Fully computable a posteriori error bounds for hybridizable discontinuous Galerkin finite element approximations

We derive a posteriori error estimates for the hybridizable discontinuous Galerkin (HDG) methods, including both the primal and mixed formulations, for the approximation of a linear second-order elliptic problem on conforming simplicial meshes in two and three dimensions. We obtain fully computable, constant free, a posteriori error bounds on the broken energy seminorm and the HDG energy (semi)norm of the error. The estimators are also shown to provide local lower bounds for the HDG energy (semi)norm of the error up to a constant and a higher-order data oscillation term. For the primal HDG methods and mixed HDG methods with an appropriate choice of stabilization parameter, the estimators are also shown to provide a lower bound for the broken energy seminorm of the error up to a constant and a higher-order data oscillation term. Numerical examples are given illustrating the theoretical results

Bayesian Model Selection Approach to Multiple Change-Points Detection with Non-Local Prior Distributions

We propose a Bayesian model selection (BMS) boundary detection procedure using non-local prior distributions for a sequence of data with multiple systematic mean changes. By using the non-local priors in the BMS framework, the BMS method can effectively suppress the non-boundary spike points with large instantaneous changes. Further, we speedup the algorithm by reducing the multiple change points to a series of single change point detection problems. We establish the consistency of the estimated number and locations of the change points under various prior distributions. From both theoretical and numerical perspectives, we show that the non-local inverse moment prior leads to the fastest convergence rate in identifying the true change points on the boundaries. Extensive simulation studies are conducted to compare the BMS with existing methods, and our method is illustrated with application to the magnetic resonance imaging guided radiation therapy data

Deep Convolutional Neural Fields for Depth Estimation from a Single Image

We consider the problem of depth estimation from a single monocular image in this work. It is a challenging task as no reliable depth cues are available, e.g., stereo correspondences, motions, etc. Previous efforts have been focusing on exploiting geometric priors or additional sources of information, with all using hand-crafted features. Recently, there is mounting evidence that features from deep convolutional neural networks (CNN) are setting new records for various vision applications. On the other hand, considering the continuous characteristic of the depth values, depth estimations can be naturally formulated into a continuous conditional random field (CRF) learning problem. Therefore, we in this paper present a deep convolutional neural field model for estimating depths from a single image, aiming to jointly explore the capacity of deep CNN and continuous CRF. Specifically, we propose a deep structured learning scheme which learns the unary and pairwise potentials of continuous CRF in a unified deep CNN framework. The proposed method can be used for depth estimations of general scenes with no geometric priors nor any extra information injected. In our case, the integral of the partition function can be analytically calculated, thus we can exactly solve the log-likelihood optimization. Moreover, solving the MAP problem for predicting depths of a new image is highly efficient as closed-form solutions exist. We experimentally demonstrate that the proposed method outperforms state-of-the-art depth estimation methods on both indoor and outdoor scene datasets.Comment: fixed some typos. in CVPR15 proceeding