Sampling-free linear Bayesian update of polynomial chaos representations

Highlights: ► We cast the probabilistic identification problem in a linear Bayesian setting based on a functional approximation. ► The update procedure does not involve sampling at any stage of the computation. ► The method can handle non-Gaussian random variables. ► It is applicable to nonlinear systems. ► The method is compared with the Ensemble Kalman Filter (EnKF). - Abstract: We present a fully deterministic approach to a probabilistic interpretation of inverse problems in which unknown quantities are represented by random fields or processes, described by possibly non-Gaussian distributions. The description of the introduced random fields is given in a "white noise" framework, which enables us to solve the stochastic forward problem through Galerkin projection onto polynomial chaos. With the help of such a representation the probabilistic identification problem is cast in a polynomial chaos expansion setting and the Baye's linear form of updating. By introducing the Hermite algebra this becomes a direct, purely algebraic way of computing the posterior, which is comparatively inexpensive to evaluate. In addition, we show that the well-known Kalman filter is the low order part of this update. The proposed method is here tested on a stationary diffusion equation with prescribed source terms, characterised by an uncertain conductivity parameter which is then identified from limited and noisy data obtained by a measurement of the diffusing quantity.