Tuesday 09:00 - 11:00 Weimar hall, Seminar room 2
Organizers: Björn Sprungk (TU Chemnitz)
Aretha Teckentrup (University of Edinburgh)
Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors
T. Sullivan (Freie Unviersität Berlin) 09:00
The framework of Bayesian inverse problems in infinite-dimensional parameter spaces, as advocated by Stuart (Acta Numer. 19:451–559, 2010) and others, is extended to the case of a heavy-tailed prior measure in the family of stable distributions, such as an infinite-dimensional Cauchy distribution, for which polynomial moments are infinite or unde-fined. It is shown that analogues of the Karhunen–Loève expansion for square-integrable random variables can be used to sample such measures on quasi-Banach spaces. Fur-thermore, under weaker regularity assumptions than those used to date, the Bayesian posterior measure is shown to depend Lipschitz continuously in the Hellinger and TV metrics upon perturbations of the misfit function and observed data.
A Transdimensional Bayesian Approach for Image Correction in Anisotropic Media
K. Tant (University of Strathclyde), A. Mulholland (University of Strathclyde), E. Galetti (University of Edinburgh), A. Curtis (University of Edinburgh)
09:20
The oil and gas, nuclear power and aerospace industries are just a subset of the sectors dependent on the routine maintenance of safety critical structures. Failure to detect structural weaknesses in components integral to the work being carried out by these industries can be catastrophic. Ultrasonic non-destructive testing is a technique which involves the transmitting of mechanical waves through the component under inspec-tion. Just like in medical ultrasound, these waves can be passed through the component and subsequently collected without disturbing its internal composition. Large networks of sensors are deployed to carry out these inspections, resulting in massive quantities of noisy data, for which mathematical algorithms are required to decipher. Thus we are presented with an inverse problem: from this observed data, can we determine the path the ultrasound waves took and characterise any obstacles that they may have intercepted? Although there already exist techniques which can pick up on large
scat-tering events within the collected ultrasonic signals, small variances in the material microstructure can distort the image and lead to inaccurate characterisation and posi-tioning of defects, particularly when the material is anisotropic or heterogeneous. This is why the development of an ultrasonic tomography system which can map out the material’s internal structure is of industrial interest. Armed with this additional infor-mation, we can compensate for variances in the wave speed and improve our ability to correctly locate and characterise a defect. The work presented here endeavours to map the locally anisotropic grain structure typically exhibited in austenitic steel welds (which are notoriously difficult to inspect) using measured time-of-flight ultrasonic data. The reversible-jump Markov chain Monte Carlo method (RJMCMC) is an ensemble inference approach within a Bayesian framework. The material geometry is initially parameterised by a Voronoi tessellation, through which the wave path is modelled. The Voronoi dia-gram is then perturbed iteratively and the results are compared to the observed data using the Metropolis-Hastings criterion. If an iteration is accepted subject to this cri-terion, the respective Voronoi diagram contributes to the final ensemble solution. This final solution is a smooth map of the varying anisotropic regions of the sample. Using this map in conjunction with existing imaging algorithms, it is shown that an improve-ment in the overall flaw placeimprove-ment and characterisation can be achieved. Additionally, uncertainty maps can also be constructed where the variance of the value at each point in the spatial domain is plotted. These can be used to quantify the reliability of the extracted map.
Spectral nonlinear Kalman filtering
B. Rosic (TU Braunschweig), H. Matthies (TU Braunschweig) 09:40
Probabilistic parameter estimation of nonlinear systems is usually considered in a Bayesian framework in a sampling manner as analytical determination of posterior distributions is only possible in special cases when considered distributions are conjugate. In this talk an algebraic way of computing posterior estimates based on conditional expectation for any distribution type is presented. In this perspective special attention is paid to the design of a new nonlinear iterative filtering formula that has a Kalman-like flavour and is realised in a spectral functional representation form. The numerical procedures will be shown on some algebraic examples, as well as linear diffusion and nonlinear von Mises elastoplasticity problems.
Sequential Design of Computer Experiments for the Solution of Bayesian Inverse Problems
M. Sinsbeck (University of Stuttgart), W. Nowak 10:00
We present a sequential sampling method for the solution of Bayesian parameter in-ference problems. The model function is assumed to be computationally expensive, so the goal is to approximate the posterior with as few function evaluations as possible.
To this end, the a priori unknown model function is described by a random field. The presented method classifies as a greedy one-step lookahead method: in each iteration, a
new point is selected such that the expected Bayes risk of the posterior distribution is minimized. Several numerical examples demonstrate that the expected Bayes risk is an appropriate refinement criterion for the solution of Bayesian inference problems. The presented approach shows to be more efficient than non-sequential sampling methods.
Uncertainty and constitutive model error quantification
I. Franck, P. Koutsourelakis 10:20
While for calibration purposes the solution of an inverse problem can almost always be achieved, underlying model inadequacies are barely considered. Traditional approaches use an additional regression model (e.g., Gaussian process) added to the model output [1], within a submodel [2] or reformulate the problem to a calibration problem (alone with regard to the model inadequacy [3]) to account for an underlying model error.
This can either violate physical constraints, is infeasible in high dimensions or does not incorporated an inverse problem with regard to model parameters.
In this work we unfold conservation and constitutive laws to estimate model discrepan-cies and to simultaneously satisfy physical constraints. Efficient Bayesian strategies are incorporated when investigating in an inverse problem from solid mechanics. A correct identification of the mechanical properties of an unknown tissue and their corresponding uncertainties leads then to an accurate noninvasive, medical diagnosis.
[1] Kennedy, M. C. and O’Hagan, A. (2001), Bayesian calibration of computer models.
Journal of the Royal Statistical Society: Series B (Statistical Methodology), 63:
425–464. doi:10.1111/1467-9868.00294
[2] Berliner, L. Mark., Jezek, K., Cressie, N., Kim, Y., Lam, C. and Van Der Veen, C. (2008). Modeling dynamic controls on ice streams: A Bayesian statistical ap-proach. Journal of Glaciology, 54 (187), 705-714.
[3] Wu, Jin-Long, Jian-Xun Wang, and Heng Xiao. ”A Bayesian Calibration–Prediction Method for Reducing Model-Form Uncertainties with Application in RANS Sim-ulations.” Flow, Turbulence and Combustion (2015): 1-26.
YR 3 : Local and Nonlocal Methods for Processing Manifolds and Point Cloud Data
Tuesday 09:00 - 11:00 Weimar hall, Wing hall
Organizers: Ronny Bergmann (TU Kaiserslautern) Daniel Tenbrinck (WWU Münster)
Introduction
09:00
Interpolation on manifolds with B-splines
P. Gousenbourger (Université catholique de Louvain) 09:20
From a set of n + 1 points pi on a manifold M associated to nodes i ∈ Z, we seek a C1 function :R → M such that B(i) = pi.
To this end, we restrict B to a family of manifold-valued piecewise-Bézier curves where the first and last segments are quadratic while the others are cubic (as in [1]). We then compute the control points of B by generalizing the Euclidean concept of natural C2-splines.
One of the benefits of this application arise in problems whose solutions (pi)ni=0depend on only one parameter and are hard to compute, but are evaluated on a manifoldM.
Hence, for a new value of the parameter, instead of solving the complicated problem, one can estimate the solution p⋆ by interpolating (pi)ni=0 on M.
The advantages of this technique are (i) a lower space complexity as the solution curve is represented by a few Bézier control points on the manifold, and (ii) a considerably simpler method that only requires two objects on the manifold: the Riemannian expo-nential and the Riemannian logarithm.
[1] P.-A. Absil, Pierre-Yves Gousenbourger, Paul Striewski, and Benedikt Wirth. Differ-entiable piecewise-B’ezier surfaces on Riemannian manifolds. SIAM Journal on Imaging Sciences, 9(4):1788–1828, 2016.
A Nonlocal Denoising Algorithm for Manifold-valued Images Using Second Order Statistics
J. Persch (Technische Universität Kaiserslautern) 09:40
Nonlocal patch-based methods, in particular the Bayes’ approach of Lebrun, Buades and Morel [2013], are considered as state-of-the-art methods for denoising (color) im-ages corrupted by white Gaussian noise of moderate variance. In this talk we present the first attempt to generalize this technique to manifold-valued images. Such images are frequently encountered in real-world applications, for example images with phase or directional entries appear in certain colormodels, or images with values in the mani-fold of symmetric positive definite matrices occur in diffusion tensor magnetic resonance imaging. We reinterpret the Bayesian approach of Lebrun et al. [2013] in terms of min-imum mean squared error estimation, which motivates our definition of a corresponding estimator on the manifold. To do so we need to generalize the normal law to manifolds, which is not canonical. Different generalizations have already been proposed in the lit-erature. Here we focus on an straightforward intrinsic model. With the estimator at hand we present a nonlocal patch-based method for the restoration of manifold-valued images. Various proof of concept examples demonstrate the potential of the proposed algorithm. This is joint work with Friederike Laus, Mila Nikolova, and Gabriele Steidl.
Retinal Image Analysis using Sub-Riemannian Geometry in SE(2)
E. Bekkers (Eindhoven University of Technology) 10:00
In our analysis of 2D Retinal images we represent the image data in higher-dimensional objects called orientation scores. Orientation scores are obtained via a wavelet-type transform with anisotropic filters. The resulting objects are densities on the (coupled) space of positions and orientations, which we identify with the roto-translation group SE(2). Due to the coupling of positions and orientations, and considering the fact that we analyze lifted (3D) representations of 2D data, we have to employ a sub-Riemannian geometry in our analyses. The sub-Riemannian geometry consists of the Lie group SE(2), a sub-bundle of the full tangent bundle which is defined via left-invariant vector fields, and Riemannian metric tensor which measures the length of vectors in this sub-bundle. In this talk I will briefly give 3 examples of how we employ a sub-Riemannian geometry in retinal image analysis:
1. Vessel tracking: Curvature penalized vessel curves are found via the computation of sub-Riemannian geodesics in SE(2). Using orientation scores we first construct a data-adaptive sub-Riemannian metric; we then solve the sub-Riemannian eikonal equation, and obtain globally optimal geodesics via backtracking on the obtained sub-Riemannian distance maps.
2. Template matching: We find anatomical landmarks via cross-correlation based templates matching in orientation scores. The templates are learned via lin-ear/logistic regression with a smoothing prior which we relate to Brownian motions SE(2).
3. Curvature biomarkers: We fit exponential curves in orientation scores which en-ables us to do pixel wise curvature measurements. These pixel wise measurements are used to compute global tortuosity measures which we show are significantly associated with hypertension and diabetes mellitus.
Robust Surface Reconstruction
V. Estellers (TUM) 10:20
We propose a method to reconstruct surfaces from oriented point clouds corrupted by er-rors arising from range imaging sensors. The core of this technique is the formulation of the problem as a convex minimization that reconstructs the indicator function of the sur-face’s interior and substitutes the usual least-squares fidelity terms by Huber penalties to be robust to outliers, recover sharp corners, and avoid the shrinking bias of least-squares models. To achieve both flexibility and accuracy, we couple an implicit parametrization that reconstructs surfaces of unknown topology with adaptive discretizations that avoid the high memory and computational cost of volumetric representations. The hierarchic structure of the discretizations speeds minimization through multiresolution, while the proposed splitting algorithm minimizes non-differentiable functionals and is easy to par-allelize. In experiments, our model improves reconstruction from synthetic and real data while the choice of discretization affects both the accuracy of the reconstruction and its computational cost.