Soft matter encompasses a wide variety of materials that are composed of mesoscopic particles in solution. The numerical study of such materials is complicated by the difference in length-scales between an atomistic, statistical mechanical model of the material and its mesoscopic structures. To facilitate simulations in such situations, one often relies on a coarse-grained description that integrates out microscopic degrees of freedom, yielding a model that is composed of mesoscopic particles with effective interactions. In most cases, this coarse-graining procedure is not exact, resulting in a biased model whose errors are difficult to assess.

In this thesis, we investigate a multilevel simulation strategy that leverages one or multiple levels of coarse graining to simulate the exact physical system of interest. This approach is inspired by the multilevel Monte Carlo method, a numerical technique in stochastic simulation typically applied to stochastic differential equations or random-coefficient partial differential equations which has seen a great deal of applications in the last decade. The idea of multilevel Monte Carlo is to estimate expectations of a system with a given target error tolerance by combining a hierarchy of resolutions, typically given by levels of discretisation of some differential equation, to efficiently estimate expectations at the finest resolution needed to reach the required accuracy.

Following this idea, we develop a two- and multilevel sequential Monte Carlo method for multi-scale physical systems which avoids the mixing problems associated with differences in length scales that make direct simulations infeasible. We use this method to investigate soft matter systems. First, we consider two models for colloid-polymer mixtures, a size-asymmetric binary hard-sphere mixture and the Asakura-Oosawa model. Using a two-level method, we demonstrate the first numerical evidence of a fluid-fluid phase separation in the binary hard-sphere mixture. Second, we develop a proof-of-concept multilevel algorithm for semidilute polymer solutions. Additionally, we present a detailed analysis of our algorithm and its asymptotic properties, including a broadly applicable central limit theorem.