nD - nD: Multidimensional dynamical systems, multi-linear numerical algorithms and tensor data

From 01-10-2015 to 30-09-2022

Description

Summary

Twenty years ago, one of the research tracks we started was the identification of dynamical models from observed data, using concepts and algorithms from numerical linear algebra. This endeavor has proven to be very rewarding, not only scientifically, but also from the point of view of engineering applications. Indeed, the last 10 years we have been diversifying into the biomedical field (signal processing, health decision support, bioinformatics), the process industry (modeling and control, data-assimilation, data-mining) and even quantum information theory. This has led to fruitful and inspiring interactions with industrial partners, and the creation of 6 spin-off companies.

With this proposal, we want to return to the basic roots of that research track, however with a fundamental twist to each of the three ingredients: The time series of the 90’s have now turned into higher-order (nD) data, lots of them (‘big data’). The concepts and algorithms of numerical linear algebra are now maturing into exciting new developments of multi-linear algebra (e.g. tensor decompositions). The ‘1D’ dynamical models, with basically only time as an independent variable, are to be generalized into multi-dimensional dynamical models, which are calculated from the nD data, using multi-linear algebra tools.

All of the applications we envisage 1 are characterized by a tsunami of data. They have in common their high dimensionality, and are described in terms of several (instead of only 1) ‘independent’ variables (e.g. time, 3 space coordinates, a frequency axis, physical quantities such as temperature, pressure, concentrations, etc.). The 2nd ingredient of this proposal concerns mathematical models of dynamical systems. These are used to simulate in silico, to design control strategies, to monitor the behavior and to predict. Increasingly, models are used to compress the tsunami of data in order to deal with the so-called ‘curse-of-dimensionality’. Indeed, nD data points tend to be correlated with neighbors in the nD data space, and by modeling this and deeper correlations by means of nD state space models, one can achieve a dramatic compression. The 3rd ingredient of the proposal describes how we develop new numerical multi-linear algebra algorithms to obtain nD models from nD data.

 The proposal is partitioned in three work packages, corresponding to the three relevant Research Challenges we have identified:

Research Challenge I: Back to the roots, or a new numerical linear algebra theory to calculate all or some of the roots of a set of multivariable polynomials. This is an important ingredient for RC II and III;

Research Challenge II: Tensor decompositions and applications in signal processing, 1D system modeling and big data analysis;

Research Challenge III: nD subspace identification for nD systems.

Team

• Lieven De Lathauwer, Co-promoter
• Bart De Moor, Promoter
• Katrien De Cock, Team member

Financing

Funding: KU Leuven - Internal Funding KU Leuven

Program/Grant Type: C1 - KU Leuven Category 1 Funding