Joint state-parameter estimation of a nonlinear stochastic energy balance model from sparse noisy data
For researchers in spatiotemporal modeling and paleoclimate reconstruction, this work provides a regularized Bayesian approach to handle ill-posed inverse problems with sparse data, though parameter estimation remains challenging.
The paper addresses joint parameter-state estimation in nonlinear stochastic PDEs from sparse noisy data, using a regularized posterior with physical priors. In a paleoclimate energy balance model, the method yields state estimates near the truth with reduced uncertainty, but parameter estimates show large uncertainty due to ill-posedness.
While nonlinear stochastic partial differential equations arise naturally in spatiotemporal modeling, inference for such systems often faces two major challenges: sparse noisy data and ill-posedness of the inverse problem of parameter estimation. To overcome the challenges, we introduce a strongly regularized posterior by normalizing the likelihood and by imposing physical constraints through priors of the parameters and states. We investigate joint parameter-state estimation by the regularized posterior in a physically motivated nonlinear stochastic energy balance model (SEBM) for paleoclimate reconstruction. The high-dimensional posterior is sampled by a particle Gibbs sampler that combines MCMC with an optimal particle filter exploiting the structure of the SEBM. In tests using either Gaussian or uniform priors based on the physical range of parameters, the regularized posteriors overcome the ill-posedness and lead to samples within physical ranges, quantifying the uncertainty in estimation. Due to the ill-posedness and the regularization, the posterior of parameters presents a relatively large uncertainty, and consequently, the maximum of the posterior, which is the minimizer in a variational approach, can have a large variation. In contrast, the posterior of states generally concentrates near the truth, substantially filtering out observation noise and reducing uncertainty in the unconstrained SEBM.