L4

Low-rank approximation of parameter-dependent matrices A(t) is an important task in the computational sciences, with applications in areas such as dynamical systems and the compression of series of images. In this talk, we introduce AdaCUR, an efficient randomised algorithm for computing low-rank approximations of parameter-dependent matrices using the CUR decomposition. The key idea of our approach is the ability to reuse column and row indices for nearby parameter values, improving efficiency. The resulting algorithm is rank-adaptive, provides error control, and has complexity that compares favourably with existing methods. This is joint work with Yuji Nakatsukasa.

Shape optimization is a rich field at the intersection of analysis, optimization, and engineering. It seeks to determine the optimal geometry of structures to minimize performance objectives, subject to physical constraints—often modeled by Partial Differential Equations (PDEs). Traditional approaches commonly assume that these constraints admit a unique solution for each candidate shape, implying a single-valued shape-to-solution map. However, many real-world structures exhibit multistability, where multiple stable configurations exist under identical physical conditions.

This research departs from the single-solution paradigm by investigating shape optimization in the presence of multiple solutions to the same PDE constraints. Focusing on a neo-Hookean hyperelastic model, we formulate an optimization problem aimed at controlling the energy jump between distinct solutions. Drawing on bifurcation theory, we develop a theoretical framework that interprets these solutions as continuous branches parameterized by shape variations. Building on this foundation, we implement a numerical optimization strategy and present numerical results that demonstrate the effectiveness of our approach.

Many applications in electromagnetism, magnetohydrodynamics, and pour media flow are well-posed in spaces from the 3D de Rham complex involving $H^1$, $H(curl)$, $H(div)$, and $L^2$. Discretizing these spaces with the usual conforming finite element spaces typically leads to discrete problems that are both structure-preserving and uniformly stable with respect to the mesh size and polynomial degree. Robust preconditioners/solvers usually require the inversion of subproblems or auxiliary problems on vertex, edge, or face patches of elements. For high-order discretizations, the cost of inverting these patch problems scales like $\mathcal{O}(p^9)$ and is thus prohibitively expensive. We propose a new set of basis functions for each of the spaces in the discrete de Rham complex that reduce the cost of the patch problems to $\mathcal{O}(p^6)$ complexity. By taking advantage of additional properties of the new basis, we propose further computationally cheaper variants of existing preconditioners. Various numerical examples demonstrate the performance of the solvers.

A beautiful example of a nonlinear eigenvalue problem is the determination of complex eigenvalues for wave scattering. This talk will show how nicely this can be done by applying AAA rational approximation to a scalarized resolvent sampled at a few real frequencies.  Even for a domain as elementary as a circle with a gap in it, such computations do not seem to have been done before. This is joint work with Oscar Bruno and Manuel Santana at Caltech.

Multicomponent fluids are mixtures of distinct chemical species (i.e. components) that interact through complex physical processes such as cross-diffusion and chemical reactions. Additional physical phenomena often must be accounted for when modelling these fluids; examples include momentum transport, thermality and (for charged species) electrical effects. Despite the ubiquity of chemical mixtures in nature and engineering, multicomponent fluids have received almost no attention from the finite element community, with many important applications remaining out of reach from numerical methods currently available in the literature. This is in spite of the fact that, in engineering applications, these fluids often reside in complicated spatial regions -- a situation where finite elements are extremely useful! In this talk, we present a novel class of high-order finite element methods for simulating cross-diffusion and momentum transport (i.e. convection) in multicomponent fluids. Our model can also incorporate local electroneutrality when the species carry electrical charge, making the numerical methods particularly desirable for simulating liquid electrolytes in electrochemical applications. We discuss challenges that arise when discretising the partial differential equations of multicomponent flow, as well as some salient theoretical properties of our numerical schemes. Finally, we present numerical simulations involving (i) the microfluidic non-ideal mixing of hydrocarbons and (ii) the transient evolution of a lithium-ion battery electrolyte in a Hull cell electrode.

Given a Hamiltonian circle action on a symplectic manifold, Fukaya and Teleman tell us that we can relate the equivariant Fukaya category to the Fukaya category of a symplectic reduction.  Yanki Lekili and I have some conjectures that extend this story - in certain special examples - to singular values of the moment map. I'll also explain the mirror symmetry picture that we use to support our conjectures, and how we interpret our claims in Teleman's framework of `topological group actions' on categories.

The motion of a particle suspended in a fluid flow is governed by hydrodynamic interactions. In this talk, I will present the rich nonlinear dynamics that arise from particle-fluid interactions for two different setups: (i) passive particles in 3D channel flows where fluid inertia is important, and (ii) active particles in 3D channel flows in the Stokes regime (i.e. without fluid inertia).

For setup (i), the particle-fluid interactions result in focusing of particles in the channel cross section, which has been exploited in biomedical microfluidic technologies to separate particles by size. I will offer insights on how dynamical system features of bifurcations and tipping phenomena might be exploited to efficiently separate particles of different sizes. For setup (ii), microswimmers routinely experience unidirectional flows in confined environment such as sperm cells swimming in fallopian tubes, pathogens moving through blood vessels, and microrobots programed for targeted drug delivery applications. I will show that our minimal model of the system exhibits rich nonlinear and chaotic dynamics resulting in a diverse set of active particle trajectories.

We investigate pattern formation for a 2D PDE-ODE bulk-cell model, where one or more bulk diffusing species are coupled to nonlinear intracellular
reactions that are confined within a disjoint collection of small compartments. The bulk species are coupled to the spatially segregated
intracellular reactions through Robin conditions across the cell boundaries. For this compartmental-reaction diffusion system, we show that
symmetry-breaking bifurcations leading to stable asymmetric steady-state patterns, as regulated by a membrane binding rate ratio, occur even when
two bulk species have equal bulk diffusivities. This result is in distinct contrast to the usual, and often biologically unrealistic, large
differential diffusivity ratio requirement for Turing pattern formation from a spatially uniform state. Secondly, for the case of one-bulk
diffusing species in R^2, we derive a new memory-dependent ODE integro-differential system that characterizes how intracellular
oscillations in the collection of cells are coupled through the PDE bulk-diffusion field. By using a fast numerical approach relying on the
``sum-of-exponentials'' method to derive a time-marching scheme for this nonlocal system, diffusion induced synchrony is examined for various
spatial arrangements of cells using the Kuramoto order parameter. This theoretical modeling framework, relevant when spatially localized nonlinear
oscillators are coupled through a PDE diffusion field, is distinct from the traditional Kuramoto paradigm for studying oscillator synchronization on
networks or graphs. (Joint work with Merlin Pelz, UBC and UMinnesota).