A random map implementation of implicit filters

Abstract

Implicit particle filters for data assimilation generate high-probability samples by representing each particle location as a separate function of a common reference variable. This representation requires that a certain underdetermined equation be solved for each particle and at each time an observation becomes available. We present a new implementation of implicit filters in which we find the solution of the equation via a random map. As examples, we assimilate data for a stochastically driven Lorenz system with sparse observations and for a stochastic Kuramoto-Sivashinski equation with observations that are sparse in both space and time.

A Survey of Implicit Particle Filters for Data Assimilation

“…Here we review the approach presented in [28] which solves (5) by the random change of variables (random map)…”

“…We refer to [28] for the details of this calculation, however note that the Jacobian is easy to evaluate, since the scalar derivative ∂ λ j /∂ ρ j can be computed efficiently by either using finite differences, or by evaluating…”

“…If we choose L j such that Σ = L T j L j , we find λ j = √ ρ j and J = | det L j |. This Jacobian is constant and need not be computed, and with this choice of L j , we sample the optimal importance function, by solving (5) with the random map (19), see [28].…”

“…This example is taken from [28]. We follow [42][43][44] and consider the stochastic Lorenz attractor [45] …”

“…In what follows, we explain how implicit particle filters [25][26][27][28][29] tackle this problem. The basic idea is to first look for regions of high probability in the target density and then focus the particles onto these regions, so that only particles with significant weights are generated and the number of particles required remains manageable even if the state dimension is large.…”