This document discusses gamma convergence and its applications. It begins with an introduction using Dr. Bruce Banner's transformation into the Hulk as an example of a natural origin story. It then provides examples of using gamma convergence to model the concentration of pollutants in water over time. Finally, it references a paper by Mariani that proposes using gamma convergence as an approach to study large deviations through notions of convergence, narrow convergence, and related inequalities.

1. Horacio G. Duhart
hgd20@bath.ac.uk
05/02/2015
G

2. Dr. Bruce Banner is bathed in the full force of
mysterious gamma rays!
As this is clearly a natural origin story for a super
hero, it is also obviously the best introduction to
the topic of Gamma convergence.

4. So, clearly, we can think of this as sequences of Hulk transformations and actors…
1 2 3
1 2 3
Convergence!
Convergence… Again!

5. G – Convergence may be phrased as the study of the asymptotic behaviour of a family of
minimum problems depending on a parameter.

6. G

7. Examples
Sequence Gamma - limit
with with

9. Concentration of pollutant in water:
PDE:
Velocity field Molecular diffusion coefficient
Advection Diffusion
Incompressible fluid
Initial condition:
Assume
Convergence in
some way
Concentration will only vary significantly on large length and time scales.
with
Effective diffusion tensor.
(Depends on b and D)

10. But…
or rather
more
specific…

11. That is…
The mythical
rate function

12. Basically we need to know what a rate function is.
We say it is a good rate function if these sets are not only closed, but also compact.
p=0.5
p=0.25
p=0.75

13. Yes, those two slides were totally the same shown on a previous talk… one year ago:
18/02/2014. But that’s not the point of this slide… more to the point:

14. The details are in the paper:
Mariani, M. A Gamma-Convergence approach
to large deviations
The idea goes as follows: “… a notion of convergence in a metric space W, containing both
probability measures and functionals.”
Large deviations
Narrow convergence
G - convergence
liminf
inequality
limsup
inequality
For all There exists

15. Mariani, M. A Gamma-Convergence approach to large deviations

16. You might be wondering when will you be able to see your favourite gamma rays
transformable super hero…

17. … remember that this 24 April 2015* the new Avengers: Age of Ultron film is coming and
is going to be awesome!
(Dear Marvel, if you’re reading this don’t forget to fund this research… please?)
*apparently.