Upcoming talks
Thursday, May 2
Speaker: Kjetil Lye (Sintef)
Title: Learning partial differential equations from observations
Abstract: In this talk we investigate the feasibility of learning partial differential equations (PDE) from data. We first focus on our newly developed framework for learning pseudo-Hamiltonian Neural Networks (PHNN) for PDEs from observations of the solution variable, and compare this approach to a baseline approach using convolutional neural networks for learning the spatial derivatives. In the PHNN approach, one is able to get a representation of the Hamiltonian and the external forces acting on the system. We furthermore look at the possibility of learning parts of a PDE, specifically source terms, with limited observation data.
Past talks
Thursday, April 18
Speaker: Kurusch Ebrahimi-Fard (NTNU)
Title: Remarks on the Magnus expansion
Abstract: In 1954, Wilhelm Magnus introduced an infinite Lie series to represent the solution of a first-order operator/matrix-valued homogeneous linear differential equation. This expansion, later named the Magnus expansion or series, has become a crucial tool utilized in various fields, including physics, chemistry, and engineering. Over the past 25 years, the Magnus expansion has undergone significant mathematical developments, uncovering intricate connections between algebra, combinatorics, and geometry. In this presentation, we will explore the Magnus expansion from the perspective of crossed morphism. If time permits, we will also discuss some recent applications in the context of Chen's signature extended to membranes (instead of paths).
Thursday, April 11
Speaker: David Cohen (Chalmers)
Title: Analysis of a positivity-preserving splitting scheme for some nonlinear
stochastic heat equations
Abstract: We construct and analyze a positivity-preserving Lie--Trotter splitting
scheme with finite difference discretization in space for approximating
the solutions to a class of nonlinear stochastic heat equations driven
by multiplicative noise.
The talk is based on joint works with Johan Ulander (Chalmers University
of Technology and University of Gothenburg) and Charles-Edouard Bréhier
(Université de Pau et des Pays de l’Adour)
Thursday, March 21
Speaker: Nicola De Nitti (EPFL)
Title: Sharp bounds on enstrophy growth for viscous scalar conservation laws
Abstract: We prove sharp bounds on the enstrophy growth in viscous scalar conservation laws. The upper bound is, up to a prefactor, the enstrophy created by the steepest viscous shock admissible by the $L^\infty$ and total variation bounds and viscosity. This answers a conjecture by D. Ayala and B. Protas (Physica D, 2011), based on numerical evidence, for the viscous Burgers equation. This talk is based on a joint work with D. Albritton [https://doi.org/10.1088/1361-6544/ad073f]
Thursday, March 14
Speaker: Peter Pang (UiO)
Title: Notions in the generic chaining
Abstract: The generic chaining is a method for bounding processes indexed by elements of metric spaces, and having exponentially decaying tails. In this talk I shall discuss an argument leading to matching upper and lower bounds for Gaussian processes and explain why these bounds are significant. This talk is based on work primarily by Michel Talagrand and Xavier Fernique, of which a reference is Talagrand's book "Upper and Lower Bounds for Stochastic Processes: Decomposition Theorems".
Thursday, February 29
Speaker: Håkon Hoel (UiO)
Title: High-order adaptive methods for exit times of diffusion processes
Abstract: We present a high-order adaptive method for strong approximations of exit times of Itô diffusion processes. The method employs a high-order Itô--Taylor scheme for simulating diffusion process paths and carefully decreases the step-size in the numerical integration as the process approaches the boundary. These techniques fit well together as high-order schemes improve the state approximation of the diffusion process at grid points, which is useful in itself and as improved feedback for controling the adaptive timestepping. We describe theoretical results on convergence rates and computational cost, and discuss how the method can be combined with multilevel Monte Carlo to produce a tractable method for estimating mean exit times.
Thursday, February 15
Speaker: Giuseppe Maria Coclite (Polytechnic University of Bari)
Title: Long-time behavior of scalar conservation laws.
Abstract: The aim of this presentation is to discuss the long-time behavior of scalar conservation laws and its viscous and non-local variants. Our focus will be on two primary examples: scalar convection- diffusion equations and a nonlocal regularization of the inviscid Burgers’ equation. In the first part of the talk, we will study the asymptotic behavior of the solutions to scalar convection- diffusion equations. When the initial datum is integrable, the mass of the solution is conserved along the evolution, leading to the long-time behavior being governed by the source-type solution of a limit equation. The form of this limit equation depends on the relative strength of convection and diffusion. In the second part, we will consider a nonlocal regularization of the Burgers' equation, where the velocity is approximated by a one-sided convolution with an exponential kernel. We will assume that the initial datum is positive, bounded, and integrable. The asymptotic profile is given by the source-type entropy solution of the Burgers' equation, commonly referred to as the "N-wave". The key components of our proofs involve suitable scaling arguments and Oleinik-type inequalities.
This talk is based on works in collaboration with N. De Nitti, A. Keimer, L. Pflug, and E. Zuazua.