Upcoming talks
Tuesday, November 19
Speaker: Steinar Evje (UiS)
Title: An example of a Hybrid PDE-NN (Neural Network) model for identification of hyperbolic conservation laws
Abstract: In this talk I try to describe some recent investigations where we combine a hyperbolic conservation law with a multilayer symbolic neural network (S-Net) to identify an unknown flux function behind some observation data. Examples include hyperbolic conservation laws and convection-diffusion equation (possibly degenerate).
I hope to shed some light on the possibilities and limitations of the approach.
This work was motivated by recent investigations where we rely on a Bayesian approach to identify large parameter vectors representing unknown aspects of a more complex system of conservation laws involved in a tumor progression model. Hence, I first describe how this Bayesian approach can be applied on a scalar hyperbolic conservation law which serves as a direct motivation for the S-Net based method.
Past talks
Tuesday, November 5
Speaker: Michael Floater (UiO)
Title: On the theory of spline interpolation of values and derivatives
Abstract: There is a remarkable and important result in univariate spline theory:
spline interpolation over a sequence of points has a unique solution if and only if the i-th point is contained in the support of the i-th B-spline. This was essentially shown by Schoenberg and Whitney in 1953 and later generalized to the (equally important) case of Hermite interpolation: the interpolation of both values and derivatives by Karlin and Ziegler in 1966. Since then various approaches have been made to simplify the proof,
by Schumaker and de Boor, among others, using techniques such as variation- diminishing, B-spline recursion and knot-insertion. I will try to give an overview of these approaches and try to land on a `simplest' proof.
Tuesday, October 22
Speaker: Ragnar Winther (UiO)
Title: The bubble transform: A new tool for analysis of finite element methods.
Abstract: In recent years Rick Falk and I have developed a theory which represents a new tool to analyze finite element methods of high polynomial degree, which we refer to as the bubble transform. For a fixed simplicial mesh we construct a decomposition of the corresponding finite element spaces into local bubbles which simultaneously covers all possible polynomial degrees. Questions on what happens with the properties of finite element methods as the polynomial degree is raised unboundedly have been asked from the late 1980s and on. But still today the so called p-method is less understood, and the corresponding analysis is less canonical, than the traditional approach of mesh refinement, i.e., the h-method. The purpose of this talk is to give a review of our theory, and to discuss potential applications.
Tuesday, October 29
Speaker: Snorre Christiansen (UiO)
Snorre will continue his presentation from October 15 about "Mixed finite elements from the point of view of category theory".
Tuesday, October 15
Speaker: Snorre Christiansen (UiO)
Title: Mixed finite elements from the point of view of category theory
Abstract: I will first give an introduction to mixed finite elements by describing the main example (developed around 1980 by Raviart, Thomas and Nédélec but closely related to constructions in differential topology by de Rham, Weil and Whitney in the 1940ies). Then I will relate how many fundamental constructions in finite element theory can be described in terms of category theory. I will not assume prior knowledge of the latter. I will argue that the concept of a sheaf provides a unified framework for studying both continuous and discretized PDEs. Sheaf theory was first introduced by Leray in the 1940ies and later clarified in the Cartan seminar in the 1950ies with in particular algebraic geometry in mind. Applied sheaf theory is now a developing subject in many fields.
Tuesday, October 8
Speaker: Erik Hiltunen (UiO)
Title: Analytical and numerical methods for spacetime metamaterials
Abstract: The ability to focus, trap, and guide the propagation of waves is of fundamental importance in physics. Specifically, the field of metamaterials aims to develop materials with, in some sense, exotic wave scattering properties. Using a carefully designed, periodic, microstructure, it is possible to design materials with effective properties beyond classical physics, such as vanishing refractive index, or to localize waves at specified positions. In this talk, we will review some of the key physical problems for metamaterials and the mathematical techniques for studying them. We will then consider the resonance and scattering properties of a composite medium of scatterers whose material properties are modulated in time. Spacetime metamaterials have seen a surge of recent interest in physics and engineering communities. Although the corresponding equations are linear, these systems may exhibit typically nonlinear phenomena such as harmonic generation, energy focusing and soliton-like (spacetime-localized) solutions.
Tuesday, October 1
Speaker: Achenef Tesfahun (Nazarbayev University, Kazakhstan)
Title: Dispersive estimate for full dispersion KP equation
Abstract: We derive some dispersive estimates for the linear part of the full dispersion Kadomtsev-Petviashvili (KP) equation introduced by David Lannes to overcome some shortcomings of the classical KP equations. The proof of these estimates combines the stationary phase method with sharp asymptotics on asymmetric Bessel functions. These estimates are used to prove local well-posedness of the Cauchy problem at a low regular Sobolev space.
This is a joint work with D. Pilod, J-C. Saut and S. Selberg.
Tuesday, August 27
Speaker: Christian Bayer (Weierstrass Institute, Berlin)
Title: A Reproducing Kernel Hilbert Space approach to singular local stochastic volatility McKean-Vlasov models
Abstract: Perfect calibration of stochastic local volatility models can be achieved by the particle method due to Guyon and Henry-Labordère. Starting from a back-bone stochastic volatility model, a local volatility factor is computed on the fly to perfectly fit market prices. Mathematically, the local volatility factor is given as a conditional expectation, which is approximated by a local regression procedure. While this procedure is quite popular among practitioners, there are substantial gaps in the theoretical understanding. Indeed, even well-posedness of the resulting singular McKean-Vlasov system is not known.
We develop a novel regularization approach based on the reproducing kernel Hilbert space technique (kernel ridge regression) and show that the regularized model is, in fact, well-posed. Furthermore, we prove propagation of chaos and provide error estimates for the numerical scheme. We demonstrate numerically that a thus regularized model is able to perfectly replicate option prices due to typical local volatility models, and demonstrate excellent performance. Our results are also applicable to more general McKean--Vlasov equations. (Joint work with Denis Belomestny, Oleg Butkovsky, and John Schoenmakers.)