Data Assimilation and Reduced Modeling for High Dimensional Problems

CIRM, Luminy, France
July 19-August 27, 2021

A component-based data assimilation strategy with applications to vascular flows

Supervisors: Fabio Nobile (EPFL), Tommaso Taddei (Inria)
Students: Open to 2 students

Project Description:

Data assimilation (DA) refers to the process of integrating information coming from a mathematical model with experimental observations for prediction: in state estimation, we are interested in estimating the system state in a region of interest; in model update, we are interested in improving (``updating'') the mathematical model of the system and/or its parameters (``model calibration''). The aim of this project is to develop an offline/online component-based DA strategy for systems governed by PDEs; our ultimate goal is to devise a robust tool to support clinical decisions for the treatment of vascular diseases.

We pursue a localised approach based on the description of physical systems as instantiations of (possibly parameterised) components from a suitable library, [2]. During the offline stage, we exploit experimental measurements and parameterised models to devise an accurate reduced-order basis (ROB) for the system state and a suitable model correction. During the online stage, we exploit the learned model and newly-available experimental observations to estimate the system state for new experimental configurations.

By learning reduced models at the component level, we aim to increase their generalization properties and ultimately their portability for new configurations. Furthermore, we envision that localised learning might also reduce sensitivity of the DA procedure to the choice of far-field boundary conditions (cf. [4]). Note that, since the available models used are local, we should exploit domain decomposition (DD) strategies to properly glue together local fields. DD should be physics-based --- it should be designed to preserve continuity of solution and fluxes at elements' interfaces.

In this project, we propose to investigate performance of the component-based DA strategy for a representative synthetic model problem. First, we adapt the state estimation strategies in [1,3] to reconstruct the local state in each component of the network based on sparse measurements; second, we aim to consider parametric and non-parametric model update to improve available models and ultimately improve online reconstruction performance. Time permitting, we also wish to tackle the online state estimation problem.

[1] Arnone, E., Azzimonti, L., Nobile, F. and Sangalli, L.M., 2019. Modeling spatially dependent functional data via regression with differential regularization. Journal of Multivariate Analysis, 170, pp.275-295.
[2] Huynh, D.B.P., Knezevic, D.J., & Patera, A.T. 2013. A static condensation reduced basis element method: approximation and a posteriori error estimation. ESAIM: Mathematical Modelling and Numerical Analysis, 47(1), 213-251
[3] Maday, Y., Patera, A.T., Penn, J.D. and Yano, M., 2015. A parameterized‐background data‐weak approach to variational data assimilation: formulation, analysis, and application to acoustics. International Journal for Numerical Methods in Engineering, 102(5), pp.933-965.
[4] Taddei, T. and Patera, A.T., 2018. A localization strategy for data assimilation; application to state estimation and parameter estimation. SIAM Journal on Scientific Computing, 40(2), pp.B611-B636.