Bayesian Reconstruction Approach Research Articles

A probabilistic data assimilation framework to reconstruct finite element error fields from sparse error estimates: Application to sub‐modeling

AbstractThe present work proposes a computational approach that recovers full finite element error fields from a small number of estimates of errors in scalar quantities of interest. The approach is weakly intrusive and is motivated by large scale industrial applications wherein modifying the finite element models is undesirable and multiple regions of interest may exist in a single model. Error estimates are developed using a Zhu‐Zienkiewicz estimator coupled with the adjoint methodology to deliver goal‐oriented results. A Bayesian probabilistic estimation framework is deployed for full field estimation. An adaptive, radial basis function based reduced order modeling strategy is implemented to reduce the cost of calculating the posterior. The Bayesian reconstruction approach, accelerated by the proposed model reduction technology, is shown to yield good probabilistic estimates of full error fields, with a computational complexity that is acceptable compared to the evaluation of the goal‐oriented error estimates. The novelty of the work is that a set of computed error estimates are considered as partial observations of an underlying error field, which is to be recovered. Future improvements of the method include the optimal selection of goal‐oriented error measures to be acquired prior to the error field reconstruction.

Stress Testing the Unknown - The Impact of Network Reconstruction on Systemic Risk Estimates

Banks connect through a complex network of interbank credit exposures that determine contagion and hence the systemic stability of the network. Yet, the bilateral linkages of these institutions and thus the true structure of the network are typically unobservable, with leading reconstruction methods, such as maximum entropy or minimum density, potentially underestimating the true contagion risk of the network. Based on publicly available data from large European institutions, this paper analyses different network reconstruction techniques and the contagion risk impact of different network structures in a stress testing context. My findings suggest that uncertainty about the true interbank network leads to substantially varying risk assessments, where a Bayesian reconstruction approach can be used to identify contagion risk boundaries. In addition, network structure metrics that explain the variation in risk assessments are identified.

Reconstructing Networks with Unknown and Heterogeneous Errors

The vast majority of network datasets contains errors and omissions, although this is rarely incorporated in traditional network analysis. Recently, an increasing effort has been made to fill this methodological gap by developing network reconstruction approaches based on Bayesian inference. These approaches, however, rely on assumptions of uniform error rates and on direct estimations of the existence of each edge via repeated measurements, something that is currently unavailable for the majority of network data. Here we develop a Bayesian reconstruction approach that lifts these limitations by not only allowing for heterogeneous errors, but also for single edge measurements without direct error estimates. Our approach works by coupling the inference approach with structured generative network models, which enable the correlations between edges to be used as reliable uncertainty estimates. Although our approach is general, we focus on the stochastic block model as the basic generative process, from which efficient nonparametric inference can be performed, and yields a principled method to infer hierarchical community structure from noisy data. We demonstrate the efficacy of our approach with a variety of empirical and artificial networks.

A Bayesian fusion model for space-time reconstruction of finely resolved velocities in turbulent flows from low resolution measurements

.The study of turbulent flows calls for measurements with high resolution in both space and time. We propose a new approach to reconstruct high-temporal–high-spatial resolution velocity fields by combining two sources of information that are well resolved either in space or in time, the low-temporal–high-spatial (LTHS) and the high-temporal–low-spatial (HTLS) resolution measurements. In the framework of co-conception between sensing and data post-processing, this work extensively investigates a Bayesian reconstruction approach using a simulated database. A Bayesian fusion model is developed to solve the inverse problem of data reconstruction. The model uses a maximum a posteriori estimate, which yields the most probable field knowing the measurements. The direct numerical simulation (DNS) of a wall-bounded turbulent flow at moderate Reynolds number is used to validate and assess the performances of the present approach. Low-resolution measurements are subsampled in time and space from the fully resolved data. Reconstructed velocities are compared to the reference DNS to estimate the reconstruction errors. The model is compared to other conventional methods such as linear stochastic estimation and cubic spline interpolation. Results show the superior accuracy of the proposed method in all configurations. Further investigations of model performances on various scales demonstrate its robustness. Numerical experiments also permit one to estimate the expected maximum information level corresponding to limitations of experimental instruments.

A full‐spectral Bayesian reconstruction approach based on the material decomposition model applied in dual‐energy computed tomography

Dual-energy computed tomography (DECT) makes it possible to get two fractions of basis materials without segmentation. One is the soft-tissue equivalent water fraction and the other is the hard-matter equivalent bone fraction. Practical DECT measurements are usually obtained with polychromatic x-ray beams. Existing reconstruction approaches based on linear forward models without counting the beam polychromaticity fail to estimate the correct decomposition fractions and result in beam-hardening artifacts (BHA). The existing BHA correction approaches either need to refer to calibration measurements or suffer from the noise amplification caused by the negative-log preprocessing and the ill-conditioned water and bone separation problem. To overcome these problems, statistical DECT reconstruction approaches based on nonlinear forward models counting the beam polychromaticity show great potential for giving accurate fraction images. This work proposes a full-spectral Bayesian reconstruction approach which allows the reconstruction of high quality fraction images from ordinary polychromatic measurements. This approach is based on a Gaussian noise model with unknown variance assigned directly to the projections without taking negative-log. Referring to Bayesian inferences, the decomposition fractions and observation variance are estimated by using the joint maximum a posteriori (MAP) estimation method. Subject to an adaptive prior model assigned to the variance, the joint estimation problem is then simplified into a single estimation problem. It transforms the joint MAP estimation problem into a minimization problem with a nonquadratic cost function. To solve it, the use of a monotone conjugate gradient algorithm with suboptimal descent steps is proposed. The performance of the proposed approach is analyzed with both simulated and experimental data. The results show that the proposed Bayesian approach is robust to noise and materials. It is also necessary to have the accurate spectrum information about the source-detector system. When dealing with experimental data, the spectrum can be predicted by a Monte Carlo simulator. For the materials between water and bone, less than 5% separation errors are observed on the estimated decomposition fractions. The proposed approach is a statistical reconstruction approach based on a nonlinear forward model counting the full beam polychromaticity and applied directly to the projections without taking negative-log. Compared to the approaches based on linear forward models and the BHA correction approaches, it has advantages in noise robustness and reconstruction accuracy.

Markov chain Monte Carlo method in Bayesian reconstruction of dynamical systems from noisy chaotic time series

The impossibility to use the MCMC (Markov chain Monte Carlo) methods for long noisy chaotic time series (TS) (due to high computational complexity) is a serious limitation for reconstruction of dynamical systems (DSs). In particular, it does not allow one to use the universal Bayesian approach for reconstruction of a DS in the most interesting case of the unknown evolution operator of the system. We propose a technique that makes it possible to use the MCMC methods for Bayesian reconstruction of a DS from noisy chaotic TS of arbitrary long duration.

Design of a CZT based breast SPECT system

A high-resolution SPECT instrument dedicated to breast imaging has been designed incorporating arrays of collimated Cadmium-Zinc-Telluride (CZT) detectors tiled on either a cylindrical surface or a hemispherical surface surrounding the breast. The performance characteristics of a three-segment rotating parallel-hole collimator as well as a stationary multiple pin-hole collimator were considered for the cylindrical system. A stationary multiple pin-hole collimated system was also considered for the hemispherical design. Monte Carlo studies suggest that at almost equal spatial resolution of 0.5 cm, the cylindrical design with parallel-hole collimator would have an approximately a factor of two higher geometrical efficiency than the hemispherical pinhole collimated system including effects of attenuation in the breast. However, whereas the parallel-hole collimator must be rotated to acquire data from multiple angles of view, the pin-hole version has the advantage of recording data from 112 views in a stationary mode. Monte Carlo studies of filtered backprojection as well as a Bayesian reconstruction approach including attenuation and scatter within the breast, where the breast was modeled as a 15 cm hemisphere of uniform activity distribution containing three spherical lesions of diameters 1.0 cm, 0.7 cm and 0.5 cm respectively suggest that the 0.5 cm could be detected with either design in a one-hour SPECT study assuming a 10:1 tumor to background ratio. The authors conclude that a high resolution breast SPECT instrument where the resolution is limited to about 0.5 cm is viable with CZT detectors.