MTH 654 - Sec 001
Numerical Methods for Inverse Problems
HW1 Discussion
"What problem are you referring to
when you say [V] 1.12?"
From here out [V] means Kurt Vogel's book, [K] means Tim Kelley's book,
[TB] is Trefethan-Bau, and [D] is Demmel. I will likely refer to [K]
often.
"For Problem 2 the condition was easy enough to show,
but I'm not sure what condition I should show for Problem 3. I've done a
similar process to Problem 2."
Rather than repeat the effort from Problem 2, use that result to show
Problem 3 (i.e., redefine "A" and "d").
You should probably convince
yourself first that the right hand expression for $T_\alpha$ makes sense,
although it is not required that you show this.
"In Problem 2 should the summation limit on the RHS of the last
equation be n, rather than m?"
d should be in R^m so I think it is correct as written.
"On Problem 6 I cannot seem to show that the inequality holds for the
function max{1,s^2/a}, infact I find the exact opposite."
I believe the solution to Problem 6 is in the Solutions to Exercises at the
author's website
http://www.math.montana.edu/~vogel/Book/.
To be precise, the answer to 6 for TSVD is in the solutions. Hints for
the other filter function can also be found in 6 for Tikhonov.
"I'm having trouble getting the 4/27 bound on the
last problem."
I expected this bound to be challenging. Feel free to show the
inequality with just alpha as the constant.
"For 4b did you mean to say min instead of max? It seems like with max the
result is unbounded."
Yes! That should be min(1, s^2/alpha) otherwise you are keeping small
singular values unchanged and amplifying large ones! Hopefully this clears
up most of the problems with 4,5, and 6.