BREAD · Breaking the curse of dimensionality: numerical challenges in high dimensional analysis and simulation

This project is concerned with problems that involve a very large number of variables, and whose efficient numerical treatment is challenged by the so-called curse of dimensionality, meaning that computational complexity increases exponentially in the variable dimension.The PI intend to establish in his host institution a scientific leadership on the mathematical understanding and numerical treatment of these problems, and to contribute to the development of this area of research through international collaborations, organization of workshops and research schools, and training of postdocs and PhD students.High dimensional problems are ubiquitous in an increasing number of areas of scientific computing, among which statistical or active learning theory, parametric and stochastic partial differential equations, parameter optimization in numerical codes. There is a high demand from the industrial world of efficient numerical methods for treating such problems.The practical success of various numerical algorithms, that have been developed in recent years in these application areas, is often limited to moderate dimensional setting.In addition, these developments tend to be, as a rule, rather problem specific and not always founded on a solid mathematical analysis.The central scientific objectives of this project are therefore: (i) to identify fundamental mathematical principles behind overcoming the curse of dimensionality, (ii) to understand how these principles enter in relevant instances of the above applications, and (iii) based on the these principles beyond particular problem classes, to develop broadly applicable numerical strategies that benefit from such mechanisms.The performances of these strategies should be provably independent of the variable dimension, and in that sense break the curse of dimensionality. They will be tested on both synthetic benchmark tests and real world problems coming from the afore-mentioned applications.""