Hi, this week will be slightly different because we will not talk about new content and new

ideas but instead we will apply, well not everything but a lot of the ideas that we

have gotten to know so far in this lecture to a specific inverse problem.

And let's talk about this inverse problem in some more detail.

So this is the usual heat equation, the stationary heat equation, so after equilibration for

a one-dimensional domain.

So as before we're essentially looking at a rod of length one and we're interested in

understanding how the stationary temperature distribution looks like for a given set of

factors.

So for example one property of the system is that the temperature at the boundary is

always set to zero.

So we're cooling down the, well what does it mean cooling down?

We're fixing the temperature to zero at the boundary of this rod and in the interior of

the domain we have some function u of x which is a heat source or heat sink depending on

whether u is positive or negative.

So this relationship, this differential equation relates a heat source with the resulting equilibrium

heat distribution.

So obviously if you apply a new heat source a physical model will take some time to evolve

towards the stationary solution but after that time evolution we will have gotten this

stationary solution which will be there.

So for example if we apply this heat source down below here to this one-dimensional rod,

so this is the domain and this is the amount of heat that we're either putting into the

system or pulling out of the system by maybe applying some probes which are hot or cold.

So for example at this point we apply let's say maybe a probe which is very hot so we're

pumping heat into the system at this point but this probe is very narrow so the actual

contact which this probe has with the domain is quite small and here we have something

slightly more weird.

So we're cooling here, we're heating on a non-negligible sub-part of the domain and

in between we're interpolating.

So this is a very complicated thing to do.

Obviously it's not easy to physically generate this kind of heat source or sink exactly but

it's just a toy model anyway.

And if we do this, if we apply such a heat profile, so we're pumping heat into the system

here and here and we're removing heat from the system here, then the resulting temperature

profile after let's say maybe five minutes or so depending on what the thermal permeability

of this rod is that we'll get something like this.

So you can see that near the parts of the domain where we apply heat, so here and here,

the resulting temperature function obviously has some local maximum.

So it's hot here and very hot here.

So although this is let's say a hotter point of contact, the larger contact area here leads

to a higher temperature in the stationary temperature distribution here.

But anyway it's hot here, it's warm here and because of this heat sink near 0.4, we have

a drop off in the temperature in between those two local maximum.

So it makes sense to have something like that and this relationship going from the heat

source to the stationary temperature function, this will be our forward problem.

And of course our inverse problem then will be to try and recover the original heat source

and heat sink, so this function here from measurement of the stationary heat equation.

So if someone gives you temperature measurements of such a rod, then their task for you is

to try and infer the original heat source and sink profile that generated this stationary

solution.