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Applied and Computational Mathematics Seminar

Spring 2024

date speaker title host(s)
Feb 2 Chris Rycroft (UW) The reference map technique for simulating complex materials and multi-body interactions
Feb 9 Scott Weady (Flatiron Institute) Entropy methods in active suspensions Saverio and Laurel
Feb 16 David Saintillan (UC San Diego) Hydrodynamics of active nematic surfaces Saverio and Tom
Feb 23 Rose Cersonsky (UW) Data-driven approaches to chemical and materials sciences Chris
Mar 1 [4:00pm Colloquium] Per-Gunnar Martinsson (UT Austin) TBA Li
Mar 8 Rogerio Jorge (UW-Madison) TBA Li
Mar 15 Di Qi (Purdue University) Statistical Reduced-Order Models and Random Batch Method for Complex Multiscale Systems Chen
Mar 22
Mar 29 Spring break
Apr 5 Jinlong Wu (UW) TBA Saverio
Apr 12 Gabriel Zayas-Caban (UW) TBA Li
Apr 19 Tony Kearsley (NIST) TBA Fabien
Apr 26 Malgorzata Peszynska (Oregon State) TBA Fabien


Chris Rycroft (UW–Madison)

Title: The reference map technique for simulating complex materials and multi-body interactions

Conventional computational methods often create a dilemma for fluid–structure interaction problems. Typically, solids are simulated using a Lagrangian approach with grid that moves with the material, whereas fluids are simulated using an Eulerian approach with a fixed spatial grid, requiring some type of interfacial coupling between the two different perspectives. Here, a fully Eulerian method for simulating structures immersed in a fluid will be presented [1]. By introducing a reference map variable to model finite-deformation constitutive relations in the structures on the same grid as the fluid, the interfacial coupling problem is highly simplified. The method is particularly well suited for simulating soft, highly-deformable materials and many-body contact problems [2], and several examples in two and three dimensions [3] will be presented.

  1. K. Kamrin, C. H. Rycroft, and J.-C. Nave, J. Mech. Phys. Solids 60, 1952–1969 (2012). [DOI link]
  2. C. H. Rycroft et al., J. Fluid Mech. 898, A9 (2020). [DOI link]
  3. Y. L. Lin, N. J. Derr, and C. H. Rycroft, Proc. Natl. Acad. Sci. 119, e2105338118 (2022). [DOI link]

Scott Weady (Flatiron Institute)

Title: Entropy methods in active suspensions

Collections of active particles, such as suspensions of E. coli or mixtures of microtubules and molecular motors, can exhibit rich non-equilibrium dynamics due to a combination of activity, hydrodynamic interactions, and steric stresses. Continuum kinetic theories, which characterize the set of particle configurations through a continuous distribution function, provide a powerful framework for analyzing such systems and connecting their micro- to macroscopic dynamics. The probabilistic formulation of kinetic theories leads naturally to a characterization in terms of entropy, whether thermodynamic or information-theoretic. In equilibrium systems, entropy strictly increases and always tends towards steady state. This no longer holds in active systems, however entropy still has a convenient mathematical structure. In this talk, we use entropy methods, specifically variational principles involving the relative entropy functional, to study the nonlinear dynamics and stability of active suspensions in the context of the Doi-Saintillan-Shelley kinetic theory. We first present a class of moment closures that arise as constrained minimizers of the relative entropy, and show these closures preserve the kinetic theory's stability and entropic structure while admitting efficient numerical simulation. We then derive variational bounds on relative entropy fluctuations for apolar active suspensions that are closely related to the moment closures. These bounds provide conditions for global stability and yield estimates of time-averaged order parameters. Finally, we discuss applications of these methods to polar active suspensions.

David Saintillan (UC San Diego)

Title: Hydrodynamics of active nematic surfaces

The dynamics of biological surfaces often involves the coupling of internal active processes with in-plane orientational order and hydrodynamic flows. Such active surfaces play a key role in various biological processes, from cytokinesis to tissue morphogenesis. In this talk, I will discuss two approaches for the modeling and simulation of active nematic surfaces. In a first model, we analyze the spontaneous dynamics of a freely-suspended viscous drop with surface nematic activity and its coupling with bulk fluid mechanics. Using a spectral boundary integral solver for Stokes flow coupled with a hydrodynamic evolution equation for the nematic tensor, numerical simulations reveal a complex interplay between the flow inside and outside the drop, the surface transport of the nematic field and surface deformations, giving rise to a sequence of self-organized behaviors and symmetry-breaking phenomena of increasing complexity, consistent with experimental observations. In the second part of the talk, I will present a novel computational approach for the simulation of active nematic fluids confined to Riemannian manifolds. The fluid velocity and nematic order parameter are represented as sections of the complex line bundle of a two-manifold. Using a geometric approach based on the Levi-Civita connection, we introduce a coordinate-free discretization method that preserves the continuous local-to-global theorems in differential geometry. Furthermore, we establish a nematic Laplacian on complex functions that can accommodate fractional topological charges through the covariant derivative on the complex nematic representation. Advection of the nematic field is formulated based on the Lie derivative, resulting in a stable geometric semi-Lagrangian discretization scheme for transport by the flow. The proposed surface-based method offers an efficient and stable means to investigate the influence of local curvature and topology on the hydrodynamics of active nematic systems, and we illustrate its capabilities by simulating active flows on a range of surfaces of increasing complexity.

Rose Cersonsky (UW–Madison)

Title: Data-driven approaches to chemical and materials sciences: the importance of data selection, representation, and interpretability

Like many other fields, there has been a recent and overwhelming wave of machine learning and artificial intelligence methods being employed in the chemical sciences. While these methods have the undoubted ability to drive innovation and capabilities, their application to chemical sciences requires a nuanced understanding of molecular representations and structure-property relationships.

In this talk, I will discuss the role of molecular featurization – how we transform atoms and molecules into mathematical signals appropriate for machine-learning thermodynamic quantities – and unsupervised analyses that allow us to easily understand and assess these so-called “featurizations” in the context of complex machine learning tasks. In doing so, I will demonstrate how linear methods – that constitute the simplest, most robust, and most transparent approaches to automatically processing large amounts of data – can be leveraged to understand molecular crystallization and aid in pharmaceutical engineering.

All methods discussed are available through the open-source scikit-matter software, an official scikit-learn companion that implement methods born out of the materials and chemistry communities.

Di Qi (Purdue)

Title: Statistical Reduced-Order Models and Random Batch Method for Complex Multiscale Systems

Abstract: The capability of using imperfect stochastic and statistical reduced-order models to capture key statistical features in multiscale nonlinear dynamical systems is investigated. A systematic framework is proposed using a high-order statistical closure enabling accurate prediction of leading-order statistical moments and probability density functions in multiscale complex turbulent systems. A new efficient ensemble forecast algorithm is developed dealing with the nonlinear multiscale coupling mechanism as a characteristic feature in high-dimensional turbulent systems. To address challenges associated with closely coupled spatio-temporal scales in turbulent states and expensive large ensemble simulation for high-dimensional complex systems, we introduce efficient computational strategies using the so-called random batch method. It is demonstrated that crucial principal statistical quantities in the most important large scales can be captured efficiently with accuracy using the new reduced-order model in various dynamical regimes of the flow field with distinct statistical structures. Finally, the proposed model is applied for a wide range of problems in uncertainty quantification, data assimilation, and control.

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