I have an open position for a Scientific Assistant/PhD student available. The salary is according to TV-L EG 13. (Don’t know what that means? Have a look here.). The position starts at 01.09.2013 (earlier start is possible) and is initially limited to two years; further extension is possible and for a PhD student, at least three years are planned.
Candidates should
- have a degree (Masters or Diploma) in mathematics above average,
- have very good knowledge in numerical mathematics and functional analysis,
- have good knowledge in scientific computing or optimization,
- know German well and
- have strong interest in applied mathematics. Also, bring a high commitment for scientific research.
The responsibilities include
- participation in teaching and
- independent but supervised research in field of applied mathematics (especially mathematical imaging and inverse problems).
Please send applications including CV, copies of certificates and letters of recommendation (if any) in electronic form directly to me. Deadline is the 30.06.2013.
If you would like to post the job advertisement at you bulletin board, here’s the pdf file.
. Some effort goes into the analysis of the models and the first step is usually to establish existence of solutions, i.e. minimizers of the respective minimization problems. The work horse is the 
, 
defined on a domain 
one seeks a flow field 
such that 
the describes the apparent motion that has happened between both images. Assuming that the points keep their gray value during motion (an assumption known as the brightness constancy constraint) and linearizing this assumption one arrives at the condition
is the time between the images 
are problematic.
weighting smoothness of 
and 
and minimize
one ensures 
. Then properness of 
, of course assuming that 
). Convexity is also clear and for lower semi-continuity one has to work a little more, but that is possible if, e.g., 
is bounded. Coercivity is not that clear and in fact 
for some 
. Then ![{\nabla u(x,y) \equiv [a\ b]^T}](http://s0.wp.com/latex.php?latex=%7B%5Cnabla+u%28x%2Cy%29+%5Cequiv+%5Ba%5C+b%5D%5ET%7D&bg=ffffff&fg=000000&s=0 "{\nabla u(x,y) \equiv [a\ b]^T}")
. Set ![{V_n(x,y) \equiv [-nb\ na]^T}](http://s0.wp.com/latex.php?latex=%7BV_n%28x%2Cy%29+%5Cequiv+%5B-nb%5C+na%5D%5ET%7D&bg=ffffff&fg=000000&s=0 "{V_n(x,y) \equiv [-nb\ na]^T}")
and note that 
while 
stays bounded (in fact constant). 
:
with 
such that 
and 
are linearly independent in 
and let 
. Then it holds that 
defined by
such that 
. Now we decompose the components of 
and 
and the “zero-mean”-part 
and 
. First consider that 
is unbounded, i.e. there is subsequence (also denoted by 
. By 
.
. Using a subsequence, we assume that 
. Now we use 
and 
are constants, by
and we conclude that 
implies that 
. Together with~(
we obtain that 
. Since for every subsequence of 
such that 
optical flow, i.e. coercivity of
and if the result on coercivity is “sharp” in the sense that linear independence of 
.
and 
defined on the same set 
, namely the 
and 
of a metric space 
and finally considered the Gromov-Hausdorff metric between two metric spaces 
and 
. Our tools have been 
, i.e. subsets 
such that for all 
there 
such that 
and for all 
and 
, i.e. metrics 
on the disjoint union of 
if 
and 
are in 
if 
and 
are in 
and 
, i.e. measures 
on 
such that 
and 
for all 
consisting a compact metric space 
? The plan is simply to augment the previous versions of the Gromov-Hausdorff distances defined in my 
of 
for any set coupling 
. Note that the obvious modification for 
leads to something very similar to the Gromov-Hausdorff metrics.
there are the measure couplings 
since this can be taken into account by the measure couplings; they do not need to have full support anyway.)
and gives an example (right after Remark 5.14) where strict inequality holds.
(with the euclidean metric) accompanied with the measures 
and 
for some points 
and 
. In this case the is only one measure coupling of 
in 
in 
such that these points are only 
-far away from each other). Hence, we see that we have
.
there is a metric coupling such that 
which shows that
we have
is either one or zero and it is one if and only if the point 
is in the set that is measured. However, we have that 
is, of course, always zero (and never larger than any 
). Hence, the measure is always zero and hence, this version of the Gromov-Prokhorov distance also gives
were easier, since one needs to know all measure couplings and no metric couplings has been involved.
and want to calculate its distance to a single point, i.e. the mm-space 
with the only possible metric and measure. This should somehow measure the “size” or “spread” of the mm-space 
are treated as if they were subsets of 
: Since 
(the element in 
defined by
although, formally 
.)
) is called the “
and 
defined of the same probability space 
with values in a metric space with metric 
)
![{I = [0,1]}](http://s0.wp.com/latex.php?latex=%7BI+%3D+%5B0%2C1%5D%7D&bg=ffffff&fg=000000&s=0 "{I = [0,1]}")
and two Radon measures 
(
there is the 
of 
denotes the 
define 
such that for measurable 
and 
located at distinct points 
in 
and 
small enough that 
, 
but 
and 
). The calculate the Prokhorov metric note that you only need to consider 
and hence, it evaluates to
and 
, namely the measure 
. Hence, we have
such that for all 
there is 
such that 
and for all 
. One may also say set coupling of 
. Three different set couplings of two subsets 
in green and two “slim” ones in red and orange. Other “slim” couplings can be obtained from surjective mappings 
by 
(or with the roles of 
of all non-empty compact subsets of a metric space 
is a compact metric space (a result, known as 
and
and 
which embed 
is the infimum over all 
such that there exists a metric space 
.
of 
and 
it holds that 
an identical copy of 
on 
that respects the metrics 
in the way a metric coupling has to.
we put a 
on all quantities from 
, 
(i.e. the distance between any two identical twins is 
that 
should fulfill
if 
leading to one specific metric coupling for any 
we do not obtain a metric but a semi-metric or pseudo-metric which is just the same as a metric but without the assumption that 
implies that 
.
and there is also a way to visualize metric couplings: When building the disjoint union of two metric spaces ![{X=Y=[0,1]}](http://s0.wp.com/latex.php?latex=%7BX%3DY%3D%5B0%2C1%5D%7D&bg=ffffff&fg=000000&s=0 "{X=Y=[0,1]}")
you can embed both into 
. A metric coupling which is similar (but not equal) to the coupling of the previous example is obtained by putting 
we would recover definition~(
from Example~
of metric couplings from Example~
for any 
. Let’s take one of the next-complicated examples and compute the distance of ![{X = [0,1]}](http://s0.wp.com/latex.php?latex=%7BX+%3D+%5B0%2C1%5D%7D&bg=ffffff&fg=000000&s=0 "{X = [0,1]}")
and ![{Y=[0,2]}](http://s0.wp.com/latex.php?latex=%7BY%3D%5B0%2C2%5D%7D&bg=ffffff&fg=000000&s=0 "{Y=[0,2]}")
, both equipped with the euclidean metric. We couple the sets 
and the respective metrics by embedding 
as follows: Put 
to 
and 
to 
:
and actually, we have equality here.
and 
have the “most equal” distance.
.
of a real vector space 
and ![{\lambda\in[0,1]}](http://s0.wp.com/latex.php?latex=%7B%5Clambda%5Cin%5B0%2C1%5D%7D&bg=ffffff&fg=000000&s=0 "{\lambda\in[0,1]}")
it holds that 
. 
for some 
in the dual space 
and 
). 
is convex if there is a convex function 
such that 
. 
from a real vector space into the real numbers is convex, if its epigraph 
is convex (as a subset of the vector space 
). 
which lie above the graph of the function and carries the same information as the function.
if one uses the right extension of the arithmetic and the obvious ordering, but we do not consider this in this post.)
with 
is also in the epigraph), and because the underlying vector space 
comes with an order in the second component, some of the definitions for convex sets from above have a specialized form:
and ![{\lambda\in [0,1]}](http://s0.wp.com/latex.php?latex=%7B%5Clambda%5Cin+%5B0%2C1%5D%7D&bg=ffffff&fg=000000&s=0 "{\lambda\in [0,1]}")
it holds that
of tuples 
it holds that
such that
be a set of real valued function on 
is said to be 
such that
on a vector space by an arbitrary collection of real valued functions 
on a set. On sees immediately that for every function 
and any real number 
the function 
is 
is a subgradient of 
it holds that 
or
if
we have
(without any additional structure) define a coupling function 
and define the 
-conjugate as
be a compact subset of 
and consider the space 
of continuous functions 
with the usual 
and the canonical duality pairing is given by
positive if 
implies that 
. If a positive measure satisfies 
(i.e. it integrates the constant function with unit value to one), we call it a probability measure and we denote with 
the set of all probability measures.
with 
induces a probability measure via
-measures: For every 
there is the 
of probability measure is the generalization of the 
is the set of convex combinations of the measures 
(
-ball). 
, 
and suppose that the subset 
such that 
and 
are convex combinations of at most 
we have
measured by 
linear measurements, there are two extremal ones which consists of 
does not make much sense, since 
for all 
In my previous post 
