MAC-MIGS afternoons continue on Friday 26th June with:

2:00pm to 2:40pm: Susana Gomes (Mathematics Institute, University of Warwick)

Parameter estimation for coupled PDE-SDE systems: an application to pedestrian dynamics

Abstract: When dealing with applications, we often have a model that describes some physical phenomenon, but we don’t know what the right parameters for our model are. If we have access to observations of the modelled phenomenon (e.g., from experiments), we can set up the inverse problem of finding the relevant parameters given the data obtained from our observations. In this talk, I will introduce a model for how pedestrians move in a crowd: in the model, the position of each pedestrian is a realisation of a stochastic differential equation (SDE) which depends on the density of pedestrians, a solution of a partial differential equation (PDE). Both the SDE and the PDE depend on an unknown parameter, the maximum speed of a pedestrian, v_max. Because of the nature of the problem, the standard way of solving the inverse problem of “finding v_max” given data by minimising the misfit does not work. I will explain why this is the case and explore different ways of solving this problem.

——————————————————————

2:50pm to 3:30pm  Emilie Chouzenoux (Inria Saclay, France)

Majorization-Minimization Subspace Algorithms for Large Scale Data Processing

Modern image recovery problems require powerful optimization frameworks to handle high dimensionality while providing reliable numerical solutions in a reasonable time. New optimization algorithms have thus to be designed, paying attention to computational complexity, scalability, and robustness. Majorization-Minimization (MM) approaches have become increasingly popular recently, in both signal/image processing and machine learning areas. My talk will present new theoretical and practical results regarding the MM Memory Gradient (3MG) algorithm [1], where the update of each iterate is restricted to a subspace of low dimension. I will show how to cast this efficient scheme into a block distributed version of it, named BD3MG, where blocks of variables are processed in an asynchronous manner, so as to take advantage of a distributed memory environment [2]. Convergence of the sequence built by the proposed BD3MG method will be analysed under mild assumptions. Application to the restoration of 3D images degraded by a depth-variant blur will show the significant computational time reduction and the great scalability potential offered by the BD3MG approach when compared to several synchronous and asynchronous competitors.

[1] E. Chouzenoux, A. Jezierska, J.-C. Pesquet and H. Talbot. A Majorize-Minimize Subspace Approach for l2-l0 Image Regularization. SIAM Journal on Imaging Science, Vol. 6, No. 1, pages 563-591, 2013.

[2] M. Chalvidal and E. Chouzenoux. Block Distributed Majorize-Minimize Memory Gradient Algorithm and its application to 3D image restoration. In Proceedings of ICIP 2020.

https://arxiv.org/abs/2002.02328

——————————————————————

3:45pm to 4:25pm  Benjamin Peherstorfer (Courant Institute of Mathematical Sciences, New York University)

Learning low-dimensional dynamical-system models from data via non-intrusive model reduction

Abstract: This work introduces a method for learning low-dimensional dynamical-system models from data of high-dimensional black-box systems. The key contribution is a data sampling scheme that introduces a re-projection step to obtain trajectories corresponding to Markovian dynamics in low-dimensional subspaces. Models fitted to re-projected trajectories exactly match reduced models that are traditionally constructed with model reduction techniques from full knowledge of the governing equations and their discrete operators of the high-dimensional systems. Building on a posteriori error estimators from traditional model reduction, we derive probabilistic bounds for the generalization error of the models learned from data. Numerical results demonstrate the workflow of the proposed approach from data to reduced models to certified predictions for establishing trust in decisions made from data.