Recent theoretical works have shown that the NSGA-II efficiently computes the full Pareto front when the population size is large enough. In this work, we study how well it approximates the Pareto front when the population size is smaller. For the OneMinMax benchmark, we point out situations in which the parents and offspring cover well the Pareto front, but the next population has large gaps on the Pareto front. Our mathematical proofs suggest as reason for this undesirable behavior that the NSGA-II in the selection stage computes the crowding distance once and then removes individuals with smallest crowding distance without considering that a removal increases the crowding distance of some individuals. We then analyze two variants not prone to this problem. For the NSGA-II that updates the crowding distance after each removal (Kukkonen and Deb (2006)) and the steady-state NSGA-II (Nebro and Durillo (2009)), we prove that the gaps in the Pareto front are never more than a small constant factor larger than the theoretical minimum. This is the first mathematical work on the approximation ability of the NSGA-II and the first runtime analysis for the steady-state NSGA-II. Experiments also show the superior approximation ability of the two NSGA-II variants.

This paper is dedicated to the study of the orthogonal decomposition of spatially and temporally distributed signals in fluid-structure interaction problems. First application is concerned with the analysis of wall-pressure distributions over bluff bodies. The need for such a tool is increasing due to the progress in data acquisition systems and in computational fluid dynamics. The classical proper orthogonal decomposition (POD) method is discussed, and it is shown that heterogeneity of the mean pressure over the structure induces difficulties in the physical interpretation. It is then proposed to use the biorthogonal decomposition (BOD) technique instead; although it appears similar to POD, it is more general and fundamentally different since this tool is deterministic rather than statistical. The BOD method is described and adapted to wall-pressure distribution, with emphasis on aerodynamic load decomposition. The second application is devoted to the generation of a spatially correlated wind velocity field which can be used for the temporal calculation of the aeroelastic behaviour of structures such as bridges. In this application, the space-time symmetry of the BOD method is absolutely necessary. Examples are provided in order to illustrate and show the satisfactory performance and the interest of the method. Extensions to other fluid-structure problems are suggested.

We introduce a systematically improvable family of variational wave functions for the simulation of strongly correlated fermionic systems. This family consists of Slater determinants in an augmented Hilbert space involving "hidden" additional fermionic degrees of freedom. These determinants are projected onto the physical Hilbert space through a constraint which is optimized, together with the single-particle orbitals, using a neural network parametrization. This construction draws inspiration from the success of hidden particle representations but overcomes the limitations associated with the mean-field treatment of the constraint often used in this context. Our construction provides an extremely expressive family of wave functions, which is proven to be universal. We apply this construction to the ground state properties of the Hubbard model on the square lattice, achieving levels of accuracy which are competitive with state-of-the-art variational methods.