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...
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| Published in | Physica. D Vol. 240; no. 4; pp. 415 - 425 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Amsterdam
Elsevier B.V
15.02.2011
Elsevier |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0167-2789 1872-8022 |
| DOI | 10.1016/j.physd.2010.10.005 |
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| Abstract | 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. |
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| AbstractList | 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. 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. |
| Author | Venturi, Daniele |
| Author_xml | – sequence: 1 givenname: Daniele surname: Venturi fullname: Venturi, Daniele email: daniele.venturi@mail.ing.unibo.it organization: Department of Energy, Nuclear and Environmental Engineering, University of Bologna, Italy |
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| Keywords | 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 |
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| Snippet | We present a general approach for nonlinear biorthogonal decomposition of random fields. The mathematical theory is developed based on a fully symmetric... |
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| SubjectTerms | 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 |
| Title | A fully symmetric nonlinear biorthogonal decomposition theory for random fields |
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