Christmas Workshop in Mathematical Modelling and Analysis

The workshop in Mathematical Modeling and Analysis is aimed at researchers, employees, and students.

Program:

13.15-14.00 Niklas Lundström, Umeå university

Millions of Growth Estimates for Subsolutions of Nonlinear PDEs

Abstract: In L22 sharp growth estimates for subsolutions in halfspaces of fully nonlinear PDEs on the form F(x,u,Du,D^2u) = 0 were proven through a characterization built upon solutions of certain ODEs. 

In this talk, we show how to generalize the main theorem of L22 to hold under weaker assumptions and derive several sharp Phragmen-Lindelöf-type theorems (growth estimates for subsolutions) as corollaries.

14.15-15.00 Suprokash Hazra, Mid Sweden University

Truncated Tube Domains with Multi-Sheeted Envelope of Holomorphy

Abstract: We define the envelope of holomorphy of a general Riemann domain over C^n and discuss the schlichtness of it with some examples. Next, we address a group of problems raised by J. Noguchi and M. Jarnicki/P. Pflug, namely whether the envelopes of holomorphy of truncated tube domains are always schlicht, that is subdomains of \C^n. By providing a counter-example diffeomorphic to 4-ball we then discuss an answer to the above problem jointly obtained by Egmont Porten and the author. Finally, we discuss a sufficient condition for schlichtness in complex dimension two.

15.30-16.15 Jonatan Vallin, Umeå university

Error Estimates for Implicit Approximation of Hypersurfaces using Deep ReLU Nets

Abstract: We develop a geometric approximation theory applicable in the classification setting for deep feed-forward neural networks with ReLU activations. Given a hypersurface in Rd represented as a level-set of a C2-function φ, we show that a deep real-valued fully-connected ReLU network of width, d+1 can implicitly construct an approximation as its zero contour - the decision boundary. Our proof is constructive and relies on a geometrical analysis of ReLU layers.

16.30-17.15 Kevin Kamm, Umeå university

On the Impact of Feeding and Biological Cost in Aquaculture Valuation

Abstract: We study the effect of stochastic feeding and biological costs on animal-based commodities with particular focus on aquaculture. More specifically, we use soybean futures to infer on the stochastic behaviour of salmon feed, which we assume to follow a Schwartz-2-factor model and use a host-parasite model to estimate mortality induced by salmon lice. We compare the decision of harvesting salmon using a decision rule assuming either deterministic or stochastic feeding costs/ mortality. We show that the harvesting decision based on this new and more realistic model can lead to a significant increase of revenue, while the additional computational costs are negligible.