Multi-resolution approximation to the stochastic inverse problem

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 ration al functions of a self-adjoint elliptic differential operator on L-2(R-d), th, ill-posed nature of the problem disappears when such operators are defi ned between appropriate fractional Sobolev spaces. In this paper, we exploi t this fact to reconstruct the input random field from the orthogonal expan sion (i.e. with uncorrelated coefficients) derived for the output random fi eld 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 t he integral transformation of the orthogonal expansion derived for the inpu t random field, in terms of an orthonormal wavelet basis, are studied. AMS 1991 Subject Classification: Primary 60G60; 60G12 Secondary 60G25; 60H15.