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This work considers sequential edge-promoting Bayesian experimental design for (discretized) linear inverse problems, exemplified by X-ray tomography. The process of computing a total variation-type reconstruction of the absorption inside the imaged body via lagged diffusivity iteration is interpreted in the Bayesian framework. Assuming a Gaussian additive noise model, this leads to an approximate Gaussian posterior with a covariance structure that contains information on the location of edges in the posterior mean. The next projection geometry is then chosen through A- or D-optimal Bayesian design, which corresponds to minimizing the trace or the determinant of the updated posterior covariance matrix that accounts for the new projection. Two- and three-dimensional numerical examples based on simulated data demonstrate the functionality of the introduced approach. [ABSTRACT FROM AUTHOR]

Given a connected compact Riemannian manifold (M , g) without boundary, dim ⁡ M ≥ 2 , we consider a space–time fractional diffusion equation with an interior source that is supported on an open subset V of the manifold. The time-fractional part of the equation is given by the Caputo derivative of order α ∈ (0 , 1 , and the space fractional part by (− Δ g) β , where β ∈ (0 , 1 and Δ g is the Laplace–Beltrami operator on the manifold. The case α = β = 1 , which corresponds to the standard heat equation on the manifold, is an important special case. We construct a specific source such that measuring the evolution of the corresponding solution on V determines the manifold up to a Riemannian isometry. [ABSTRACT FROM AUTHOR]

This paper concerns the random source problems for the time-harmonic acoustic and elastic wave equations in two and three dimensions. The goal is to determine the compactly supported external force from the radiated wave field measured in a domain away from the source region. The source is assumed to be a microlocally isotropic generalized Gaussian random function such that its covariance operator is a classical pseudo-differential operator. Given such a distributional source, the direct problem is shown to have a unique solution by using an integral equation approach and the Sobolev embedding theorem. For the inverse problem, we demonstrate that the amplitude of the scattering field averaged over the frequency band, obtained from a single realization of the random source, determines uniquely the principle symbol of the covariance operator. The analysis employs asymptotic expansions of the Green functions and microlocal analysis of the Fourier integral operators associated with the Helmholtz and Navier equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider an inverse problem for the n -dimensional random Schrödinger equation (Δ − q + k 2) u = 0. We study the scattering of plane waves in the presence of a potential q which is assumed to be a Gaussian random function such that its covariance is described by a pseudodifferential operator. Our main result is as follows: given the backscattered far field, obtained from a single realization of the random potential q , we uniquely determine the principal symbol of the covariance operator of q. Especially, for n = 3 this result is obtained for the full nonlinear inverse backscattering problem. Finally, we present a physical scaling regime where the method is of practical importance. [ABSTRACT FROM AUTHOR]

In this paper we consider variational regularization methods for inverse problems with large noise that is in general unbounded in the image space of the forward operator. We introduce a Banach space setting that allows to define a reasonable notion of solutions for more general noise in a larger space provided that one has sufficient mapping properties of the forward operators. A key observation, which guides us through the subsequent analysis, is that such a general noise model can be understood with the same setting as approximate source conditions (while a standard model of bounded noise is related directly to classical source conditions). Based on this insight we obtain a quite general existence result for regularized variational problems and derive error estimates in terms of Bregman distances. The latter is specialized for the particularly important cases of one- and |$p$| -homogeneous regularization functionals. As a natural further step we study stochastic noise models and in particular white noise for which we derive error estimates in terms of the expectation of the Bregman distance. The finiteness of certain expectations leads to a novel class of abstract smoothness conditions on the forward operator, which can be easily interpreted in the Hilbert space case. We finally exemplify the approach and in particular the conditions for popular examples of regularization functionals given by squared norm, Besov norm and total variation. [ABSTRACT FROM AUTHOR]

Modern ground-based telescopes rely on a technology called adaptive optics in order to compensate for the loss of angular resolution caused by atmospheric turbulence. Next-generation adaptive optics systems designed for a wide field of view require a stable and high-resolution reconstruction of the turbulent atmosphere. By introducing a novel Bayesian method, we address the problem via reconstructing the atmospheric turbulence strength profile and the turbulent layers simultaneously, where we only use wavefront measurements of incoming light from guide stars. Most importantly, we demonstrate how this method can be used for model optimization as well. We propose two different algorithms for solving the maximum a posteriori estimate: the first approach is based on alternating minimization and has the advantage of integrability into existing atmospheric tomography methods. In the second approach, we formulate a convex non-differentiable optimization problem, which is solved by an iterative thresholding method. This approach clearly illustrates the underlying sparsity-enforcing mechanism for the strength profile. By introducing a tuning/regularization parameter, an automated model reduction of the layer structure of the atmosphere is achieved. Using numerical simulations, we demonstrate the performance of our method in practice. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We study an inverse acoustic scattering problem in half-space with a probabilistic impedance boundary value condition. The Robin coefficient (surface impedance) is assumed to be a Gaussian random function with a pseudodifferential operator describing the covariance. We measure the amplitude of the backscattered field averaged over the frequency band and assume that the data is generated by a single realization of λ . Our main result is to show that under certain conditions the principal symbol of the covariance operator of λ is uniquely determined. Most importantly, no approximations are needed and we can solve the full non-linear inverse problem. We concentrate on anisotropic models for the principal symbol, which leads to the analysis of a novel anisotropic spherical Radon transform and its invertibility. [ABSTRACT FROM AUTHOR]

We consider a smooth Riemannian metric tensor g on $${\mathbb{R}^n}$$ and study the stochastic wave equation for the Laplace-Beltrami operator $${\partial_t^2 u - \Delta_g u = F}$$ . Here, F = F( t, x, ω) is a random source that has white noise distribution supported on the boundary of some smooth compact domain $${M \subset \mathbb{R}^n}$$ . We study the following formally posed inverse problem with only one measurement. Suppose that g is known only outside of a compact subset of M and that a solution $${u(t, x, \omega_0)}$$ is produced by a single realization of the source $${F(t, x, \omega_0)}$$ . We ask what information regarding g can be recovered by measuring $${u(t, x, \omega_0)}$$ on $${\mathbb{R}_+ \times \partial M}$$ ? We prove that such measurement together with the realization of the source determine the scattering relation of the Riemannian manifold ( M, g) with probability one. That is, for all geodesics passing through M, the travel times together with the entering and exit points and directions are determined. In particular, if ( M, g) is a simple Riemannian manifold and g is conformally Euclidian in M, the measurement determines the metric g in M. [ABSTRACT FROM AUTHOR]

• We propose a method to model internal knot locations based on laser scans of logs. • The method is faster and cheaper than methods using computer tomography scan. • Dataset consisting of 100 logs and resulting boards was collected in a real sawmill. • The method allows estimating knot locations in the resulting boards. • The method could be used for sawing angle optimization and process control. The quality of the end product in the sawmill industry is highly dependent on the distribution of knots. If the internal structure of the logs used as raw material were known it would be possible to optimize the sawing process by controlling the locations of individual knots in the resulting boards. Methods such as Computer Tomography or Magnetic Resonance Imaging can be used to gain reliable information about the internal structure of the logs prior to sawing. Such scanners are, however, expensive or slow, which renders the use of these methods ill-suited to online integration in sawmill operations. Laser range scanners, on the other hand, are very fast but provide only external information. A method to estimate the internal log structure using only external information provided by a laser range scanner is proposed in this paper. The method comprises five major steps: point cloud filtering and centreline estimation, log surface heightmap generation, knot segmentation, volumetric reconstruction of knots and virtual sawing. The end result is the generation of images of virtual boards for the given sawing parameters. The method was evaluated on debarked softwood logs. The experimental part of the work demonstrates that the pixel intensities of the virtual boards generated using the proposed method correlate well with the probabilities of knots appearing in the corresponding locations in the final product. Virtual sawing allows exhaustive evaluation of different sawing parameters, thus making it possible to optimize sawing parameters prior to sawing. [ABSTRACT FROM AUTHOR]