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Subsequent extensions to unperturbed three-body and four-body regularization proved beneficial in dealing with multiple interactions. Fortuitously, the elegant Kustaanheimo-Stiefel two-body regularization soon permitted small star clusters to be studied (NBODY3).

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The first code (NBODY1) was based on the simple concepts of a force polynomial combined with individual time steps, where numerical problems due to close encounters were avoided by a softened potential. I review the development of direct N-body codes at Cambridge over nearly 40 years, highlighting the main stepping stones. Ultimately, such a method applied to strong lensing clusters with a very large number of multiple images may provide unique geometrical constraints on cosmology. We confirm the ability of strong lensing to constrain the mass profile in the central region of galaxy clusters in this way.

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Finally, we describe how to rank models quantitatively using the Bayesian evidence. The mass of the galaxies is generally recovered with a 20% error, largely due to the poorly constrained cut-off radius. However, we find that the mass of the galaxies is strongly degenerated with the cluster mass when no multiple images appear in the cluster centre. We find that, for typical Hubble Space Telescope (HST)-quality imaging data, the total mass in the Einstein radius is recovered with ~1-5% error according to the considered lensing configuration. We ray-trace three light beams through each model to produce a catalogue of multiple images, and then use the MCMC sampler to recover the model parameters in the three different lensing configurations. To illustrate the power of the MCMC technique, we simulate three clusters of galaxies, each composed of a cluster-scale halo and a set of perturbing galaxy-scale subhalos. We use a publicly available Markov chain Monte-Carlo (MCMC) sampler ('Bayesys'), allowing us to avoid local minima in the likelihood functions. We describe a procedure for modelling strong lensing galaxy clusters with parametric methods, and to rank models quantitatively using the Bayesian evidence. In the former cases, we show that PBL is better able to identify cusps and substructures than are grid-based reconstructions, and in the latter case, we show that PBL is able to identify substructure in the Bullet Cluster without even exploiting strong lensing measurements. In this paper, however, we describe a shear-only reconstruction, and apply the method to several test cases, including simulated lensing clusters, as well as the well-studied ``Bullet Cluster'' (1E0657-56). PBL is designed to include all lensing observables, including multiple image positions and fluxes from strong lensing, as well as weak lensing signals including shear and flexion. In this way, we can tune a reconstruction to produce constant signal-noise throughout, and maximally exploit regions of high information density. In PBL, we treat each of the lensed galaxies as a particle and then reconstruct the potential by smoothing over a local kernel with variable smoothing scale. We approach the problem from a different perspective, motivated by the success of multi-scale analysis in smoothed particle hydrodynamics.

Traditionally, most methods have employed either a finite inversion or gridding to turn observational lensed galaxy ellipticities into an estimate of the surface mass density of a galaxy cluster. We present Particle-Based Lensing (PBL), a new technique for gravitational lensing mass reconstructions of galaxy clusters.

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