A Summer 2015 Mathematics A To Z: well-posed problem


Well-Posed Problem.

This is another mathematical term almost explained by what the words mean in English. Probably you’d guess a well-posed problem to be a question whose answer you can successfully find. This also implies that there is an answer, and that it can be found by some method other than guessing luckily.

Mathematicians demand three things of a problem to call it “well-posed”. The first is that a solution exists. The second is that a solution has to be unique. It’s imaginable there might be several answers that answer a problem. In that case we weren’t specific enough about what we’re looking for. Or we should have been looking for a set of answers instead of a single answer.

The third requirement takes some time to understand. It’s that the solution has to vary continuously with the initial conditions. That is, suppose we started with a slightly different problem. If the answer would look about the same, then the problem was well-posed to begin with. Suppose we’re looking at the problem of how a block of ice gets melted by a heater set in its center. The way that melts won’t change much if the heater is a little bit hotter, or if it’s moved a little bit off center. This heating problem is well-posed.

There are problems that don’t have this continuous variation, though. Typically these are “inverse problems”. That is, they’re problems in which you look at the outcome of something and try to say what caused it. That would be looking at the puddle of melted water and the heater and trying to say what the original block of ice looked like. There are a lot of blocks of ice that all look about the same once melted, and there’s no way of telling which was the one you started with.

You might think of these conditions as “there’s an answer, there’s only one answer, and you can find it”. That’s good enough as a memory aid, but it isn’t quite so. A problem’s solution might have this continuous variation, but still be “numerically unstable”. This is a difficulty you can run across when you try doing calculations on a computer.

You know the thing where on a calculator you type in 1 / 3 and get back 0.333333? And you multiply that by three and get 0.999999 instead of exactly 1? That’s the thing that underlies numerical instability. We want to work with numbers, but the calculator or computer will let us work with only an approximation to them. 0.333333 is close to 1/3, but isn’t exactly that.

For many calculations the difference doesn’t matter. 0.999999 is really quite close to 1. If you lost 0.000001 parts of every dollar you earned there’s a fine chance you’d never even notice. But in some calculations, numerically unstable ones, that difference matters. It gets magnified until the error created by the difference between the number you want and the number you can calculate with is too big to ignore. In that case we call the calculation we’re doing “ill-conditioned”.

And it’s possible for a problem to be well-posed but ill-conditioned. This is annoying and is why numerical mathematicians earn the big money, or will tell you they should. Trying to calculate the answer will be so likely to give something meaningless that we can’t trust the work that’s done. But often it’s possible to rework a calculation into something equivalent but well-conditioned. And a well-posed, well-conditioned problem is great. Not only can we find its solution, but we can usually have a computer do the calculations, and that’s a great breakthrough.

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