Information Technology Reference
In-Depth Information
Furthermore, if a very small amount of noise is added to g ,say
g.x/ D g.x/ C 10 20 sin .3x/ D .1 C 10 20 / sin .3x/;
f of (9.24) changes dramatically, i.e.
then the corresponding solution
f.x/D .1 C 10 20 /e 3 2 2 sin .3x/ f.x/C 3:77 10 18 sin .3x/:
In fact,
f.x/ f.x/ D 3:77 10 18 sin .3x/;
even though
g.x/ g.x/ D 10 20 sin .3x/:
The problem is extremely unstable: Very small changes in the observation data g
can lead to huge changes in the solution f of the problem.
One can therefore argue that it is almost impossible to estimate the temperature
distribution backward in time by only using the present temperature and the diffu-
sion equation. Further information is needed. This issue has led mathematicians to
develop various techniques for incorporating a priori data, for example, that f.x/
should be almost constant. More precisely, a number of methods for approximating
unstable problems with stable equations have been proposed, commonly referred to
as regularization techniques .
Do the mathematical considerations presented above agree with our practical
experiences with diffusion? For example, is it possible to track the temperature dis-
tribution backward in time in the room you are sitting? What kind of information
do you need to do so? If you want to pursue a career in applied math, these are the
kind of issues that you must consider carefully.
The estimation of spatially and/or temporally dependent parameters in differen-
tial equations often leads to unstable problems. It is an active research field called
inverse problems . (You can think of the backward diffusion equation as the inverse
of forecasting the future temperature.)
9.3
Estimating the Diffusion Coefficient
The examples discussed above are rather simple, since explicit formulas for the
solutions of the involved differential equations are known. This is, of course, not
always the case, and we will now briefly consider such a problem.
Assume that one wants to use surface measurements of the temperature to com-
pute a possibly non-constant diffusion coefficient k D k.x/ inside a medium. With
our notation, the output least squares formulation of this task is
" Z T
. u .1; t I k/ h 2 .t // 2 dt #
. u .0; t I k/ h 1 .t // 2 dt C Z T
0
min
k
0
Search WWH ::




Custom Search