Annual Report 2008.pdf - SAMSI

other. A core problem is how to infer the distribution of the projected (three-dimensional) size of
large inclusions from measurements made by either of the methods on sectional (twodimensional)
sizes. A simple new approximate solution to this stereological problem is proposed
and is compared to existing approaches. This area abounds with challenging problems. Some
examples are models for wave forms of loads, global modeling of extreme waves, how loads are
transferred though structures, and simplification of methods to make them useful in enginerring
praxis. Exciting work is being done by Anderson, Leadbetter, de Mare, Rychlik, and coworkers.
Pitting Corrosion: We discuss how Extreme Value Statistics can be used to to validate and
improve designed experiments with extremal responses, and to extrapolate and compare results.
A main motivation is corrosion tests: Localized, or "pitting", corrosion can limit the usefulness
of aluminum, magnesium and other new lightweight materials. It makes judicious choice of
alloys and surface treatments necessary. Standard methods to evaluate corrosion test are based on
weight loss due to corrosion and ANOVA. These methods fail in two ways. The first is that it
usually is not weight loss but the risk of perforation, i.e. the depth of the deepest pit which is of
interest. The second is that the standard ANOVA assumption of homogeneity of variances
typically is not satisfied by pit depth measurements, and that normality doesn't give credible
extrapolation into extreme tails.
The challenge is to develop a full theory of design and analysis of experiments with extreme
value distributed responses - at present we are far from this goal.
This is joint work with Clive Anderson, Anne-Laure Fougµeres, Sture Holm, Jacques de Mare,
John Nolan, Olivier Perrin, and Roger Taessler.
Gennady Samorodnitsky
Cornell University
gennady@orie.cornell.edu
“Inverse Problems for Regular Variation of Linear Filters”
We study a group of related problems: the extent to which the presence of regular variation in the
tail of certain $\sigma$-finite measures at the output of a linear filter determines the
corresponding regular variation of a measure at the input to the filter. This turns out to be related
to the presence of a particular cancellation property in $\sigma$-finite measures,which, in turn, is
related to the uniqueness of the solution of certain
functional equations. The techniques we develop are applied to weighted sums of iid random
variables, to products of independent random variables, and to stochastic integrals with respect to
L\'evy motions.
Elaine Spiller
SAMSI
spiller@math.duke.edu
“Constructing a Risk Map for Pyroclastic Flows: Using Simulations and Data to Predict Rare
Events”