We discussed an application of the X method for the Y model.
I would not say:
The X method to the Y model. (this might work in a different context)
but
The X method for the Y model. (a purpose is involved, that method is for this model)
Would you mind judging this sentence?
They have created a method to improving the performance of this model.
Many thanks.
We are talking about 'the X method' and not just 'method', but there is something in what you're saying. OK, I'll have to revise and say, in 99% of cases (especially here because you do apply methods to model!) the usage would be:
'discuss the application of something to something'
so: 'discuss the application of the X method to the Y model'
but, yes, I suppose the X method could be an existing element of the Y model and not something introduced by application, so:
'discuss the application of the X method for the Y model'
might just be possible but even here I'd use 'used in' (or possibly 'used for' or 'used under').
Anyway, I think I've exhausted myself on this thread so will leave it there for now.
I agree with bertietheblue; further discussion is not necessary. But the following can make it clearer:
As I mentioned, apply + to is a correct formula.
If you change apply into a noun, its preposition will be still the same. This process is referred to as abstract noun phrase, as a result of which a verb is turned into a noun while preserving its verbal function. Of course, the abstract noun is usually followed by of and preceded by the:
the + application + of + something + to + something
The above long phrase is the direct object in the sentence:
we + discussed + [the + application + of + X method + to + Y model]
It's the most detailed explanation that I can come up with now.
Bye,