People use recipe books. This is 2nd year undergrad stuff.
Up to about 15 years ago almost all PDE models were in ADI with Crank Nicolson. I introduced ADE, Soviet Spliting and Method of Lines (MOL) here.
It is amazing (till today) that quant finance community is totally unaware of these methods.
Huge lacunae in the knowledge.
We are speaking about different subjects: you are talking about approximation and prediction (extrapolating) of a function, if and only if a 100% assurance is given that this function reflects some process that can be fit into differential equation (for example, options trading). If you are not sure that PDE is applicable to your process then you got nothing to do with all those methods.
I am talking about decent (accurate) approximation and profitable extrapolation (prediction) of ANY SAMPLED function (including inner parts and/or formulas of all your methods). Can you see the difference, Cuch?
Hmmm... It would seem your method also requires "100% assurance" that the stencils and the linear methods reflects the process you are modeling.
Although stencil-based systems might be fine for interpolation (even there they can fail badly), extrapolation requires much more assurance of the validity of the methods.
1). Yes, you understand it correctly.
I wrote above:
"In short: getting more samples is not a problem, the problems are:
(1). to find proper stencil functions with decent relation to a real process;
and
(2). to fit them into samples array with minimal error."
The simple Taylor interpolation (even 7...9 order) works fine for sin/cosin regardless of REAL relation between polynom and SIN.
2). QR decomposition gets much more adequate parameters of model than traditional Gauss (Least Squares). This allows more precise extrapolation of unknown function. Sure, the extrapolation is always a tricky process, but in modern math (if we are talking about price time series), the only thing you can be sure about (you can 100% extrapolate) - is periodical sinus/cosinus functions. Any other extrapolation is not flawless.
3). It is not "my" method. It is a generally - accepted practice in science (except FinTech quant finance, weird) in all cases of poorly-defined (poorly-conditioned) stencil matrix. Read here as "when you are not sure about the nature of your process, or when the practice shows up that it perfectly fits into a set of more simple functions". Nothing different of Taylor approximation of sin/cosin.
