Presentation080715

Basic Problem Varying Backside Case Missing Nodes Case
Finding the Shape of a Resistor Grid from
Boundary Measurements
Esther Chiew, University of Illinois
Vincent Selhorst-Jones, Pomona College
Rose-Hulman REU, Summer 2008
July 23, 2008
Finding the Shape of a Resistor Grid from Boundary Measurements Chiew & Selhorst-Jones

Basic Problem
Introduction
Fudamental Equations
Example of Propagation
Remarks on Propagation
Varying Backside Case
The Model
The Algorithm
Instability
Missing Nodes Case

Introduction
Basic Problem
Given (some of) the
boundary data on a grid of
uniformly 1 ohm resistors,
what can we determine
about the shape of the grid?
For example, might know
voltage and current on front
nodes while knowing the
sides are insulated and the
back nodes are at 0 volts.
What is the height of each
column of the grid?
Figure: A rectangular resistor grid.

Introduction
Compare the below. Let sides be insulated. If we set same current
and voltage on the bottoms, we’ll get diﬀering results on the tops.
Figure: An undamaged grid. Figure: One missing node.

Fundamental Equations
The resistor network is governed by two laws:
Ohm’s Law: V = IR,
Kirchoﬀ’s Current Law: node i = 0.

Looking at any inner node (recall R = 1 uniformly), we have:
v1 − v = i1
v2 − v = i2
v3 − v = i3
v4 − v = i4
node i = 0.
Summing the center equations, this gives the relation:
v1 + v2 + v3 + v4
4
= v,
i.e., every inner node is the average of its surrounding nodes.

The next few slides are an example of how data is propagated
inwards. For it, we will assume
Know voltage and current on front nodes,
Sides are insulated (notice that this is equivalent to simply
omitting the resistors sticking out of the sides).

From voltage and
current on front
nodes, can use
Ohm’s Law to obtain
voltage on next row.

Can obtain new
voltage by using rule
that each node is the
average of its
surrounding nodes.

Do this this for the
enitre row.

Continue to
propagate this
information upwards.

Information can be
pushed up arbitrarily
high by this method.

Remarks on Propagation
The following things should be noted about how the propagation
works:
Each row is entirely determined by the two rows under it.
This process uses only linear operations, and thus can be
represented by a propagation matrix acting on a voltage
vector whose length is twice the width of the grid.
The process is highly unstable, i.e. small diﬀerences in initial
data will give wildly varying results. This comes from the fact
that the propagation matrix has eigenvalues greater than 1.

The Model
Varying Backside Case
For recovering a varying backside, we use the following model:
We know the width of the grid.
Sides of grid are insulated.
Voltage and current known on
front nodes.
Every node above the “height” of a
column is at 0 volts.
Current can leave through 0 volt
nodes in either direction. (They
still aﬀect adjacent inner nodes.)
Figure: Heights: 3, 3, 2, 3, 4.

The Algorithm
Fundamental Theorem
To implement the algorithm we will use, we need the following:
Theorem
If the voltage of every node on the front is greater (less) than 0,
then the voltage of all nodes below the backside must be as well.
Proof
Furthermore, it’s reasonable to expect we have control over the
front voltages (say, because we’re the ones inducing them with a
voltage generator), so we can assume the front voltages are strictly
positive for purposes of the algorithm.

The Algorithm
The Algorithm
1. Use the voltage and current for the front nodes to determine
the voltages on the ﬁrst inner row.
2. Using the top two rows found so far and the fact that each
node is the average of its surrounding nodes, calculate the
voltages for the next row above.
3. If any of the nodes in the latest row are less than or equal to 0
(or some error term, say 5 · 10−3), set that node and all the
above nodes in that column to 0.
4. Repeat from Step 2 until you obtain a row that is all 0’s.
5. The ﬁrst row a column contains a 0 in is the column’s height.
Example

Instability
Given starting data with no error and a computational method that
introduces no error, the algorithm works perfectly. However, since
we’re dealing with the real world, let’s look at what error does to
the algorithm:

Instability
The following is an example of a ten-wide grid with 10 volts put on
each front node and the following column heights:
9 10 9 8 6 5 6 7 8 9
Putting voltage into the system induces currents on the front.
Feed the voltage and currents into the algorithm, we get:

Instability
9 10 9 8 6 5 6 7 8 9
9 ∞ 9 8 6 5 6 7 9 9
Not bad. But what if we introduce a tiny error of, say, 0.01% to
the reading of the voltage from the eighth node?

Instability
9 10 9 8 6 5 6 7 8 9
9 ∞ 9 8 6 5 6 7 9 9
Not bad. But what if we introduce a tiny error of, say, 0.01% to
the reading of the voltage from the eighth node?
9 ∞ 8 ∞ 6 6 6 ∞ 6 ∞
Not good.

Instability
Clearly, small errors can cause extreme damage to the algorithm.
With every new row that is generated, the initial errors become
considerably greater. So what can we do about this instability?
Shape: The more rows the data has to propagate through, the
worse the error gets. Thus, short, squat grids can be
recovered, but tall ones work much less well.

Instability
Regulation: The growth in error is caused by the large
eigenvalues of the propagation matrix. If we manually shrink
these eigenvalues, error propagation decreases.

Instability
Regulation: The growth in error is caused by the large
eigenvalues of the propagation matrix. If we manually shrink
these eigenvalues, error propagation decreases.
Voltage: Choosing the initial voltage well can allow for better
seeing. If we have an expected backside, we can use that to
generate a voltage pattern that might let us recover more.

Instability
Regulation is far from perfect, though. Heavy regulation wipes out
large eigenvalues – Not a problem for perfectly rectangular grids,
but large eigenvalues are what primarily propagates the information
of a varied backside.
Nonetheless, the process can still be improved through the use of
light regulation. In addition, we can use the algorithm to obtain a
hypothetical backside, from which we can ﬁnd a voltage vector
that gives a more “distinct” current. Using this new current, we
can repeat the whole process a few times and improve the
algorithm slightly.

The End! Thanks for listening!

Proof of Theorem Example Use of Algorithm
Appendix Index
Proof of Backside Theorem
Example of Backside Algorithm

Theorem
If the voltage of every node on the front is greater (less) than 0,
then the voltage of all nodes below the backside must be as well.
Proof.
Let the voltage of every front node be greater than 0. Let p denote
an interior node with the lowest voltage obtained by an interior
node. Assume p is non-positive. Then since p is the average of its
surrounding nodes and also has the lowest possible voltage, every
node surrounding p must have equal voltage. Moving down the
column p is contained in, we eventually hit the front, implying that
one of the front nodes is not strictly positive. ⇒⇐
Back Appendix Index

Let’s look at an example
of using the backside
algorithm. To begin with,
recall that we know the
current and voltage on the
front nodes.

Using this information, we
can determine the voltages
in the second row as well.

As noted earlier, knowing
two rows gives the next
one above.

Continue to propagate in
this manner until we ﬁnd a
node with a voltage of 0.

By theorem, we know that
node is the “top” of its
column. Thus we set all
above nodes to 0 as well.

Continue to propagate
upwards, using this
information.

When new 0’s are
encountered, repeat the
process of setting above
nodes to 0.

Repeat until the top of
each column is found.

A column’s height is the
ﬁrst row in it that has a 0.

The algorithm is complete.
We now have the height of
each column.
Back
Appendix Index

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