Direct Bayesian Update of Polynomial Chaos Representations

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 a non-Gaussian prior distribution. 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 representation, the probabilistic identification problem is cast in a polynomial chaos expansion setting and the linear Bayesian form of updating. This representation leads to a corresponding new formulation of the minimum squared error estimator, obtained by its additional projection onto the polynomial chaos basis. By introducing the Hermite algebra this becomes a direct, purely algebraic way of computing the posterior, which is inexpensive to evaluate. In addition, we show that the well-known Kalman filter method is the low order part of this update. The proposed method has been 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.