Integration with Diagrammatic Monte-Carlo

A very simple yet instructive example for applying Diagrammatic Monte-Carlo to solve an oscillating integral numerically.

The main topic of my PhD thesis is a discussion of applying the methods of Diagrammatic Monte Carlo (DiagMC) and Bold Diagrammatic Monte Carlo (BDMC) to Quantum Field theories, such as QED or QCD.

There are several reasons why this could work under certain conditions, however, there are also reasons why this is not a good idea. In the end the thesis will give hints on why it does not work and what would be a proper way to do it. Right now it seems that the way to do it, might also require automatic regularization by discretization using a lattice. This makes using diagrammatic methods practically worthless, as lattice QCD (LQCD) has already a quite strong community with many achievements. The scaling would then also be not beneficial resulting in an algorithm that is not as mature and does not contain advantageous properties compared to usual LQCD methods.

A subset of my thesis is the presentation and discussion of applications for DiagMC / BDMC. A simple example for BDMC is the S-wave scattering. This example is already online. One could have picked any integral equation instead. An instructive example for DiagMC is even simpler. Here we just use a simple integral that features high oscillation depending on some parameter.

Why is this example so interesting? First, we can study how the method behaves for highly oscillating integrals. In the end we want to study theories with actions that imply high oscillations (for example when introducing a finite chemical potential). The question is if only the BDMC algorithm can treat such oscillations properly, or if already DiagMC is already sanitizing the result.

Second, we can basically design the diagrammatic space. For this example we stayed with the given "first order diagram", which is just the integral. Additionally, as with the S-wave example, we introduce an artificial zero order diagram, which is just a constant. This makes normalization and measuring quite easy, and let's use run the simulation. With just a single level we cannot use update steps like adding or removing a diagram. However, those are the only two update steps we are interested in (at least in this simple example). One might introduce additional steps like changing the variable value, but as this is the (only) update step of MC integration, we'd like to distinguish our method from MC integration.

Finally, we can of course learn something about general features like auto-correlation, error estimation and other important quantities for statistics.

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