A fully symmetric nonlinear biorthogonal decomposition theory for random fields

We present a general approach for nonlinear biorthogonal decomposition of random fields. The mathematical theory is developed based on a fully symmetric operator framework that unifies different types of expansions and allows for a simple formulation of necessary and sufficient conditions for their completeness. The key idea of the method relies on an equivalence between nonlinear mappings of Hilbert spaces and local inner products, i.e. inner products that may be functionals of the random field being decomposed. This extends previous work on the subject and allows for an effective formulation of field-dependent and field-independent representations. The proposed new methodology can be applied in many areas of mathematical physics, for stochastic low-dimensional modelling of partial differential equations and dimensionality reduction of complex nonlinear phenomena. An application to a transient stochastic heat conduction problem in a one-dimensional infinite medium is presented and discussed.