Dynamic Data-Driven Adaptive Observations in Data Assimilation for Multi-scale Systems

Rough McKean-Vlasov dynamics for robust ensemble Kalman filtering

“…Similar results in a rough-path context can be found in [69]. The homogenised dynamics (101) encapsulates the rugged landscape described by p in the diffusion-mass matrix K. Like in the previous experiments, we consider the task of estimating the parameter θ from noisy observations of (100), using the EnKF and RP-EnKF based on (101). We choose V (z) = 1 2 |z| 2 , p1(x) = cos(x), p2(x) = 1 2 cos(x) and σ = 1.…”

“…We would also like to mention the works [38,57] that allow treating the Zakai SPDE (governing the unnormalised filtering distribution [10, Section 3.5]) in a rough paths framework, however noticing that the numerical treatment of SPDEs is faced with enormous challenges, in particular in high-dimensional settings. Some other works addressing issues in robust or multiscale filtering include [3,4] (assuming uncertainty in the coefficients) as well as a sequence of works by N. Perkowski and coworkers in the context of averaging and homogenisation [14,12,13,59,60,73,72,101,102].…”

“…Data from the two-scale dynamics (100) is simulated for ε = 10 −2 , θtrue = 1 using an Euler-Maruyama discretisation with time step ∆t = 10 −4 . With the same values for θtrue, ∆t and σ, we simulate data from the reduced model (101), where K ≈ diag(0.62386, 0.884176) has been obtained by numerical integration in (102). Both ( 100) and ( 101) are perturbed by noise according to (98) with R = 10 −2 and, as in Sections 6.1 and 6.2, the resulting observation paths are used in the EnKF-and RP-EnKF schemes (see equations ( 17) and ( 14)) to estimate θtrue.…”