Understanding Sensitivity Analysis by ParametricNDSolve[]

Questions
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1

Dear Community,

I have been going over the documentation for the use of ParametricNDSolve[], and was wondering about the meaning of one of the basic examples provided in the documentation.

In the example, we are trying to show the sensitivity of a solution provided by ParametricNDSolve[], of a differential equation with two-parameter inputs. The equation is:

sol = ParametricNDSolve[{y''[t] + a y[t] == 0, y[0] == b, y'[0] == 0},
   y, {t, 0, 20}, {a, b}];

If looking at the sensitivity of variable a with respect to time t, the code is:

Plot[Evaluate[(y[1, 1][t] + {0, .1, -.1} D[y[a, 1], a][t] /. 
     a -> 1) /. sol], {t, 0, 20}, Filling -> {2 -> {3}}]

Yielding the following plot:

plot1.jpg
plot1.jpg360×216 11.4 KB

The same is done for the analysis of b (which I left out, as I suspect the above plot will be enough for a helpful soul to dumb it down to my level of understanding :slight_smile: ).

My question is, what exactly is shown? Lets for instance say that y(t) is the population of ticks within a defined area. The population is dependent on the temperature (say, variable a), and the humidity (variable b). Then:

y[1,1][t]

would correspond to evaluating a base case scenario where a=b=1. To this expression we then add:

{0, .1, -.1} D[y[a, 1], a][t] /. a -> 1

When multiplying the differential by 0, we get the base case, but is correctly understood that {.1,-.1} multiplied with the differential, gives the base case value, with the effect of a 10% change in a at time t, by finding the rate of change of y(t), with respect to a, and then evaluating the rate (or slope) at a=1.

But even so, does this tell me that the effect of a 10% change of a would be more significant at later times, rather than at the very beginning? or am I way off?

Sorry for the long post!

Best Regards

2

It is telling you what you suspected, to wit, that lthe effect in a sense widens over time (very literally I suppose, since it is changing the frequency). You can see the same, in this simple example, from the analytic solution.

DSolve[{y''[t] + a y[t] == 0, y[0] == b, y'[0] == 0},  y[t], t]

(* Out[272]= {{y[t] -> b Cos[Sqrt[a] t]}} *)

One thing though is that using the first order change in your plot means it also strays over time. The variant below might be a better way to go.

Plot[Evaluate[{y[1.1, 1][t], y[1.0, 1][t], y[0.9, 1][t]} /. sol], {t, 
  0, 20}, Filling -> {2 -> {3}}]

pndsolve.png
pndsolve.png360×210 25.7 KB

Among other things, the peak magnitudes do not (incorrectly) change with variation in the a parameter.