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MATH 745 [Fall 2023] »

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equationsAndAIMSLab

Research Activities

[Emerging Research Directions]

  • Applications of the Riemann-Hilbert Problem in Fluid Mechanics
  • Computational Nonequilibrium Thermodynamics
  • Bayesian Inversion and Uncertainty Quantification

    • [Main Research Themes (past & present)]

    • Systematic Search for Extreme (Singular?) Behavior in Fluid-Mechanics Models -- This research program, summarized in Kang, Yun & Protas (2019); Pocas & Protas (2018); Ayala & Protas (2017); Yun & Protas (2017); Ayala & Protas (2014a); Ayala & Protas (2014b), has opened a "computational window"' into some basic questions of the mathematical fluid dynamics. We have been interested in the problem whether certain fundamental estimates concerning the maximum possible growth of various quadratic quantities in hydrodynamics obtained using rigorous methods of mathematical analysis are sharp and can be realized in actual flows. Originally formulated by Charles Doering of the University of Michigan, these questions are intrinsically related to the problem of the finite-time singularity formation in the 3D Navier-Stokes equation, an issue recognized by the Clay Mathematics Institute as one of the "Millennium Problems" for the mathematics community. We have developed a versatile framework wherein such questions can be studied by systematically searching for the most singular solutions using methods of numerical optimization applied to suitably-defined variational optimization problems. This research has provided new insights about the sharpness of key estimates on the maximum growth of different Sobolev norms of the solutions (referred to as enstrophy and palinstrophy) in 1D, 2D and 3D flows as well as about the precise nature of the solutions which saturate these estimates. These findings have complemented rigorous analysis by demonstrating which estimates are already sharp and cannot be improved and if they can, what form they could possibly take, thus establishing a first of its kind bridge between theory and computation. In particular, we showed that even in the worst-case scenario the largest possible growth of enstrophy in 3D Navier-Stokes flows remains finite, indicating that there is no evidence for singularity formation in finite time (Kang, Yun & Protas, 2019). While computational studies of potential singularity formation have had a long history in hydrodynamics, what sets our approach aside is that we methodically search for worst behaved solutions using optimization methods. These questions were among the main themes of the Thematic Program on Multiscale Scientific Computing and a follow-up Workshop on Extreme Events and Criticality in Fluid Mechanics which took place at the Fields Institute in Toronto in January-April 2016 and in April 2019. They were also the subject of my former student Diego Ayala's Ph.D. thesis for which in 2015 he received the prestigious Cecil Graham Dissertation Award of the Canadian Applied and Industrial Mathematics Society. Some of these results have already inspired new developments in the mathematical analysis of this problem and partly motivated recent activities on the complementary problem of finding tight polynomial bounds.

    • Existence and Stability of Vortex Equilibria -- Problems of vortex dynamics are among the most classical in fluid mechanics and have been studied for a long time. My research has offered a new perspective on a number of basic questions concerning the existence and stability of inviscid vortex equilibria. The main mathematical difficulty presented by these problems is that they are often of the free-boundary type and to address this issue, in collaboration with the late Professor Elcrat of Wichita State University, we developed a general framework for studying vortex stability based on modern tools from differential geometry known as the "shape calculus" (Elcrat & Protas, 2013). This approach was then applied to resolve long-standing open problems in theoretical fluid mechanics concerning the stability of Hill's and Norbury's vortices with respect to axisymmetric perturbations (Protas & Elcrat, 2016; Protas, 2019). These results represent a stepping stone towards obtaining a full understanding of the stability properties of inviscid vortex rings, which is another main question to be addressed in the proposed research. In a research effort related to yet another classical problem in fluid mechanics, we demonstrated that the equilibria of inviscid vortex sheets can in fact be stabilized using methods of modern control theory (Protas & Sakajo, 2018). Vortex sheets are governed by the notoriously ill-posed Birkhoff-Rott equation and are known to be among the most unstable objects in fluid mechanics. More recently, we used methods of complex analysis, namely, the theory of the Riemann-Hilbert problem, to construct new families of equilibrium solutions to Euler equations involving vortex sheets (Protas & Sakajo, 2019), which is one of the first constructions of such kind. In 2017 I was awarded the prestigious Invitational Fellowship of the Japan Society for the Promotion of Science to pursue joint research on these problems together with Professor Sakajo at Kyoto University. The techniques we developed to study these problems are being adopted by other researchers.

    • Optimal Identification of Constitutive Relations -- In this research effort we have developed an efficient computational framework, relying on a variational formulation, which allows one to optimally identify key elements of the structure of a governing system (such as, e.g., the dependence of the nonlinear fluxes on the state variables, etc.) based on some available measurements. In contrast to earlier studies on similar inverse problems, our formulation is developed in the continuous (infinite-dimensional) setting and hence its key ingredients are independent of the particular discretization used and subject to a minimum number of assumptions. This work was the subject of my former student Vlad Bukshtynov's Ph.D. dissertation for which in 2013 he received the prestigious Cecil Graham Dissertation Award of the Canadian Applied and Industrial Mathematics Society. The approach we developed saw several important applications, e.g., to optimal identification of inertial manifolds in reduced-order models (Protas, Noack & Morzynski, 2014), to inferring the structure of a vortex from partial velocity data (Danaila & Protas, 2015) and to inverse problems in electrochemistry (Sethurajan et al., 2015). An important open problem of this type which has many practical applications in Computational Fluid Dynamics concerns the design of optimal turbulence closures in hydrodynamic models. We have successfully studied this problem in the context of simplified flow models obtaining a convincing proof of the concept for this approach (Protas, Noack & Oesth, 2015; Matharu & Protas, 2019). Upscaling it to more realistic problems, such as systems governed by the 3D Navier-Stokes equation, is the third main theme of the present application. In recent years I been invited to the Chinese University of Hong Kong and to the Universities of Poitiers and Rouen in France to collaborate on various extensions of this approach.

    • Numerical Optimization of Partial Differential Equations (PDEs) -- The advances reported above would not have been possible without continued research on improving methods for the numerical solution of PDE optimization problems. As one key enabler, Sobolev gradients are used to ensure that the computed solutions possess the required regularity properties and in our recent work (Novruzi & Protas, 2019) we have introduced a comprehensive framework allowing one to determine Sobolev gradients in an optimal manner. In the paper Danaila & Protas, 2017 we combined concepts from Riemannian geometry and the theory of Sobolev gradients to derive a new conjugate gradient method for direct minimization of the Gross-Pitaevskii energy functional with rotation. Ground states computed in this way accurately describe the Bose-Einstein condensate observed in a number of recent experiments. I conducted a part of this work during several invited visits at Universite de Rouen in France where I was also invited to lecture on these topics to graduate students during an "autumn school" Rencontres Normandes sur les aspects theoriques et numeriques des EDP.

    • Mathematical Modelling in Electrochemistry -- Fueled by advances made through our basic research, such as the work on optimal identification of constitutive relations described above, this research effort has focused on applied problems arising in industry such as modeling mechanical processes and transport of charges species in materials used in modern Li-ion batteries. A hallmark of this research is elegant integration of experimental data with mathematical modeling, asymptotic analysis and scientific computation, which made it possible for us to obtain one of the first systematic solutions of an inverse problem in the field of electrochemistry (Sethurajan et al., 2015) together with careful quantification of uncertainties (Sethurajan et al., 2019a). These methodological advances in turn allowed us to obtain new insights about the validity of the Planck-Nernst equation which is the main mathematical model used in electrochemistry (Sethurajan et al., 2019b). In this fast-moving field our work has been published in high-impact journals: Harris et al., 2017 in Chemistry of Materials and Lui et al., 2016; Foster et al., 2017; Font et al., 2018 in the Journal of Power Sources. This interdisciplinary research involves close collaboration with chemists, material scientists as well as research engineers from industrial partner organizations.

    [Current Research Collaborations]

    • Miguel Bustamante (School of Mathematics and Statistics, University College Dublin, Ireland)
    • Ionut Danaila (Laboratoire de Math�matiques Rapha�l Salem, Universit� de Rouen, France)
    • Charles Doering (Departments of Physics, Mathematics, and Complex Systems, University of Michigan, Ann Arbor, USA)
    • Jamie Foster (School of Mathematics & Physics, University of Portsmouth, UK)
    • Gillian Goward (Department of Chemistry & Chemical Biology, McMaster University, Canada)
    • Nicholas Kevlahan (Department of Mathematics and Statistics, McMaster University, Canada)
    • Bernd Noack (LIMSI, CNRS, Université Paris-Saclay, France)
    • Giles Richardson (Mathematical Sciences, University of Southamptom, UK)
    • Takashi Sakajo (Department of Mathematics, Kyoto University, Japan)

    [Research Funding]