MAC-MIGS Research Afternoon – Friday 8th July 2022

** Intersections in Optimisation and Fluid Mechanics **

The workshop will take place on the 8th July, 11am-4pm. This is a hybrid event with limited in-person spaces at Bayes Centre, University of Edinburgh. To register, please complete the  Registration Form

Fluid Mechanics and Optimisation are two branches of applied mathematics which have been extensively studied, and which have many interesting connections. For example, an active area of research in fluid mechanics is turbulence, including the development of numerics and analytics to describe turbulent behavior. Modern techniques often employ a blending of optimisation techniques as we try to understand how to control turbulence. This MAC-MIGS research afternoon aims to cover topics which lie at the intersection of optimisation and fluid mechanics, bringing together problems from turbulent flow, machine learning, and other modern topics within these fields.

Timetable

Organisers: Anastasia Istratuca, Théo Lavier, Moritz Linkmann, Jacques Vanneste, John Pearson

Abstracts:

Camilla Nobili : The role of boundary conditions on scaling laws in the Rayleigh-Bénard convection problem

Rayleigh-Bénard convection is the buoyancy-driven flow of a fluid heated from below and cooled from above and is a paradigm for nonlinear dynamics with important applications in meteorology, oceanography and engineering. We are interested in obtaining quantitative bounds on the Nusselt number, the vertical heat transport enhancement factor. The Nusselt number, besides being an interesting quantity for engineering applications, is the natural quantity to measure the intensity and effectiveness of the motion. For this reason, we are interested in proving (upper) bounds which catch the relation between the Nusselt number and the (nondimensional control parameter) Rayleigh number, in turbulent regimes. Despite great scientific developments in this field in the last 30 years, it is still not clear what role the boundary conditions play in the scaling laws for the Nusselt number. In this talk we address this problem, establishing rigorous bounds for the Rayleigh-Bénard convection problem with Navier-slip boundary conditions for the velocity. We employ the background field method and deal with a careful PDE analysis, due to the production of vorticity at the walls. In conclusion, we relate this result to other bounds derived for no-slip and stress-free boundary conditions and discuss open problems.

Nicolas Gauger: Linking classical optimization with PDEs/ODEs to the training of ANNs

with contributions from Emre Özkaya, Rohit Pochampalli, Guillermo Suarez from TUK, Germany and Stefanie Günther from LLNL, USA

The so-called “grey-box modeling” approach cleverly combines insights from simulations (“white-box modeling”) on high-performance computing (HPC) architectures with data-driven approaches (“black-box modeling”) using artificial intelligence (AI) methods.

As an example, we consider here the so-called “field inversion” method in data-driven turbulence modeling for the Navier-Stokes equations. For the field inversion, classical methods from the area of optimization with partial differential equations are used. Furthermore, in the data-driven part, the training of certain artificial neural networks shows analogies to the optimization for ordinary differential equations. In all approaches, we keep an eye on automation and parallelization on the computing systems to be used.

Coralia Cartis : Optimization with expensive and uncertain data – challenges and improvements

Real-life applications often require the optimization of nonlinear functions with several unknowns or parameters – where the function is the result of highly expensive and complex model simulations involving noisy data (such as climate or financial models, chemical experiments), or the output of a black-box or legacy code, that prevent the numerical analyst from looking inside to find out or calculate problem information such as derivatives. Thus, classical optimization algorithms, that use derivatives (steepest descent, Newton’s methods) often fail or are entirely inapplicable in this context. Efficient derivative-free optimization algorithms have been developed in the last 15 years in response to these imperative practical requirements. As even approximate derivatives may be unavailable, these methods must explore the landscape differently and more creatively. In state-of-the-art techniques, clouds of points are generated judiciously and sporadically updated to capture local geometries as inexpensively as possible; local function models around these points are built using techniques from approximation theory and carefully optimised over a local neighbourhood (a trust region) to give a better solution estimate.

In this talk, I will describe our implementations and improvements to state-of-the-art methods. In the context of the ubiquitous data fitting/least-squares applications, we have developed a simplified approach that is as efficient as state of the art in terms of budget use, while achieving better scalability. Furthermore, we substantially improved the robustness of derivative-free methods in the presence of noisy evaluations. Theoretical guarantees of these methods will also be provided.