While going through some C++ code about stochastic processes, I came across this concept of discretization repeatedly. I have checked the Wikipedia link but description goes into deeper details too quickly for a non-mathematician like me. Can someone explain in simple terms what the concept means, what is its use and how it is applied?
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Mathematical models of physical phenomena (planetary motion, fluid dynamics, heat transfer, etc.) are in terms of continuous variables, like time, distances, velocities, temperatures. In the so-called constituent equations (ODE's, PDE's) of these models the quantities under discussion appear as arguments of nice functions, like $\sin$ or $\exp$, and therefore one expects that the solutions are nice, i.e., continuous, functions as well. If we can determine these solutions "explicitly", i.e., in terms of finite formulas, all is fine. But in most cases we can't, and we have to resort to numerical methods. This means that in the end we will know the values of the solution $t\mapsto x(t)$ (or similar) only at a finite number of points.
Now it is the essence of continuity that a continuous function can be recovered (or produced) completely if you are able to compute its values, at least in thought, on an arbitrarily dense grid of points. That's where discretization comes in: We are willing to settle for the values of the solution on a "reasonably" dense grid of points. The problem now arises to "simulate" the exact constituent equations (e.g., the law of gravity) in a gridlike environment, such that as little extraneous effects enter the picture as possible. This is the central problem of discretization; it has nothing to do with rounding errors.
I am not sure if this is what you are looking for, but I'll try......(It would have been better, if you'd mentioned what you understood, and what you're looking for.)
First of all, you should know the difference between a continuous variable and a discrete variable.
A continuous variable is a variable, which can take an infinity of values between any two chosen points. A discrete variable, on the other hand is one, which can take only a fixed number of values, between any two chosen points.
The process of discretization involves converting a continuous variable in to a discrete variable. This has the advantage of reducing the computational complexity of the resulting model(fewer values mean fewer calculations). The exact process of how it is done, depends on the phenomenon we're modelling.In engineering applications, for example it involves discarding values that are insignificant (i.e which introduce only a small error, when we physically realize the model). There are two key things to keep in mind while creating a discrete model -
As you can imagine, we have to maintain an optimal tradeoff between the two. Increasing one will decrease the other. The bulk of the process of discretization, centers on this optimization process.
The meaning of the word accuracy needs to be explained here - you are using the discrete model as a substitute for the continuous(original) model. So we should be able to extract the main features of the original model from the discrete model. (To be able to do this while retaining computational efficiency is where the ability of the mathematician/engineer/scientist who's creating the model, lies.)
This is a general explanation of discretization, without any specific field of application in mind.(the engineering reference was because of my own background.)
I hope it helps.... if not, please let me know.