The first step is to compute the singular value decomposition.

For that you can also again use a NumPy module.

NumPy linear algebra SVD gives you the matrices U, the singular values S, and the matrix V,

each transposed.

And if we now make a semi-log plot of all the singular values which are in this diagonal

matrix S. You can see that well we're in a thousand dimensions so we have obviously

a thousand singular values and the magnitude varies between roughly 10 to the minus 1 and

10 to the minus 12 probably and that is a very large gap so that means that the ratio

between largest and smallest singular value is on the order of magnitude of 10 to the

13 which is a very very bad condition number so this is why this is a heart inverse problem.

Okay so what we do one thing that we have talked about before in this course was truncated

SVD. The formula you have seen before in a different application you just take 1 over

S we can do this because S doesn't actually have any zero singular values they're just

very small and you cut off this spectrum by at some threshold alpha that you can choose

then you return V times this kind of regularized pseudo inverse of the diagonal matrix S in

this form times U transpose times the data set and now you can do that and if you do

nothing at all then you get something very similar to the E3 construction which is a

lot of oscillations more noise than data if you take a very large cutoff parameter which

means that you're cutting off everything essentially everything then your reconstruction is essentially

almost a constant and if you're somewhere in between you cut off something but not everything

so for example here you take the first I don't know let's say 20 singular values or so and

you ignore the rest so this is what this orange line says you take the first 20 or so and

as soon as they drop below some threshold you ignore the rest this gives you this reconstruction

so this seems to be slightly too regular so let's go a bit smaller here okay so maybe

maybe this letter of taste this is probably the best you can do with TSVD and Tikhanov

works very similarly here again as a formula for Tikhanov reconstruction that you've seen

before as well so I won't go into details here and if you do this you can play around

with Tikhanov no regularization means that very high frequency reconstruction a lot of

regularization means that it's essentially a constant reconstruction and somewhere in

between you get well this is probably a bit too small but you can see nicely how Tikhanov

doesn't cut off at some point like oh the labeling is off here sorry so here blue is

the under regularized one here it's orange so that's what you have to look at so this

gives you this decay that Tikhanov does so maybe let's do a bit more and well this is

quite nice let's go with more maybe that's a nice reconstruction and in my opinion here

Tikhanov works a lot better than TSVD TSVD this sharp cut off that's just a bit too much

and that is almost able to recover this sharp interface here well yeah maybe that's alright

that's a good reconstruction and that is how you do that's how you do TSVD and Tikhanov