Below you will find the discussion and relevant materials for the ES 105 homework assignment due on March 02, 2009.
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.
* along Y=0.20: X=-0.6, -0.3, 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.
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
* 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:
* 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.
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