A fully symmetric nonlinear biorthogonal decomposition theory for random fields

• Published in: Physica. D Vol. 240; no. 4; pp. 415 - 425
• Main Author: Venturi, Daniele
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 15.02.2011; Elsevier
• Subjects: Completeness; Decomposition; Exact sciences and technology; Hilbert space; Mathematical models; Nonlinear biorthogonal decomposition; Nonlinear phenomena; Nonlinearity; Physics; Random fields; Reduced-order stochastic modelling; Reduction; Representations; Stochasticity; Reduced-order stochastic modelling; Random fields; Nonlinear biorthogonal decomposition; Operator; Dimensionality; Partial differential equations; Necessary and sufficient condition; Non linear phenomenon; Random field; Equivalence; Functionals; Space of mappings; Non linear theory; Inner product; Hilbert space; Infinite dimension; Mathematical physics; Thermal conduction
• ISSN: 0167-2789; 1872-8022
• DOI: 10.1016/j.physd.2010.10.005

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. ► Nonlinear biorthogonal decomposition of random fields.► Dimensionality reduction and representation of complex nonlinear phenomena.► Stochastic low-dimensional modelling of partial differential equations.► Field-dependent and field-independent representations of random spaces.