Discretization-invariant Bayesian inversion and Besov space priors

By Matti Lassas. Eero Saksman, Samuli Siltanen. 2009

203 citations recorded by the source corpus.

Bayesian solution of an inverse problem for indirect measurement $M = AU + {\mathcal{E}}$ is considered, where $U$ is a function on a domain of $R^d$. Here $A$ is a smoothing linear operator and $ {\mathcal{E}}$ is Gaussian white noise. The data is a realization $m_k$ of the random variable $M_k = P_kA U+P_k {\mathcal{E}}$, where $P_k$ is a linear, finite dimensional operator related to measurement device. To allow computerized inversion, the unknown is discretized as $U_n=T_nU$, where $T_n$ is a finite dimensional projection, leading to the computational measurement model $M_{kn}=P_k A U_n + P_k {\mathcal{E}}$. Bayes formula gives then the posterior distribution $\pi_{kn}(u_n | m_{kn})\sim\pi_n(u_n) \exp(-{1/2}\|m_{kn} - P_kA u_n\|_2^2)$ in $R^d$, and the mean $U^{CM}_{kn}:=\int u_n \pi_{kn}(u_n | m_k) du_n$ is considered as the reconstruction of $U$. We discuss a systematic way of choosing prior distributions $\prior_n$ for all $n\geq n_0>0$ by achieving them as projections of a distribution in a infinite-dimensional limit case. Such choice of prior distributions is {\em discretization-invariant} in the sense that $\prior_n$ represent the same {\em a priori} information for all $n$ and that the mean $U^{CM}_{kn}$ converges to a limit estimate as $k,n\to\infty$. Gaussian smoothness priors and wavelet-based Besov space priors are shown to be discretization invariant. In particular, Bayesian inversion in dimension two with $B^1_{11}$ prior is related to penalizing the $\ell^1$ norm of the wavelet coefficients of $U$.

Subfields: Computational Mechanics, Computer Vision and Pattern Recognition

View Paper: https://www.semanticscholar.org/paper/78cf9fad267d156060096d153b549f6f42185109

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Citations in the corpus

Spectral gaps for a Metropolis–Hastings algorithm in infinite dimensions (2011, background): “Other classes of target measure, such as those arising from Besov prior measures [28, 14], or the uniform measures in [47], would also provide interesting applications.”
A Computational Framework for Infinite-Dimensional Bayesian Inverse Problems Part I: The Linearized Case, with Application to Global Seismic Inversion (2013, unknown): “If discretization is not performed carefully at each step, it is unlikely that the discrete solutions will converge to the desired infinite-dimensional solution in a meaningful way [32, 42].”
Inverse problems: A Bayesian perspective (2010, background): “…problems on function space, including the early study by Franklin (1970), and the subsequent papers by Mandelbaum (1984), Lehtinen, Paivarinta and Somersalo (1989) and Fitzpatrick (1991); the paper by Lassas et al. (2009) contains a good literature review of this material, and further references.”
Optimal Scaling and Diffusion Limits for the Langevin Algorithm in High Dimensions (2011, methodology): “Particularly, the Besov priors of [18] provide an interesting class of such reference measures, and the paper [13] provides a machinery for analyzing change of measure from the Besov prior, analogous to that used here in the Gaussian case.”
Hyperpriors for Matérn fields with applications in Bayesian inversion (2016, methodology): “Constructions of proposed non-Gaussian priors, which do not converge to Gaussian fields in the discretisation limit, include the Besov space priors (Lassas et al. 2009) [23], which are constructed on a wavelet basis, hierarchical Mumford-Shah priors (Helin and Lassas, 2011) [18], and recently…”
The Bayesian Approach to Inverse Problems (2013, methodology): “There is some current interest in extending prior measures beyond the Gaussian case and a particularly promising avenue is the develpment of so-called Besov priors, tuned to wavelet representation of functions, and introduced recently in [LSS09]; see [DHS] for further analaysis of these priors, and ”