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Computation and analysis of statistical solutions of fluid flow

Periodic Reporting for period 3 - COMANFLO (Computation and analysis of statistical solutions of fluid flow)

Reporting period: 2021-08-01 to 2023-01-31

The project COMANFLO focuses on the analysis and computation of statistical solutions for the nonlinear partial differential equations (PDEs) which model the flow of fluids. The study of fluid flow is crucial importance in several areas of
science and engineering. Currently, there is only partial understanding of the complex dynamics of fluids. In particular, notions of existence, uniqueness, stability and computability of solutions of the governing equations of fluids such as
hyperbolic systems of conservation laws and the incompressible Euler and Navier-Stokes equations. This project investigates a recently proposed solution concept, the so-called statistical solutions, which are time-parameterized probability
measures on infinite-dimensional function spaces, as an appropriate framework for describing complex fluid motion. We study whether these solutions exist, are unique and can be computable. We also seek to design efficient algorithms to
compute these solutions. At the mid-term stage of the project, several of these objectives have already been met and there is a very high probability for a successful completion of most of the aims of the project. If successful, the mathematical
analysis in the project will further validate the use of the nonlinear PDEs governing fluid dynamics in several contexts and the algorithms developed in the project can be used for efficient computation of fluid flows.
At the mid-term stage of the project, we have successfully completed several of the sub-projects and many goals have already been met. In particular, we have designed Monte Carlo based algorithms (together with
Finite Volumes or spectral viscosity methods) for the computation of statistical solutions of systems of conservation laws and incompressible Euler equations. These algorithms have been proved to converge, under certain
natural and verifiable hypotheses, to a statistical solution. Moreover, a large number of numerical experiments have been performed to illustrate both the algorithms as well as different qualitative aspects of statistical
solutions. Similarly, partial results on the existence, uniqueness of statistical solutions and on their connection with turbulence models have been obtained. Finally, significant progress has been made in the design of
novel algorithms that accelerate the base Monte Carlo algorithm for computing statistical solutions. In particular, the use of novel machine learning techniques has made great impact.
The results achieved so far within the purview of the project go significantly beyond the state of the art and push the frontiers of our knowledge about
nonlinear PDEs governing fluid flows and their computation. In particular, we were able to obtain the first convergence results for numerical algorithms to the statistical
solutions of multi-dimensional hyperbolic systems of conservation laws and incompressible Euler equations. Moreover, several novel algorithms for computing these statistical
solutions efficiently, based on the use of deep neural networks, have been developed and tested. Looking into the future, one expects further progress on the hard theoretical
questions of global existence and uniqueness of statistical solutions, as well as on the further acceleration of computation of these solutions with novel machine learning algorithms.
Moreover, we expect to also significantly advance our understanding and computation of Bayesian Inverse problems for these equations