Having no practical experience of tuning, still a thought appears
in my engineer's mind on how to reduce the number of tuning rounds
after restringing a piano.

For the matter of explanation I will use a few very round figures:
On one hand, the total stretching force on all strings in a piano is
to the order of 4 tons.  Assuming their combined cross sectional area
to be 2 cm^2 and a 20*10^10 Pa Young's modulus (steel) means each tuned
string is elongated about 0.1% from its relaxed length.

On the other hand the cast iron plate is not infinitely stiff.  When
you tension the strings, then you will also compress the plate.  A 40
cm^2 total area of the plate beams and a 10*10^10 Pa Young's modulus
(cast iron) then means the plate is compressed some 0.01%.  This is
about 10% of the string elongation.  This means, for example, that if
you remove most of the strings and leave just a few, then the plate
will expand and stretch these to raise them something like a semitone.
The effect is most probably even stronger than this, because the string
support is more compliant than the plate alone, for instance because of
deformation of the soundboard due to the downbearing; experts might
know.

Reading Art Reblitz' piano book I understand that a discarded set of
strings should do their last service as an ordered set of templates for
the string maker.  What I suggest here is that they could also do an
additional, next to last service.

To install new strings you no doubt have a plan for in which order to
put them in.  When you take out the old strings, I suggest you bring
a tuner device and write a protocol.  I assume that the old strings are
roughly in tune.  You take them out in _reverse_ order to that of
installation.  As you remove a number of strings the force on the plate
will reduce, it will expand and stretch the remaining strings.  For
each six or twelve notes, record from the tuner what is the deviation
(e.g. in cents) of the next note you are going to remove.  The less
number of strings remaining, they should be progressively too high.

When you install the new strings, coarse tune them immediately
according to the protocol, that will always be too high.  As the
following strings are added and tuned the earlier strings will
gradually sink.  Theoretically, when you are finished, the original
tuning should be automatically reproduced.

Johan Liljencrants, Stockholm, Sweden