Friday, May 10, 2019 3:00 pm
-
3:00 pm
EDT (GMT -04:00)
Jaspar Wiart, RICAM, Austrian Academy of Sciences
"An
Unexpected
Appearance
of
the
Shift"
Monte
Carlo
integration
is
a
method
of
estimating
integrals
particularly
well-suited
to
high
dimensions.
Simply
pick
a
number
of
random
points,
evaluate
the
function
at
those
points,
and
average
the
results.
Randomized
quasi-Monte
Carlo
seeks
to
improve
upon
this
method
by
choosing
the
points
in
a
clever
way.
We
pick
the
points
so
that
each
point
is
uniformly
distributed
but
with
a
dependence
structure
that
prevents
clustering.
Scrambled
(0,m,s)-nets
are
a
class
of
point
sets
with
very
nice
equidistribution
properties.
We
recently
found
that
these
point
sets
are
negatively
lower
orthant
dependent.
This
result
is
equivalent
to
showing
that
the
restriction
of
a
certain
continuous
linear
functional
on
l1
to
a
particular
subset
has
norm
one.
The
norm
calculation
involved
a
convexity
argument
and
a
convenient
but
unexpected
appearance
of
the
shift.
Knowing
that
these
nets
satisfy
this
dependence
condition
allowed
us
to
gain
a
deeper
understanding
of
when
these
point
sets
preform
better
than
Monte
Carlo.
With
the
help
of
plenty
of
pictures
I
will
introduce
scrambled
(0,m,s)-nets
and
negative
lower
orthant
dependence
as
well
as
explain
the
absurd
appearance
of
the
shift
in
this
project.
MC
5479