We analyse the problem of inference about a latent signal governing the dynamics of a system given only the observed noisy data. We adopt the discrete-time state space approach due to the wide range of problems it can capture. Because in general no closed-form solution are available in this framework, we discuss the class of methods used for approximating of the posterior state distributions, called Sequential Monte Carlo. These methods are based on the Dirac-measures which stem from the draws (particles) from the distribution constructed in the previous iteration. A special attention is devoted to the filtering problem, where one is interested in the estimation of the current state of the system given the current system measurements. We derive theoretical forms of the particle filters, which we then use to construct algorithms suitable for numerical analysis. We discuss the degeneracy problem, inherent to the sequential importance sampling and selected methods to tackle it. The basic convergence results in the context of particle filters are presents. Finally, we consider three numerical application.

Reconfiguration of a combinatorial problem is the application of small transformations to a solution of the problem in a way that preserves the property of being a solution. This thesis studies how the structure of the underlying problem can be used to find sequences of such transformations. A more general view on the reconfiguration of any problem described by a homomorphism allows to formally describe certain patterns and helps to understand exceptions known before. We show that among parameters that describe the sparsity of a graph, only bounded treedepth leads to effective reconfiguration algorithms. In contrast, we give an algorithm which uses the special structure of the so-called claw-free graphs to find reconfiguration sequences between independent sets. Finally, for the reconfiguration of 3-colorings, which somewhat surprisingly has been shown easier than the problem of finding 3-colorings, we give a new proof of this fact which highlights the role of one assumption that suffices to carry out most of the proof in the more general context of homomorphisms.