1,108 research outputs found
Semivariogram methods for modeling Whittle-Mat\'ern priors in Bayesian inverse problems
We present a new technique, based on semivariogram methodology, for obtaining
point estimates for use in prior modeling for solving Bayesian inverse
problems. This method requires a connection between Gaussian processes with
covariance operators defined by the Mat\'ern covariance function and Gaussian
processes with precision (inverse-covariance) operators defined by the Green's
functions of a class of elliptic stochastic partial differential equations
(SPDEs). We present a detailed mathematical description of this connection. We
will show that there is an equivalence between these two Gaussian processes
when the domain is infinite -- for us, -- which breaks down when
the domain is finite due to the effect of boundary conditions on Green's
functions of PDEs. We show how this connection can be re-established using
extended domains. We then introduce the semivariogram method for estimating the
Mat\'ern covariance parameters, which specify the Gaussian prior needed for
stabilizing the inverse problem. Results are extended from the isotropic case
to the anisotropic case where the correlation length in one direction is larger
than another. Finally, we consider the situation where the correlation length
is spatially dependent rather than constant. We implement each method in
two-dimensional image inpainting test cases to show that it works on practical
examples
Analysis of the Gibbs sampler for hierarchical inverse problems
Many inverse problems arising in applications come from continuum models
where the unknown parameter is a field. In practice the unknown field is
discretized resulting in a problem in , with an understanding
that refining the discretization, that is increasing , will often be
desirable. In the context of Bayesian inversion this situation suggests the
importance of two issues: (i) defining hyper-parameters in such a way that they
are interpretable in the continuum limit and so that their
values may be compared between different discretization levels; (ii)
understanding the efficiency of algorithms for probing the posterior
distribution, as a function of large Here we address these two issues in
the context of linear inverse problems subject to additive Gaussian noise
within a hierarchical modelling framework based on a Gaussian prior for the
unknown field and an inverse-gamma prior for a hyper-parameter, namely the
amplitude of the prior variance. The structure of the model is such that the
Gibbs sampler can be easily implemented for probing the posterior distribution.
Subscribing to the dogma that one should think infinite-dimensionally before
implementing in finite dimensions, we present function space intuition and
provide rigorous theory showing that as increases, the component of the
Gibbs sampler for sampling the amplitude of the prior variance becomes
increasingly slower. We discuss a reparametrization of the prior variance that
is robust with respect to the increase in dimension; we give numerical
experiments which exhibit that our reparametrization prevents the slowing down.
Our intuition on the behaviour of the prior hyper-parameter, with and without
reparametrization, is sufficiently general to include a broad class of
nonlinear inverse problems as well as other families of hyper-priors.Comment: to appear, SIAM/ASA Journal on Uncertainty Quantificatio
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