Lecture 4 Solving the forward problem

Today's topic is solving the forward problem. For the discussion of FEM I will refer to my book chapter in Holder's book with Nick Polydorides and Andrea Borsic (see preprint version). Theses are also useful, Marko Vaukhonen's and Nick Polydorides' (see reading list)

The video (a single AVI) download here

The "cot formula" that relates FEM to resistor networks in 3D appears in our paper from the Florida EIT meeting William R. B. Lionheart and Kyriakos Paridis, Finite Elements and Anisotropic EIT reconstruction. Journal of Physics: Conference Series, vol. 224, no. 1, p. 012022, 2010

Lecture 3 videos now here

This is just here to trigger your RSS feed or whatever, I'll delete it when people have noticed...I am not yet convinced of this "email lists are so last century" idea and new fangled things like RSS feeds and blogger followers, well we will see how it works.

Yes it is Friday at 11am

We are continuing on Fridays at 11-12, in the same room, except for Nov 26th. We probably wont have a lecture then as the room is booked by someone else and one of my students is being viva'd. We will endeavour to put the lectures on the web the same day but some times (eg this time) it might slip to Monday

Lecture 3

Time for some vector calculus and linear algebra. I started by explaining the so-called Louiville transformation between the conductivity and stationary Schrödinger equation. Then we went on to least squares solution of over determined linear systems, Tikhonov regularization and the SVD.

This material is covered in more detail in my lecture notes for a course we used to teach use the username inverse and the password crime to access this page and look at chapter 3 of the notes.

Video as one avi file, or
part1 part2 part3 part4 part5

Or on YouTube Part 1,

Updates

We have added links to the lectures in avi files split up in to (less than) 15 min chunks. I am experimenting with Youtube. Please post in comments to let us know if this works for you. Also I have put up some homework problems on the blog post for lecture 2.

Lecture 2 the linearization

In this lecture we go through Calderón's Born Neumann series for the forward problem and the linearization that results from truncating the higher order terms. The Fréchet derivative of the Neuman-to-Dirichlet map is derived from Calderón's calculation of the Fréchet derivative of the Dirichlet-toNeumann map with respect to conductivity.

The (high resolution) video is here Lower resolution in chunks part1, part2,part3,part4.

Some helpful resources:

• Calderón's foundational paper you can see the original paper scanned here as well as a reprint with slightly different typing errors.
• If you are confused by the difference between little-o and big-O Wikipedia may help, also definition of the Fréchet derivative
• Martin Hanke has formalised the method of finding an inclusion from Cauchy data at the boundary and describes in his paper On real-time algorithms for the location search of discontinuous conductivities with one measurement

Some homework suggestions

1. Have a go at deriving the Born-Neumann series and directly the linearization for the Neumann-to-Dirichlet case. To make it simpler assume that the conductivity does not change in a neighbourhood of the boundary
2. How does the argument change for the case of complex conductivity? Clearly you need some complex conjugates around.
   
3. Find the sensitivity/Fréchet derivative for the stationary Schrödinger/variable wave speed Helmholtz equation -∇² u + c u=0. If c is allowed to be negative you have to assume that it is "non resonant", that is you avoid eigenvalues, then there is a well defined Green's operator. This problem is relevant to ultrasound/seismic imaging, diffuse optical tomography, quantum scattering etc.
   

Lecture 1 Introduction to the Forward Problem

The video of the first lecture is available as a download (about 628MBytes). Kyriakos has made a smaller avi file or you can see it in 15min chunks part1, part2,part3,part4. We are trying Youtube (not yet sure resolution is enough) part1

There is a home work exercise implicit in the lecture. That is to check the calculation of the Dirichlet to Neumann map for the concentric anomaly by separation of variables. Another interesting thing to do is do some calculations of current and voltage patterns, and see how they change with rho and mu.

If you get anywhere with this post it in the comments too!