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Multi-resolution approximation to the stochastic inverse problem

Part of: Stochastic processes Stochastic analysis

Published online by Cambridge University Press:  01 July 2016

J. M. Angulo  and
M. D. Ruiz-Medina
J. M. Angulo*
Affiliation:
University of Granada
M. D. Ruiz-Medina*
Affiliation:
University of Granada
*
∗ Postal address: Departamento de Estadística e Investigación Operativa, Facultad de Ciencias, Campus Fuentenueva s/n, Universidad de Granada, E-18071 Granada, Spain.
∗ Postal address: Departamento de Estadística e Investigación Operativa, Facultad de Ciencias, Campus Fuentenueva s/n, Universidad de Granada, E-18071 Granada, Spain.
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Abstract

The linear inverse problem of estimating the input random field in a first-kind stochastic integral equation relating two random fields is considered. For a wide class of integral operators, which includes the positive rational functions of a self-adjoint elliptic differential operator on L2(ℝd), the ill-posed nature of the problem disappears when such operators are defined between appropriate fractional Sobolev spaces. In this paper, we exploit this fact to reconstruct the input random field from the orthogonal expansion (i.e. with uncorrelated coefficients) derived for the output random field in terms of wavelet bases, transformed by a linear operator factorizing the output covariance operator. More specifically, conditions under which the direct orthogonal expansion of the output random field coincides with the integral transformation of the orthogonal expansion derived for the input random field, in terms of an orthonormal wavelet basis, are studied.

Keywords

Covariance factorizationmean-square linear predictionmulti-resolution approximationorthogonal expansionorthonormal wavelet basisstochastic inverse problem

MSC classification

Primary: 60G60: Random fields 60G12: General second-order processes
Secondary: 60G25: Prediction theory 60H15: Stochastic partial differential equations
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 31 , Issue 4 , December 1999 , pp. 1039 - 1057
Copyright
Copyright © Applied Probability Trust 1999 

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