Abstract : Many interesting evolutionary problems in nature can be described by variational principles like gradient flows or Hamiltonian dynamics. Recent results have shown that exploiting the variational structure of the evolution equation provides a fruitful research area combining applied analysis and stochastic modeling. For real-world multi-scale problems, focusing on the variational structure is particularly vital as it forms a physically motivated basis.
The aim of this two-part minisymposium is inspiring and bringing together researchers interested in calculus of variations, PDEs and stochastic analysis for starting collaborations. Focus is placed on interacting particle systems and discrete-to-continuous limit passages, e.g. by evolutionary Gamma-convergence.
Organizer(s) : Yuan Gao, Matthias Liero, Artur Stephan
Abstract : In this talk, we study a class of variational problems for regularized conservation laws with Lax's entropy-entropy flux pairs. We first introduce a modified optimal transport space based on conservation laws with diffusion. Using this space, we demonstrate that conservation laws with diffusion are "flux--gradient flows." We next construct variational problems for these flows, for which we derive dual PDE systems for regularized conservation laws. Several examples, including traffic flow and Burgers' equation, are presented. We successfully compute the control of conservation laws by incorporating both primal-dual algorithms and monotone schemes. This is based on joint work with Siting Liu and Stanley Osher.
[03821] Transport problems with non linear mobilities: a particle approximation result.
Format : Talk at Waseda University
Author(s) :
Lorenzo Portinale (Hausdorff Center for Mathematics, Bonn)
Simone Di Marino ( Università di Genova )
Emanuela Radici ( Università degli Studi dell’Aquila)
Abstract : We study discretisation of generalised Wasserstein distances with non linear mobilities on the real line via a Riemannian metrics on the space of N ordered particles. In particular, we provide a Γ-convergence result for the associated discrete metrics as N → ∞ to the continuous one and discuss applications to the approximation of one-dimensional conservation laws (of gradient flow type) via the so-called generalised minimising movements (or JKO scheme).
[04444] Variational convergence for irreversible population dynamics
Format : Talk at Waseda University
Author(s) :
Jasper Hoeksema (Eindhoven University of Technology)
Abstract : We consider the forward Kolmogorov equations corresponding to measure-valued processes stemming from a class of interacting particle systems in population dynamics. In contrast to previous work, where we assumed detailed balance, we will now treat the irreversible case. We exchange gradient structures for more general dissipation structures, and show convergence of these structures in the large population limit. In particular we obtain convergence to the mean-field limit and establish entropic propagation of chaos.
[03883] Mathematical modeling of structured magnesium alloys
Format : Talk at Waseda University
Author(s) :
Karel Svadlenka (Kyoto University)
Abstract : Structured materials, such as metallic alloys with atomic-scale layers, show peculiar deformation patterns, which may have significant implications on material properties. In this talk, I will discuss one possible approach to modeling of this kind of pattern formation through the so-called rate-independent evolution in the variational setting of finite-strain elasto-plasticity. Besides mentioning connections to homogenization via Gamma-convergence, I will present the underlying mathematical theory and show numerical simulations in comparison to experimental measurements.
[04828] Variational numerical schemes for gradient flows
Format : Online Talk on Zoom
Author(s) :
Yiwei Wang (University of California, Riverside)
Chun Liu (Illinois Institute of Technology)
Abstract : We'll present a numerical framework for developing structure-preserving variational schemes for various types of gradient flows. The numerical approach starts with the energy-dissipation law of the underlying system and can combine different spatial discretizations, including Eulerian, Lagrangian, particle, and neural-network-based approaches. The numerical procedure guarantees the developed schemes are energy stable and can preserve the intrinsic physical constraints. Several applications and theoretical justifications will be discussed.
[04971] Quantitative coarse-graining of Markov chains
Format : Online Talk on Zoom
Author(s) :
Upanshu Sharma (UNSW Sydney)
Bastian Hilder (Lund University)
Abstract : Coarse-graining is the procedure of approximating large and complex systems by simpler and lower-dimensional ones. It is typically characterised by a mapping which projects the full state of the system onto a smaller set; this mapping captures the relevant (often slow) features of the system. Starting from a (non-reversible) continuous-time Markov chain and such a mapping, I will discuss an effective dynamics which approximates the true projected Markov chain and present error estimates on the approximation error.
[04391] Variational convergence from mean-field stochastic particle systems to the exchange-driven growth model
Format : Online Talk on Zoom
Author(s) :
Chun Yin Lam (Universität Münster)
André Schlichting (Universität Münster)
Abstract : We consider the hydrodynamic limit of mean-field stochastic particle systems on a complete graph using variational methods.
The evolution is driven by particle exchanges with its rate depending on the population of the initial and final vertices. This model is a generalisation of the zero-range process and has applications in cloud formation, polymerization, and wealth exchange.
Under detailed balance conditions, the evolution equation has a gradient structure motivated by the Large Deviations Principle. The variational formulation is based on the LDP rate function.
[05130] On time-splitting methods for gradient flows with two dissipation mechanisms
Format : Talk at Waseda University
Author(s) :
Artur Stephan (Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany)
Abstract : A gradient system consists of a state space $X$, an energy functional $E:X\to\mathbb{R}\cup\{\infty\}$ and a dissipation potential $R:X\to[0,\infty[$ and defines a gradient-flow equation.
Considering the case where the dual dissipation potential $R^*$ is given by the sum $R^*=R_1^*+R_2^*$, we show how convergence of a time-splitting method where the solution of the combined gradient system is approximated by concatenating the separate gradient-flows.
This is joint work with Alexander Mielke (Berlin) and Riccarda Rossi (Brescia).
