5

I have come accross a lot of articles that talk about inverse problems. However, I dont really appreciate the uses due to my poor understanding of the notion.

From the mathematics point of view, when does a problem qualify to be called an inverse problem.

OKPALA MMADUABUCHI
  • 1,063
  • 6
  • 16
  • Example, Problem: given all the divisors of anumber $n$, find $n$. Inverse problem: given $n$ find all its divisirs. More genarly, problem: deduce conclusion $Y$ from the data $X$. Inverse problem: deduce conclusion $X$ from the data $Y$. – Ofir Schnabel Jul 30 '15 at 13:41
  • 2
    …there's a large wikipedia article about this exact topic. – Columbo Jul 30 '15 at 13:48

2 Answers2

4

I will give an example from my field. Suppose you have a signal corrupted with additive noise: $$ y = x + w$$

You receive the noisy signal $y$. For to be any good use, you need to de-noise the received signal to get the approximation to the sent signal $x$: $$ \hat{x} = f(y) $$

by using some operation $f$.

This is essentially 'inverse problem' since you need to estimate the original signal from the corrupted one. Design of a good $f$ will reduce the risk or error in some sense, say mean square error:

$$ \mathbb{C} = \mathbb{E}|| \hat{x} - x||^2 $$

The one with lowest $\mathbb{C}$ wins the race. I hope you see how its different from the forward problem where you are given $x$ and you apply some operation to find a good $y$ in some sense.

To put it very intutively:

Forward: What will you do to get a certain type of observation?

Inverse: What happened at source which generated this type of an observation?

further explanation is on wiki as pointed out. here

Ashutosh Gupta
  • 442
1

Why is the following answer being deleted Jose Carlos Santos (from University of Porto) and Harish Chandra Rajpoot (from IIT Bombay)?

Forward problem is when you try to solve for output y, given an input x to a known/fixed function f(x).

Inverse problem is when you try to solve for the input x, given the output to a known/fixed function f(x).

So in the equation y = f(x), forward problem is when you know x and you want to obtain the output y. Inverse problem is when you know y and you want to obtain what was the input x that produced the output y.

In real applications, the physical model that describes the measurements y (which you can measure using tools) and the physical parameters x that you actually want to identify, are often given in the form y = f(x).

So many physical problems naturally arise as the solving of "inverse problems", i.e., given that we know the "output" identify the related "input".

  • I am unsure Jose Carlos Santos and Harish Chandra Rajpoot, if you are aware of inverse problems... but do let me know if you have arguments. I've reuploaded the answer a 3rd time, because you have again not given any reasons. If you have time to delete this, I'm sure you'll have time to give some reasons which I'm not aware of at the moment.

    E.g., Inverse problem does not require there to be noises.

     –  Dec 17 '23 at 01:55