When your life does not converge

What to do and not to.

Everyone has convergence problems of some kind. I have, and have had many. Very much many. So, I would like to talk about convergence in this first post.

Convergence comes from the Latin convergĕre, composed by con[1]- (with, together) and vergĕre (bend towards, incline).

Literally, it means “incline together”. Actually, it means moving together from different directions and meeting at a point As when the waters of a small river join the dark ones of the ocean, or the lines meet at the vanishing point in the Architectural Veduta.

Architectural Veduta (Berlin), (ca. 1495), Gemäldegalerie.

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When dealing with molecular simulations or (almost) any kind of numerical calculations, convergence implies that “nothing changes”. It might be a bit disappointing, after the beautiful converging patterns of nature and art.   However, the mathematical implication is not that much different from the idea of meeting at a point. You just have to think slightly differently.

Imagine you got a bunch of numbers (or pictures, or cookies) that you order in a certain way. You start looking at them and realize that the every next number (or picture, or cookie) is closer to a limit. The n-th number is almost the same as the limit; it differs from the limit by a (small) quantity ε. If the small quantity is small enough ε (for you), then you can say that your bunch of numbers (or pictures, or cookies) has converged to the limit. If you further take the n+1-th number (or picture, or cookie), it will not change anymore, or just it changes by a quantity that you do not care about.

Written in symbols: |bn −b|< ε for all n ≥ n0, with bn being the n-th element of the bunch of numbers (or pictures, or cookies), and b being your limit.

So far, convergence sounds easy. I guarantee that it is not.

To understand why it is difficult to let a system converge, take two people in big city. To become friends, they have to meet in the Friendship Square, stay there, and chat. How can they meet? There are basically two problems they will face; first, in the city there are so many streets and not all of them lead to the Friendship Square; second, the two people have to walk with steps big enough to find the Friendship Square, and not to get stuck in a street, but small enough not to miss the Square. One of them maybe takes a street and never finds the Friendship Square. And then he/she/it keeps going in that street and goes far away from the Friendship Square. That’s sad.

Now, imagine that you have a molecule with some atoms that can do more or less what they want. Atoms in a computer are very undisciplined. However, to make it easier, your molecule has only two atoms.  Your job is to make these atoms meet in the configuration with lowest energy (aka Friendship Square) and stay there. You face again the same problems of before:

1. How to make the atoms “taking the right streets” .

2. How to not to miss the structure with by jumping too much.

For the first problem, there are tons of algorithms that compute where and when the energy is lower.  For the second problem, what you do is to adjust the step size of your algorithm.

This is the basic idea about convergence, meeting at a point. It becomes very hard and/or impossible for complex systems; but when a calculation did not converge, you want to find out why, and try again with something new. At the end, this is not that different than what we should do when life does not converge.

Thank you for reading.

Useless info: there exist a lot of "Friendship Squares”, for real! I know one in Berlin, such an amusing place!

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[1] In case you are wondering, it is con and not cum since it is late Latin.