This chapter is devoted to the optimal particle filter (OPF). Like the bootstrap particle filter (BPF) from the previous chapter, the OPF approximates the filtering distribution by a sum of Dirac masses. But while the BPF is conceptually derived by factorizing the update of the filtering distribution into a prediction and an analysis step, the OPF uses a different factorization which can result in improved performance.

Recall the stochastic dynamics and data models introduced in the previous Chapter.

In this chapter we introduce data assimilation problems in which the model of interest, and the data associated with it, have a time-ordered nature.We distinguish between the filtering problem (on-line) in which the data is incorporated sequentially as it comes in, and the smoothing problem (off-line) which is a specific instance of the inverse problems that have been the subject of the preceding chapters.

In this chapter we study the linear-Gaussian setting, where the forward model (·)is linear and both the prior on 𝑢 and the distribution of the observation noise 𝜂 are Gaussian. This setting is highly amenable to analysis and arises frequently in applications. Moreover, as we will see throughout these notes, many methods employed in nonlinear or non-Gaussian settings build on ideas from the linear- Gaussian case by performing linearization or invoking Gaussian approximations.

In this chapterwe describe the Extended Kalman Filter (ExKF)1 and the Ensemble Kalman Filter (EnKF). The ExKF approximates the predictive covariance by linearization, while the EnKF approximates it by the empirical covariance of a collection of particles. The ExKF is a provably accurate approximation of the filtering distribution if the dynamics are approximately linear and small noise is present in both signal and data, in which case the filtering distribution is well approximated by a Gaussian.

In this chapter we explore the properties of Bayesian inversion from the perspective of an optimization problem which corresponds to maximizing the posterior probability; that is, to finding a maximum a posteriori (MAP) estimator, or mode of the posterior distribution. We demonstrate the properties of the point estimator resulting from this optimization problem, showing its positive and negative attributes, the latter motivating our work in the following three chapters. We also introduce, and study, basic gradient-based optimization algorithms.

In this chapter we introduce Monte Carlo sampling and importance sampling. These are two general techniques for estimating expectations with respect to a given pdf π. Monte Carlo generates independent samples from π and combines them with equal weights, whilst importance sampling uses independent samples, weighted appropriately, from a different distribution. In quantifying the error in Monte Carlo and importance sampling, we will use a distance on random probability measures that reduces to total variation in the case of deterministic probability measures; and we will introduce the χ2 divergence.

In this chapter we introduce the Bayesian approach to inverse problems in which the unknown parameter and the observed data are viewed as random variables. In this probabilistic formulation, the solution of the inverse problem is the posterior distribution on the parameter given the data. We will show that the Bayesian formulation leads to a form of well-posedness: small perturbations of the forward model or the observed data translate into small perturbations of the posterior distribution. Well-posedness requires a notion of distance between probability measures. We introduce the total variation and Hellinger distances, giving characterizations of them, and bounds relating them, that will be used throughout these notes. We prove well-posedness in the Hellinger distance.

The aim of these notes is to provide a clear and concise mathematical introduction to the subjects of Inverse Problems and Data Assimilation, and their interrelations, together with bibliographic pointers to literature in this area that goes into greater depth. The target audiences are advanced undergraduates and beginning graduate students in the mathematical sciences, together with researchers in the sciences and engineering who are interested in the systematic underpinnings of methodologies widely used in their disciplines.

In this chapter we introduce the Bayesian approach to inverse problems in which the unknown parameter and the observed data are viewed as random variables. In this probabilistic formulation, the solution of the inverse problem is the posterior distribution on the parameter given the data. We will show that the Bayesian formulation leads to a form of well-posedness: small perturbations of the forward model or the observed data translate into small perturbations of the posterior distribution. Well-posedness requires a notion of distance between probability measures. We introduce the total variation and Hellinger distances, giving characterizations of them, and bounds relating them, that will be used throughout these notes. We prove well-posedness in the Hellinger distance.

This chapter demonstrates the use of optimization, namely the 3DVAR and 4DVAR methodologies, to obtain information from the filtering and smoothing distributions. We emphasize that the methods we present in this chapter do not provide approximations of the filtering and smoothing distributions; they simply provide estimates of the signal, given data, in the filtering (on-line) and smoothing (off-line) data scenarios.

This chapter is devoted to the particle filter, a method that approximates the filtering distribution by a sum of Dirac masses. Particle filters provably converge to the filtering distribution as the number of particles, and hence the number of Dirac masses, approaches infinity. We focus on the bootstrap particle filter (BPF), also known as sequential importance resampling; it is linked to the material on Monte Carlo and importance sampling described in Chapter 5.