Discussion

Our homework assignment for this week, Problem#44, asked us to solve the above system. We were provided: 1) an element file, 2) a node file, 3) a file with boundary nodes and their values, and 4) a README file with details on the problem set.

For this week's homework we were asked to work on the previous weeks problem using the REPRESENTER's method for Inverting data.

Note: All (x,y) units in this problem are stated in the units of the .nod file. The length scales indicated on the sketch are not to be used.
Find the Sources
Given the standard solution to problem 44a. This is ``TRUTH''. Measurements of voltage are available at the following 12 locations:

* along Y=0.20: X=-0.6, -0.3, 0.0, 0.3, 0.6
* along Y=0.40: X=-0.3, 0.0, 0.3, 0.6
* along Y=0.60: X= 0.0, 0.3, 0.6

These data are available with perfect precision. There is no other data. You want to reconstruct the full solution i.e. the voltage everywhere. Neither the boundary conditions along the bottom "ground plane", nor the location or strength of the sources, are known. The Neumann BC is thought to be perfectly insulated. Prior assumptions:

* Along the Dirichlet boundary: the ground condition is known pretty well, within 0.1 volt. The correlation length for the ground condition is relatively short, L=0.4.
* Interior sources are presumed to be present but could be anywhere (at any or all nodes). Their expected size is 1.0; they are assumed to have correlation length scale L = 0.3 but they are known to be uncorrelated with the ground.
* Model-Data Misfit is expected to be of order .05 volts with zero mean, and no correlation to anything.

Essentially, you want to find the source distribution and the ground voltage, and use that forcing to construct the voltage everywhere.

* Problem 1: TRUTH. Sample the truth to obtain the data. These are perfect data. So a perfect solution exists, if you can find it!

* Problem 2: Invert the perfect data using Representers. Make maps of the source distribution, the potential, and the difference between TRUTH and the inverse truth. The latter is an ERROR map -- in this case, you know TRUTH through a special secret arrangement, so the error is computable. Because the data is perfect, this is a map of BIAS in your least-squares estimator. Compute the RMS of this bias; the RMS of the model-data misfit; the RMS of the nodal sources; and the RMS of the Dirichlet BC's.

RMS OF LSS BIAS: 0.387160610259675
RMS MODEL-DATA MISFIT: 6.374936321789749E-005
RMS OF NODALSOURCES: 1.981924686657907E-002
RMS OF DIRICHLET BC'S: 1.426607344241128E-005

* Problem 3: Your solution is dependent on the error models used in the inversion. Suppose you are unsure about two of those parameters. Show the sensitivity of your inverse bias map to each separately:
1. The model-data misfit variance.
2. The correlation lenght of the interior sources is hypothesized to be smaller.

* Problem 4: Compute the inverse noise assuming observational error will be totally random, uncorrelated, with variance (0.05 volts)^2. Plot contours of the square root of the diagonals of the inverse noise covariance matrix. This is a measure of the IMPRECISION in your estimator.

Plots

Solution using SOLVE.F

Solution using REPRESENTER's to INVERT DATA

Difference Map between Solutions using SOLVE.F and REPRESENTERs

RHS using REPRESENTER's to INVERT DATA

INVERSE NOISE using REPRESENTER's to INVERT DATA

MODEL DATA MISFIT VARIANCE CHANGED TO 500 V FROM 0.05 V

CORRELATION OF INTERNAL SOURCES CHANGED TO 1E-6