3. Sensitivity and identifiability

Two explanations for one temperature field

Suppose you observe the entire steady field in the previous chapter. Can you recover both the source strength $s$ and conductivity $a$?

Recall the solution on the one-metre square:

$$u=b+\frac{sL^2}{2a\pi^2}\,\psi,\qquad \psi(x,y)=\sin(\pi x/L)\sin(\pi y/L),$$

where $L=1\,\mathrm m$. The response depends on the ratio $s/a$. Multiplying both parameters by the same positive factor leaves $u$ unchanged everywhere. More temperature sensors do not remove this particular ambiguity, even with exact measurements.

Source scale $s$ in $\mathrm m^{-2}$	Conductivity ratio $a$	Centre temperature, $b=0$
$2\pi^2$	1	1
$4\pi^2$	2	1
$\pi^2$	0.5	1

These are analytically identical predictions. A numerical search can return one row with an excellent fit while the others remain equally compatible with the data.

Sensitivity asks about a small change

A parameter sensitivity is the derivative of the observation with respect to a parameter. At fixed position,

$$\frac{\partial u}{\partial s}=\frac{L^2\psi}{2a\pi^2},\qquad \frac{\partial u}{\partial a}=-\frac{sL^2\psi}{2a^2\pi^2},\qquad \frac{\partial u}{\partial b}=1.$$

Let $J$ collect these derivatives for all observed positions. For a small parameter change $\delta p$,

$$\delta y\approx J\delta p.$$

The source and conductivity columns are proportional: $J_a=-(s/a)J_s$. In direction $\delta p=(s,a,0)$, they cancel. The measurements have no first-order response in that direction, consistent with the exact scaling ambiguity above.

Identifiability asks whether observations distinguish the unknown parameters. Column dependence exposes a local ambiguity in a differentiable model. Here we also have the stronger, explicit family of exactly equivalent solutions. The linear-dependence language is developed in Boyd and Vandenberghe, chapter 5 of Introduction to Applied Linear Algebra.

Try the cancellation in Eqiora

The Poisson script orders its inputs as source, conductivity, boundary value. Replace its direction array with:

python

direction = np.array([2.0 * np.pi**2, 1.0, 0.0], dtype=np.float64)

The program.jvp(direction) call returns the field change in that direction. Predict a near-zero tangent. Compare it with the source-only direction np.array([1.0, 0.0, 0.0], dtype=np.float64), which produces a nonzero field change in the interior. Both experiments reuse the same compiled mathematical component and the same numerical plan.

For a finite change, compare program.evaluate(nominal) with program.evaluate(np.array([4.0 * np.pi**2, 2.0, 0.0], dtype=np.float64)). Their primary fields should agree up to the solution error.

Design an observation that adds information

Which change helps? An independent conductivity measurement fixes $a$, so the field can determine $s$. Alternatively measure heat flux as well as temperature. In our normalized model the flux is $\boldsymbol q=-a\nabla u$, so substitution gives

$$\boldsymbol q=-\frac{sL^2}{2\pi^2}\nabla\psi.$$

Its dependence on $a$ has cancelled. A suitably located nonzero flux measurement can determine $s$, while temperature then determines $a$. A flux measurement at the centre is unhelpful because the gradient vanishes there. The instrument and its placement matter as much as the number of readings.

If $b$ is also unknown, include information that separates a uniform offset from the sinusoidal response. A boundary-temperature reading directly observes $b$; two distinct interior response levels can also separate the offset and amplitude. Repeating the centre reading reduces random noise but does not create a second spatial response pattern.

Exercises

1. Derive the three sensitivity formulas without looking at them above.
2. Give two distinct $(s,a)$ pairs with centre temperature 0.75 and $b=0$.
3. At $(x,y)=(0,0.5)$, calculate the normalized outward flux from the analytic solution. Which unknown does it identify? Keep the outward normal sign explicit.
4. Suppose $a$ is known but $s$ and $b$ are unknown. Compare two centre readings with one centre and one boundary reading by writing their $2\times2$ Jacobians.
5. Explain why reducing the optimizer’s stopping tolerance cannot remove the source/conductivity ambiguity.

Check your reasoning: on the left edge at its midpoint the outward flux is $sL/(2\pi)$ in the normalized units; the normal points in the negative $x$ direction.

Reading

Stephen Boyd and Lieven Vandenberghe, Introduction to Applied Linear Algebra, Cambridge University Press, 2018, chapters 5 and 12. Authors’ book.

Previous: Recovering a heat source · Book map · Next: Differentiating a solved model