6 research outputs found
Parameter estimation for semilinear SPDEs from local measurements
This work contributes to the limited literature on estimating the diffusivity
or drift coefficient of nonlinear SPDEs driven by additive noise. Assuming that
the solution is measured locally in space and over a finite time interval, we
show that the augmented maximum likelihood estimator introduced in Altmeyer,
Reiss (2020) retains its asymptotic properties when used for semilinear SPDEs
that satisfy some abstract, and verifiable, conditions. The proofs of
asymptotic results are based on splitting the solution in linear and nonlinear
parts and fine regularity properties in -spaces. The obtained general
results are applied to particular classes of equations, including stochastic
reaction-diffusion equations. The stochastic Burgers equation, as an example
with first order nonlinearity, is an interesting borderline case of the general
results, and is treated by a Wiener chaos expansion. We conclude with numerical
examples that validate the theoretical results.Comment: corrected versio
Data assimilation: the Schrödinger perspective
Data assimilation addresses the general problem of how to combine model-based predictions with partial and noisy observations of the process in an optimal manner. This survey focuses on sequential data assimilation techniques using probabilistic particle-based algorithms. In addition to surveying recent developments for discrete- and continuous-time data assimilation, both in terms of mathematical foundations and algorithmic implementations, we also provide a unifying framework from the perspective of coupling of measures, and Schrödinger’s boundary value problem for stochastic processes in particular
Rough McKean-Vlasov dynamics for robust ensemble Kalman filtering
Motivated by the challenge of incorporating data into misspecified and multiscale dynamical models, we
study a McKean-Vlasov equation that contains the data stream as a common driving rough path. This setting
allows us to prove well-posedness as well as continuity with respect to the driver in an appropriate rough-path
topology. The latter property is key in our subsequent development of a robust data assimilation methodology:
We establish propagation of chaos for the associated interacting particle system, which in turn is suggestive of
a numerical scheme that can be viewed as an extension of the ensemble Kalman filter to a rough-path framework.
Finally, we discuss a data-driven method based on subsampling to construct suitable rough path lifts and
demonstrate the robustness of our scheme in a number of numerical experiments related to parameter estimation
problems in multiscale contexts
State and Parameter Estimation from Observed Signal Increments
The success of the ensemble Kalman filter has triggered a strong interest in expanding its scope beyond classical state estimation problems. In this paper, we focus on continuous-time data assimilation where the model and measurement errors are correlated and both states and parameters need to be identified. Such scenarios arise from noisy and partial observations of Lagrangian particles which move under a stochastic velocity field involving unknown parameters. We take an appropriate class of McKean−Vlasov equations as the starting point to derive ensemble Kalman−Bucy filter algorithms for combined state and parameter estimation. We demonstrate their performance through a series of increasingly complex multi-scale model systems