arXiv subjects

Sari Lasanen

Publications and source records attributed to Sari Lasanen.

8 recordsLinked to original sources

arXiv2020

Non-Stationary Multi-layered Gaussian Priors for Bayesian Inversion

In this article, we study Bayesian inverse problems with multi-layered Gaussian priors. We first describe the conditionally Gaussian layers in terms of a system of stochastic partial differential equations. We build the computational inference method using a finite-dimensional Galerkin method. We show that the proposed approximation has a convergence-in-probability property to the solution of the original multi-layered model. We then carry out Bayesian inference using the preconditioned Crank--Nicolson algorithm which is modified to work with multi-layered Gaussian fields. We show via numerical experiments in signal deconvolution and computerized X-ray tomography problems that the proposed method can offer both smoothing and edge preservation at the same time.

math.ST↗

arXiv2020

Enhancing Industrial X-ray Tomography by Data-Centric Statistical Methods

X-ray tomography has applications in various industrial fields such as sawmill industry, oil and gas industry, chemical engineering, and geotechnical engineering. In this article, we study Bayesian methods for the X-ray tomography reconstruction. In Bayesian methods, the inverse problem of tomographic reconstruction is solved with help of a statistical prior distribution which encodes the possible internal structures by assigning probabilities for smoothness and edge distribution of the object. We compare Gaussian random field priors, that favour smoothness, to non-Gaussian total variation, Besov, and Cauchy priors which promote sharp edges and high-contrast and low-contrast areas in the object. We also present computational schemes for solving the resulting high-dimensional Bayesian inverse problem with 100,000-1,000,000 unknowns. In particular, we study the applicability of a no-U-turn variant of Hamiltonian Monte Carlo methods and of a more classical adaptive Metropolis-within-Gibbs algorithm for this purpose. These methods also enable full uncertainty quantification of the reconstructions. For faster computations, we use maximum a posteriori estimates with limited-memory BFGS optimisation algorithm. As the first industrial application, we consider sawmill industry X-ray log tomography. The logs have knots, rotten parts, and even possibly metallic pieces, making them good examples for non-Gaussian priors. Secondly, we study drill-core rock sample tomography, an example from oil and gas industry. We show that Cauchy priors produce smaller number of artefacts than other choices, especially with sparse high-noise measurements, and choosing Hamiltonian Monte Carlo enables systematic uncertainty quantification.

cs.CE↗

arXiv2019

Posterior Convergence Analysis of $α$-Stable Sheets

This paper is concerned with the theoretical understanding of $α$-stable sheets $U$ on $\mathbb{R}^d$. Our motivation for this is in the context of Bayesian inverse problems, where we consider these processes as prior distributions, aiming to quantify information of the posterior. We derive convergence results referring to finite-dimensional approximations of infinite-dimensional random variables. In doing so we use a number of variants which these sheets can take, such as a stochastic integral representation, but also random series expansions through Poisson processes. Our proofs will rely on the fact of whether $U$ can omit $L^p$-sample paths. To aid with the convergence of the finite approximations we provide a natural discretization to represent the prior. Aside from convergence of these stable sheets we address whether both well-posedness and well-definiteness of the inverse problem can be attained.

math.PR↗

arXiv2016

Hyperpriors for Matérn fields with applications in Bayesian inversion

We introduce non-stationary Matérn field priors with stochastic partial differential equations, and construct correlation length-scaling with hyperpriors. We model both the hyperprior and the Matérn prior as continuous-parameter random fields. As hypermodels, we use Cauchy and Gaussian random fields, which we map suitably to a desired correlation length-scaling range. For computations, we discretise the models with finite difference methods. We consider the convergence of the discretised prior and posterior to the discretisation limit. We apply the developed methodology to certain interpolation and numerical differentiation problems, and show numerically that we can make Bayesian inversion which promotes competing constraints of smoothness and edge-preservation. For computing the conditional mean estimator of the posterior distribution, we use a combination of Gibbs and Metropolis-within-Gibbs sampling algorithms.

math.ST↗

arXiv2016

Cauchy difference priors for edge-preserving Bayesian inversion with an application to X-ray tomography

We study Cauchy-distributed difference priors for edge-preserving Bayesian statistical inverse problems. On the contrary to the well-known total variation priors, one-dimensional Cauchy priors are non-Gaussian priors also in the discretization limit. Cauchy priors have independent and identically distributed increments. One-dimensional Cauchy and Gaussian random walks are special cases of Lévy $α$-stable random walks with $α=1$ and $α=2$, respectively. Both random walks can be written in closed-form, and as priors, they provide smoothing and edge-preserving properties. We briefly discuss also continuous and discrete Lévy $α$-stable random walks, and generalize the methodology to two-dimensional priors. We apply the developed algorithm to one-dimensional deconvolution and two-dimensional X-ray tomography problems. We compute conditional mean estimates with single-component Metropolis-Hastings and maximum a posteriori estimates with Gauss-Newton-type optimization method. We compare the proposed tomography reconstruction method to filtered back-projection estimate and conditional mean estimates with Gaussian and total variation priors.

math.ST↗

arXiv2016

Elliptic boundary value problems with Gaussian white noise loads

Linear second order elliptic boundary value problems (BVP) on bounded Lipschitz domains are studied in the case of Gaussian white noise loads. Especially, Neumann and Robin BVPs are considered. The main obstacle for applying the usual variational approach is that the Gaussian white noise has irregular realizations. In particular, the corresponding Neumann boundary values are not well-defined in the ordinary sense. In this work, the BVP is reformulated by replacing the continuity of the boundary trace mapping with measurability. Instead of using variational methods alone, the reformulation of the BVP derives also from Cameron-Martin space techniques. The reformulation essentially returns the study of irregular white noise loads to study of $L^2$-loads. Admissibility of the reformulation is demonstrated by showing that usual finite element approximations of the BVP with discretized white noise loads converge to the solution of the reformulated problem. For Neumann and Robin BVPs, the finite dimensional approximations have been utilized before. However, here also the infinite-dimensional limit is considered.

math.PR↗

arXiv2014

Sparse approximations of fractional Matérn fields

We consider a fast approximation method for a solution of a certain stochastic non-local pseudodifferential equation. This equation defines a Matérn class random field. The approximation method is based on the spectral compactness of the solution. We approximate the pseudodifferential operator with a Taylor expansion. By truncating the expansion, we can construct an approximation with Gaussian Markov random fields. We show that the solution of the truncated version can be constructed with an over-determined system of stochastic matrix equations with sparse matrices. We solve the system of equations with a sparse Cholesky decomposition. We consider the convergence of the discrete approximation of the solution to the continuous one. Finally numerical examples are given.

math.ST↗

arXiv2011

Posterior convergence for approximated unknowns in non-Gaussian statistical inverse problems

The statistical inverse problem of estimating the probability distribution of an infinite-dimensional unknown given its noisy indirect observation is studied in the Bayesian framework. In practice, one often considers only finite-dimensional unknowns and investigates numerically their probabilities. As many unknowns are function-valued, it is of interest to know whether the estimated probabilities converge when the finite-dimensional approximations of the unknown are refined. In this work, the generalized Bayes formula is shown to be a powerful tool in the convergence studies. With the help of the generalized Bayes formula, the question of convergence of the posterior distributions is returned to the convergence of the finite-dimensional (or any other) approximations of the unknown. The approach allows many prior distributions while the restrictions are mainly for the noise model and the direct theory. Three modes of convergence of posterior distributions are considered -- weak convergence, setwise convergence and convergence in variation. The convergence of conditional mean estimates is studied. Several examples of applicable infinite-dimensional non-Gaussian noise models are provided, including a generalization of the Cameron-Martin formula for certain non-Gaussian measures. Also, the well-posedness of Bayesian statistical inverse problems is studied.

math.ST↗