Reduction for Data Assimilation and Inverse Problems

### Mario Ohlberger

### University of Muenster, Germany - mario.ohlberger@uni-muenster.de

Data assimilation and large scale inverse problems pose sever challenges to numerical methods due to their high computational complexity. In this contribution we present novel model reduction approaches to handle such problems. In particular, we focus on localized model reduction with online enrichment, as well as on combined parameter and state reduction.

Joint work with Christian Himpe (MPI Magdeburg), Stephan Rave (University of Muenster), Felix Schindler (University of Muenster).

July 12, 15:30 ~ 16:00 - Room T1

## Continuous time filtering algorithms

### Sebastian Reich

### University of Potdam and University of Reading, Germany - sereich@uni-potsdam.de

I will review recent work on interacting particle representations of the continuous time filtering problem. The starting point is provided by the feedback particle filter and its particle approximations. The ensemble Kalman-Bucy filter emerges as a special and I will summarize results on its accuracy and stability. I will conclude with some open problems.

Joint work with Jana de Wiljes (University of Potsdam), Wilhelm Stannat (TU Berlin), Prashant Mehta (University of Illinois at Urbana-Champaign) and Amirhossein Taghvaei (University of Illinois at Urbana-Champaign).