Thu, 05 May 2022

14:00 - 15:00
L3

### Finite elements for metrics and curvature

Snorre Christiansen
(University of Oslo)
Abstract
In space dimension 2 we present a finite element complex for the deformation operator acting on vectorfields and the linearized curvature operator acting on symmetric 2 by 2 matrices. We also present the tools that were used in the construction, namely the BGG diagram chase and the framework of finite element systems. For this general framework we can prove a de Rham theorem on cohomology groups in the flat case and a Bianchi identity in the case with curvature.
Thu, 20 May 2021
14:00
Virtual

### The bubble transform and the de Rham complex

Ragnar Winther
(University of Oslo)
Abstract

The bubble transform was a concept introduced by Richard Falk and me in a paper published in The Foundations of Computational Mathematics in 2016. From a simplicial mesh of a bounded domain in $R^n$ we constructed a map which decomposes scalar valued functions into a sum of local bubbles supported on appropriate macroelements.The construction is done without reference to any finite element space, but has the property that the standard continuous piecewise polynomial spaces are invariant. Furthermore, the transform is bounded in $L^2$ and $H^1$, and as a consequence we obtained a new tool for the understanding of finite element spaces of arbitrary polynomial order. The purpose of this talk is to review the previous results, and to discuss how to generalize the construction to differential forms such that the corresponding properties hold. In particular, the generalized transform will be defined such that it commutes with the exterior derivative.

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.

Mon, 25 May 2020

16:00 - 17:00

### Infinitely regularizing paths, and regularization by noise.

Fabian Harang
(University of Oslo)
Abstract

Abstract:

In this talk I will discuss regularization by noise from a pathwise perspective using non-linear Young integration, and discuss the relations with occupation measures and local times. This methodology of pathwise regularization by noise was originally proposed by Gubinelli and Catellier (2016), who use the concept of averaging operators and non-linear Young integration to give meaning to certain ill posed SDEs.

In a recent work together with   Nicolas Perkowski we show that there exists a class of paths with exceptional regularizing effects on ODEs, using the framework of Gubinelli and Catellier. In particular we prove existence and uniqueness of ODEs perturbed by such a path, even when the drift is given as a Scwartz distribution. Moreover, the flow associated to such ODEs are proven to be infinitely differentiable. Our analysis can be seen as purely pathwise, and is only depending on the existence of a sufficiently regular occupation measure associated to the path added to the ODE.

As an example, we show that a certain type of Gaussian processes has infinitely differentiable local times, whose paths then can be used to obtain the infinitely regularizing effect on ODEs. This gives insight into the powerful effect that noise may have on certain equations. I will also discuss an ongoing extension of these results towards regularization of certain PDE/SPDEs by noise.​

Mon, 29 Oct 2018

14:15 - 15:15
L3

### Extensions of the sewing lemma to Multi-parameter Holder fields

FABIAN ANDSEM HARANG
(University of Oslo)
Abstract

In this seminar we will look at an extension of the well known sewing lemma from rough path theory to fields on [0; 1]k. We will first introduce a framework suitable to study such fields, and then find a criterion for convergence of multiple Riemann type sums of a class of abstract integrands. A simple application of this extension is construct the Young integral for fields.Furthermore, we will discuss the use of this theorem to study integration of fields of lower regularity by using ideas familiar from rough path theory. Moreover, we will discuss difficulties we face by looking at “multi-parameter ODE's” both from an existence and uniqueness point of view.

Thu, 22 Nov 2018

14:00 - 15:00
L4

### Some new finding for preconditioning of elliptic problems

Prof Kent-Andre Mardal
(University of Oslo)
Abstract

In this talk I will present two recent findings concerning the preconditioning of elliptic problems. The first result concerns preconditioning of elliptic problems with variable coefficient K by an inverse Laplacian. Here we show that there is a close relationship between the eigenvalues of the preconditioned system and K.
The second results concern the problem on mixed form where K approaches zero. Here, we show a uniform inf-sup condition and corresponding robust preconditioning.

Thu, 19 May 2016
12:00
L6

### Stochastic Conservation Laws

Kenneth Karlsen
(University of Oslo)
Abstract
Stochastic partial differential equations arise in many fields, such as biology, physics, engineering, and economics, in which random phenomena play a crucial role. Recently many researchers have been interested in studying the effect of stochastic perturbations on hyperbolic conservation laws and other related nonlinear PDEs possessing shock wave solutions, with particular emphasis on existence and uniqueness questions (well-posedness). In this talk I will attempt to review parts of this activity.
Thu, 09 Oct 2008

14:00 - 15:00
Comlab

### Barycentric coordinates and transfinite interpolation

Prof Michael Floater
(University of Oslo)
Abstract

Recent generalizations of barycentric coordinates to polygons and polyhedra, such as Wachspress and mean value coordinates, have been used to construct smooth mappings that are easier to compute than harmonic amd conformal mappings, and have been applied to curve and surface modelling.

We will summarize some of these developments and then discuss how these coordinates naturally lead to smooth transfinite interpolants over curved domains, and how one can also match derivative data on the domain boundary.

Tue, 05 Feb 2008
13:15
Oxford-Man Institute

### "A mathematical equilibrium model for insider trading in finance"

Professor Bernt Oksendal
(University of Oslo)
Abstract
A trader in finance is called an insider if she (or he) knows more about the prices in the market than can be obtained from the market history itself. This is the case if, for example, the trader knows something about the future price/value of a stock. We discuss the following question: What is the optimal portfolio of an insider who wants to maximize her expected profit at a given future time? The problem is that heavy trading by the insider will reveal parts of her inside price information to the market and thereby reduce her information advantage.

We will solve this problem by presenting a general anticipative stochastic calculus model for insider trading. Our results generalize equilibrium results due to Kyle (1985) and Back (1992).

The presentation is partly based on recent joint work with Knut Aase and Terje Bjuland, both at the Norwegian School of Economics and Business Administration (NHH).

Fri, 09 Mar 2007
14:15
DH 2nd floor SR

### Electricity spot price modelling with non-Gaussian Ornstein-Uhlenbeck processes.

Dr Thilo Meyer-Brandis
(University of Oslo)
Thu, 10 Mar 2005
14:00
Comlab

### Backward error analysis, a new view and further improvements

Dr Per Christian Moan
(University of Oslo)
Abstract

When studying invariant quantities and stability of discretization schemes for time-dependent differential equations(ODEs), Backward error analysis (BEA) has proven itself an invaluable tool. Although the established results give very accurate estimates, the known results are generally given for "worst case" scenarios. By taking into account the structure of the differential equations themselves further improvements on the estimates can be established, and sharper estimates on invariant quantities and stability can be established. In the talk I will give an overview of BEA, and its applications as it stands emphasizing the shortcoming in the estimates. An alternative strategy is then proposed overcoming these shortcomings, resulting in a tool which when used in connection with results from dynamical systems theory gives a very good insight into the dynamics of discretized differential equations.

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