We present a numerical method to simulate thick elastic curves that accounts for self-contact and container constraints under large deformations (the motivating model is DNA packing). The base model includes bending, torsion, and inextensibility. A minimizing movements, descent scheme is proposed for computing energy minimizers, under the non-convex inextensibility, self-contact, and container constraints (if the container is non-convex). At each pseudo time-step of the scheme, the constraints are linearized, which yields a convex minimization problem (at every time-step) with affine equality and inequality constraints. First order conditions are established for the descent scheme at each time-step, under reasonable assumptions on the admissible set. Furthermore, under a mild time-step restriction, we prove energy decrease for the descent scheme, and show that all constraints are satisfied to second order in the time-step, regardless of the total number of time-steps taken.

We also give a modification of the scheme that regularizes the inequality constraints, and establish convergence of the regularized solution. We then discretize the regularized problem with a finite element method using Hermite and Lagrange elements. Several numerical experiments are shown to illustrate the method, including an example that exhibits massive amounts of self-contact for a tightly packed curve inside a sphere.  We also demonstrate the effect of parameter choices on packing configurations.

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In several areas of mathematics, including probability theory, statistics and asymptotic convex geometry, one is interested in high-dimensional objects, such as measures, data or convex bodies.  One common theme is to try to understand what lower-dimensional projections can say about the corresponding high-dimensional objects.   I will describe how this line of inquiry leads to geometric generalizations of some classical results in probability related to tails of random projections, both in commutative and non-commutative settings, and also discuss their relation to some long standing open problems in convex geometry.

Harnessing the demonstrated effectiveness of Bayesian Additive Regression Tree (BART) principles, we develop a fully Bayesian procedure to train an ensemble of small neural networks for regression tasks.  We describe how BART samples from a Bayesian posterior of an ensemble of decision trees, and then adapt this method to neural networks.  Using Markov Chain Monte Carlo, Bayesian Additive Regression Networks (BARN) samples from the space of single hidden layer neural networks that are conditioned on their fit to data.  To create an ensemble of networks, we apply Gibbs' sampling to update each network against the residual target value (i.e. subtracting the effect of the other networks).  We examine the test performance of BARN on several benchmark regression tasks, comparing it to equivalent neuron count single neural networks as well as equivalent tree count BART.  We also use BARN to model an applied problem and compare against the state of the art modeling methods for that specific domain.  BARN provides more consistent and often more accurate results, with a mean root mean square error just 5% higher than the best (or next best) method across 9 datasets with different ``best methods'' in each.  This comes at the cost of significantly greater computation time (minutes vs seconds).  But this may be surmountable with more clever programming, and errors may further shrink with a hyperparameter grid search.

A traditional model (1910) is the Gouy-Chapman model, which predicts the distribution of ions and the shape of a poential in a solution, assumed to be semi-infinite. This model, refined by Stern in 1924, takes into account the Debye length. We propose a variational formulation of the Poisson-Boltzmann system, both for electroneutral and non electroneutral cells. We prove the existence and uniqueness of a solution in the 1d setup (radial solutions in 2d and 3d are included), give ideas for the existence and uniqueness of a solution in higher dimension, and obtain an analytic solution in the 1d set-up for non electroneutral cells. This analytic solution relies on incomplete Jacobi elliptic functions, and the unique solution is obtained by solving a system of equations in $\R_+^2$ (two equations with two unknowns)

Joint work with Clair Poignar (Inria Bordeaux Sud Ouest)

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Please join us for an informal discussion with Michell Avent, Applied Information Sciences, Sandia National Laboratories for a discussion of what it’s like to work at a national laboratory, student internships, and career opportunities for current students as well as new graduates. 

Place:  ENR2, S395 

In this talk, I will present a model to describe cell migration. Cell migration plays a key role in many physiological processes, such as embryogenesis, wound repair or metastasis formation. It is the result of a complex activity that involves different time and space scales. I will first detail the construction of the model and then present rigorous results and numerical simulations.

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Abstract: TBA

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https://math.mit.edu/~dunkel/

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The stochastic epidemic is a classic example of probabilistic modelling dating back to work of Daniel Bernoulli in 1760, if not earlier. The problem becomes much harder when the model is "spatial" and the disease results in future immunity/death.

Inspired in part by a personal encounter with Covid-19, we consider two models for the spread of infection about a population of diffusing individuals in Euclidean space. The emphasis is upon the case of post-infection immunity. Partial results for the existence (or not) of a pandemic may sometimes be proved via comparisons with branching processes and percolation processes. The principal difficulties lie in the combination of movement and immunity.

The presentation will include summaries of the percolation and contact models, and an outline of their uses in epidemic theory. (Joint work with Zhongyang Li.)

Recently, there has been a growing interest in approximating nonlinear functions and PDEs on tensor manifolds. The reason is simple: tensors can drastically reduce the computational cost of high-dimensional problems when the solution has a low-rank structure. In this talk, I will review recent developments on rank-adaptive algorithms for temporal integration of PDEs on tensor manifolds. Such algorithms combine functional tensor train (FTT) series expansions, operator splitting time integration, and an appropriate criterion to adaptively add or remove tensor modes from the FTT representation of the PDE solution as time integration proceeds. I will also present a new tensor rank reduction method that leverages coordinate flows. The idea is very simple: given a multivariate function, determine a coordinate transformation so that the function in the new coordinate system has a smaller tensor rank. I will restrict the analysis to linear coordinate transformations, which give rise to a new class of functions that we refer to as tensor ridge functions. Numerical applications are presented and discussed for linear and nonlinear advection equations, and for the Fokker-Planck equation.

SHORT BIO: Dr. Venturi is a professor of Applied Mathematics in the Baskin School of Engineering at UC Santa Cruz. He received his MS in Mechanical Engineering in 2002, and his PhD in Applied Physics in 2006 from the University of Bologna (Italy). His research activity has been recently focused on the numerical approximation of PDEs on tensor manifolds, including high-dimensional PDEs arising from the discretization of functional differential equations (infinite-dimensional PDEs).

WEBPAGE:  https://venturi.soe.ucsc.edu

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We study reinforcement learning (RL) in a setting with a network of agents whose states and actions interact in a local manner where the objective is to find policies such that the (discounted) global reward is maximized. A fundamental challenge in this setting is that the state-action space size scales exponentially in the number of agents, rendering the problem intractable for large networks. In this paper, we present our framework that exploits the network structure to conduct reinforcement learning in a scalable manner. The key feature in our framework is that we prove spatial decay properties for the Q function and the policy, meaning their dependence on faraway agents decays when the distance increases. Such spatial decay properties enable approximations by truncating the Q functions and policies to local neighborhoods, hence drastically reducing the dimension and avoiding the exponential blow-up in the number of agents.

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https://engineering.cmu.edu/directory/bios/qu-guannan.html

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To design and control magnetic confinement fusion reactors, one must compute the geometry of the confined plasma at equilibrium. In this talk, I present a new method for solving this free-boundary problem through integral equations and PDE-constrained optimization. I detail a formulation of the equilibrium equations that couples the Grad-Shafranov equation for the interior magnetic field with an integral equation for the exterior vacuum field. I introduce high order numerical methods for solving the associated integral equation, including a novel application of the Kapur-Rokhlin quadrature rule for singular integrals in axisymmetric magnetic confinement systems. Finally, I frame the coupled equations in the larger PDE-constrained optimization framework and discuss gradient descent methods for the optimization iteration.

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The discovery in 1998 of a link between the Wasserstein-2 metric, entropy, and the heat equation, by Jordan, Kinderlehrer, and Otto, precipitated the increasing relevance of optimal mass transport in the evolving theory of finite-time thermodynamics, aka stochastic energetics. Specifically, dissipation in finite-time thermodynamic transitions for Langevin models of colloidal particles can be measured in terms of the Wasserstein length of trajectories. This enabling new insight has led to quantifying power and efficiency of thermodynamic cycles that supersede classical quasi-static Carnot engine concepts that alternate their contact between heat baths of different temperatures. Indeed, naturally occurring processes often harvest energy from temperature or chemical gradients, where the enabling mechanism responsible for transduction of energy relies on non-equilibrium steady states and finite-time cycling. Optimal mass transport provides the geometric structure on the manifold of thermodynamic states for studying energy harvesting mechanisms. In this, dissipation and work output can be expressed as path and area integrals, and fundamental limitations on power and eficiency in geometric terms lead to isoperimetric problems. The analysis presented provides guiding principles for building autonomous engines that extract work from thermal or chemical anisotropy in the environment. Based on joint work with Olga Movilla Miangolarra, Amirhossein Taghvaei, Rui Fu, and Yongxin Chen

Bio:  Tryphon T. Georgiou was educated at the National Technical University of Athens, Greece (Diploma 1979) and the University of Florida, Gainesville (PhD. 1983).  He is a Distinguished Professor at the Department of Mechanical and Aerospace Engineering, University of California, Irvine, and Professor Emeritus at the University of Minnesota. He is a Fellow of IEEE, SIAM, IFAC, AAAS and a Foreign Member of the Royal Swedish Academy of Engineering Sciences (IVA).

https://georgiou.eng.uci.edu/    

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