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In this paper is proposed a novel incremental iterativeGauss-Newton-Markov-Kalman filter method for state estimation of dynamic modelsgiven noisy measurements. The mathematical formulation of the proposed filteris based on the construction of an optimal nonlinear map between the observableand parameter (state) spaces via a convergent sequence of linear maps obtainedby successive linearisation of the observation operator in a Gauss-Newton-likeform. To allow automatic linearisation of the dynamical system in a sparseform, the smoother is designed in a hierarchical setting such that the forwardmap and its linearised counterpart are estimated in a Bayesian manner given aforecasted data set. To improve the algorithm convergence, the smoother isfurther reformulated in its incremental form in which the current andintermediate states are assimilated before the initial one, and thecorresponding posterior estimates are taken as pseudo-measurements. As thelatter ones are random variables, and not deterministic any more, the novelstochastic iterative filter is designed to take this into account. To correctthe bias in the posterior outcome, the procedure is built in apredictor-corrector form in which the predictor phase is used to assimilatenoisy measurement data, whereas the corrector phase is constructed to correctthe mean bias. The resulting filter is further discretised via time-adaptingsparse polynomial chaos expansions obtained either via modified Gram-Schmidtorthogonalisation or by a carefully chosen nonlinear mapping, both of which areestimated in a Bayesian manner by promoting the sparsity of the outcomes. Thetime adaptive basis with non-Gaussian arguments is further mapped to thepolynomial chaos one by a suitably chosen isoprobabilistic transformation.Finally, the proposed method is tested on a chaotic nonlinear Lorenz 1984system.

Stochastic state estimation via incremental iterative sparse polynomial chaos based Bayesian-Gauss-Newton-Markov-Kalman filter