Regularizer – more than you ever wanted to know

Dynamic Light Scattering can extract a particle size distribution from intensity statistics. Occasionally we get asked: How exactly does that work? In this post, I try to guide you towards understanding the intricacies of the process: from an ill-posed problem to a solution – in a hand-waving way. And with a regularizer – the magic parameter for finding the solution. So how do we get the distribution?

What’s an ill-posed problem?

You may have read or heard about this. Once we obtain a correlation function we then have to convert it to a size distribution, and this mathematical problem is also called an ill-posed problem. It is ill-posed because there is not a single direct solution for this. Other terms for this situation are improperly-posed problem, on in a specific situation also an “inverse problem”. The core issue is: There are several potential solutions that can explain the observed situation or parameter, and now we are stuck with trying to find “the right one”.

To be even more specific, there is a range of potential size distributions that can explain an experimentally observed correlation function. Out of all these distribution functions, which one should we pick as a result?

Narrow the space a little…

A common concept in the approach to the solution is to make the potential space of solutions smaller. One very reasonable concept is to require that real solutions should only have contributions from positive sizes. In other words, we cannot have particles with negative sizes, like minus 20nm radius. Only positive radius sizes are allowed. This constraint has the indication Non-Negative. An algorithm in the Zetasizer is the NNLS, which stands for non-negative least squares. Here, the error between the proposed solution (the size distribution, ergo the correlation function that it would produce) and the observation (the correlation function we measured) is minimized. But that’s not yet enough.

And make it smooth – with a regularizer!

The non-negative constraint still leaves a lot of potential solutions (just half as many as before). To arrive at one solution, we need to apply a further constraint: This one is a smoothness parameter. From the fact that nature prefers smooth solutions, we stipulate that the obtained result should have a certain smoothness. With smooth, we mean the opposite of very many, very sharp, very thin peaks. Instead fewer, not-so-sharp, broader peaks are what we prefer as outcome.

This can mathematically be handled with an addition of the second derivative to the equation we want to solve. Metaphorically, we then solve this for various situations depending on whether we force the solution to be very smooth, or not so smooth. The parameter that determines this is a factor (in front of the second derivative) with the name regularizer. It is a strange factor, and yet a very important one: if you select a different regularizer you obtain a different result. The key then is to find a suitable regularizer. [Another term for this parameter is also the alpha parameter, or α-parameter.]

General Purpose – one parameter fits all

Well almost all. The default regularizer (“General Purpose Algorithm”) in the Zetasizer software is such, that it fits a very wide range of experimental situations: it should give a relatively smooth distribution, yet also show separate peaks when they are present (and far enough apart = away from each other). Details of the mathematics are in an older white paper on DLS Deconvolution Algorithms. Other potential algorithms and choices in the field are Multiple Narrow Mode, L-curve and historically CONTIN, DYNALS, REPES, etc.

Improve even more with MADLS

An interesting concept to expand the resolving power of dynamic light scattering is to simultaneously look at the result from several angles (multi-angle dynamic light scattering or MADLS). This can under ideal conditions lead to an improved resolvability, in some cases peaks as close as 2:1 can be separated with the algorithm. The idea here is that one uses a regularization not only on a single data set (single correlation function) but on more data sets (three correlation functions) with physically realistic weighting (according to Mie theory) built in. And that brings us back to the image at the top of this blog, where the three correlation functions lead to a single size distribution.

But that is probably more than you wanted to read about regularization, the math behind inversions, ill-posed problems and complicated mathematics anyway. The latest Zetasizer generation and software has this built in, so no need to do this in a spreadsheet.

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Have any questions? Please email me ulf.nobbmann@malvernpanalytical.com – Thanks! Opinions are those of the author. Our editorial team modifies them occasionally.