Posteriori Estimates Research Articles

Hierarchical A Posteriori Error Estimators for the Mimetic Discretization of Elliptic Problems

We present an a posteriori error estimate of hierarchical type for the mimetic discretization of elliptic problems. Under a saturation assumption, the global reliability and efficiency of the proposed a posteriori estimator are proved. Several numerical experiments assess the actual performance of the local error indicators in driving adaptive mesh refinement algorithms based on different marking strategies. Finally, we analyze and test an inexpensive variant of the proposed error estimator which drastically reduces the overall computational cost of the adaptive procedures.

A Probabilistic Approach to Spectral Graph Matching

Spectral Matching (SM) is a computationally efficient approach to approximate the solution of pairwise matching problems that are np-hard. In this paper, we present a probabilistic interpretation of spectral matching schemes and derive a novel Probabilistic Matching (PM) scheme that is shown to outperform previous approaches. We show that spectral matching can be interpreted as a Maximum Likelihood (ML) estimate of the assignment probabilities and that the Graduated Assignment (GA) algorithm can be cast as a Maximum a Posteriori (MAP) estimator. Based on this analysis, we derive a ranking scheme for spectral matchings based on their reliability, and propose a novel iterative probabilistic matching algorithm that relaxes some of the implicit assumptions used in prior works. We experimentally show our approaches to outperform previous schemes when applied to exhaustive synthetic tests as well as the analysis of real image sequences.

A Statistical Model and Estimation of Disease Rates as Functions of Age and Time

Estimating incidence, remission, and mortality rates as functions of age and time is an important public health problem. We model disease rates as smooth with respect to both age and time. This is combined with an ordinary differential equation for the healthy and infected population, where date of birth (cohort) is fixed and age is the independent variable. The result is a multivariate Gauss--Markov random field model for the healthy population, infected population, and the disease rates. The corresponding inverse covariance matrix (precision matrix) is sparse. We model some standard public health measurements in terms of this random field. An offset parameter controls the level at which the residual transitions between being lognormal and normal. This enables the combination of measurement values that are zero with measurements that are lognormal for large values. The freely available nonlinear programming solver Ipopt, which takes advantage of the sparsity, is used to obtain the maximum a posteriori estimate for the healthy population, infected population, and the disease rates as a function of age and cohort. A simulation example is used to demonstrate that a small amount of prevalence data can recover the shape of the incidence rate as a function of age and time. A diabetes example is used to demonstrate how one applies this model to real world public health data.

Bayesian Stereo Matching Method Based on Edge Constraints.

A new global stereo matching method is presented that focuses on the handling of disparity, discontinuity and occlusion. The Bayesian approach is utilized for dense stereo matching problem formulated as a maximum a posteriori Markov Random Field (MAP-MRF) problem. In order to improve stereo matching performance, edges are incorporated into the Bayesian model as a soft constraint. Accelerated belief propagation is applied to obtain the maximum a posteriori estimates in the Markov random field. The proposed algorithm is evaluated using the Middlebury stereo benchmark. Our experimental results comparing with some state-of-the-art stereo matching methods demonstrate that the proposed method provides superior disparity maps with a subpixel precision.

A framework for robust eigenvalue and eigenvector error estimation and Ritz value convergence enhancement

We present a general framework for the a posteriori estimation and enhancement of error in eigenvalue/eigenvector computations for symmetric and elliptic eigenvalue problems, and provide detailed analysis of a specific and important example within this framework—finite element methods with continuous, affine elements. A distinguishing feature of the proposed approach is that it provides provably efficient and reliable error estimation under very realistic assumptions, not only for single, simple eigenvalues, but also for clusters which may contain degenerate eigenvalues. We reduce the study of the eigenvalue/eigenvector error estimators to the study of associated boundary value problems, and make use of the wealth of knowledge available for such problems. Our choice of a posteriori error estimator, computed using hierarchical bases, very naturally offers a means not only for estimating error in eigenvalue/eigenvector computations, but also cheaply accelerating the convergence of these computations—sometimes with convergence rates which are nearly twice that of the unaccelerated approximations.

Extra-optimal methods for solving ill-posed problems

Abstract. The problem of a posteriori accuracy estimation for approximate solutions of ill-posed extremal problem is under investigation. We present a new approach to a posteriori estimation along with the study of principal properties of the used estimator. The numerical algorithm for the solution of central extremal problem of a posteriori accuracy estimation is formulated as well. We analyze the optimality in order of obtained a posteriori accuracy estimates and introduce a new important notion of extra-optimal regularizing algorithm as a method for the solution of ill-posed problems having optimal in order a posteriori accuracy estimate. Finally, we demonstrate the results of numerical experiments on applications of extra-optimal regularizing methods, constructed on the base of Tikhonov and Lavrent'ev regularization, for calculating a posteriori accuracy estimates.

A new approach of formation navigation derived from multi-robots cooperative online FastSLAM

A new formation navigation approach derived from multi-robots cooperative online FastSLAM is proposed. In this approach, the leader and follower robots are defined. The posteriori estimation of the leader robot state is treated as a relative reference for all follower robots to correct their state priori estimations. The control volume of individual follower will be achieved from the results of the corrected estimation. All robots are observed as landmarks with known associations by the others and are considered in their landmarks updating. By the method, the errors of the robot posterior estimations are reduced and the formation is well kept. The simulation and physical experiment results show that the multi-robots relative localization accuracy is improved and the formation navigation control is more stable and efficient than normal leader-following strategy. The algorithm is easy in implementation.

New detection algorithms based on the jointly Gaussian approach and successive interference cancelation for iterative MIMO systems

SUMMARYA new detection algorithm based on the jointly Gaussian approach (JGA) and successive interference cancelation (SIC), named the SIC‐JGA algorithm, is proposed for iterative MIMO systems. Both the a priori symbol estimates of the undetected layers and the a posteriori symbol estimates of the previously detected layers are used in the SIC process. To reduce the complexity of covariance matrix inversion in the proposed algorithm, simplified algorithms named S‐SIC‐JGA are investigated for both the single path and multipath block fading channels under M Phase Shift Keying modulations. A detailed complexity analysis is presented for both the SIC‐JGA and S‐SIC‐JGA algorithms. Numerical results exhibit that the system using the SIC‐JGA algorithm outperforms the JGA algorithm, with an additional complexity for determining the detection order of SIC. The S‐SIC‐JGA algorithm has a much lower complexity compared with the SIC‐JGA algorithm while experiencing an acceptable system performance degradation. Proper detection algorithms can be chosen in different iterations to balance the system performance and the detection complexity. Copyright © 2012 John Wiley & Sons, Ltd.

Personal Genomic Measurements: The Opportunity for Information Integration

High-throughput genomic measurements initially emerged for research purposes but are now entering the clinic. The challenge for clinicians is to integrate imperfect genomic measurements with other information sources so as to estimate as closely as possible the probabilities of clinical events (diagnoses, treatment responses, prognoses). Population-based data provide a priori probabilities that can be combined with individual measurements to compute a posteriori estimates using Bayes' rule. Thus, the integration of population science with individual genomic measurements will enable the practice of personalized medicine.

A HYBRID APPROACH TO EXTRACTION AND REFINEMENT OF BUILDING FOOTPRINTS FROM AIRBORNE LIDAR DATA

Abstract. This work presents a combined bottom-up and top-down approach to extraction and refinement of building footprints from airborne LIDAR data. Building footprints are interesting for many applications in urban planning. The cadastral maps, however, may be limited for certain areas or not be updated frequently. Airborne laser scanning data is therefore considered by many people in the last decade as an important alternative data for change detection and update of building footprints. Laser scanning data of city scenes, however, often shows noise and incompleteness because of, e.g., the clutter by vegetation and the reflection of windows/waterlogged depressions on the roof. Results of the bottom-up detection may thus be limited to incomplete or irregular polygons. We employ 3D Hough transform to detect the building points. An improved joint multiple-plane detection scheme is proposed to find and label the laser points on multiple roof facets synchronously. The bottom-up processing provides not only a rough point segmentation but also additional 3D information, e.g., roof heights and horizontal ridges. Using these as priors, a top-down reconstruction is conducted via generative models. We consider the building footprint as an assembly of regular primitives. A statistical search by means of Reversible Jump Markov Chain Monte Carlo and Maximum A Posteriori estimation is implemented to find the optimal configuration of the footprint. By these means a robust and plausible reconstruction is guaranteed. First results on point clouds with various resolutions show the potential of this approach.

A primal–dual interior-point framework for using the L1 or L2 norm on the data and regularization terms of inverse problems

Maximum a posteriori estimates in inverse problems are often based on quadratic formulations, corresponding to a least-squares fitting of the data and to the use of the L2 norm on the regularization term. While the implementation of this estimation is straightforward and usually based on the Gauss–Newton method, resulting estimates are sensitive to outliers and result in spatial distributions of the estimates that are smooth. As an alternative, the use of the L1 norm on the data term renders the estimation robust to outliers, and the use of the L1 norm on the regularization term allows the reconstruction of sharp spatial profiles. The ability therefore to use the L1 norm either on the data term, on the regularization term, or on both is desirable, though the use of this norm results in non-smooth objective functions which require more sophisticated implementations compared to quadratic algorithms. Methods for L1-norm minimization have been studied in a number of contexts, including in the recently popular total variation regularization. Different approaches have been used and methods based on primal–dual interior-point methods (PD-IPMs) have been shown to be particularly efficient. In this paper we derive a PD-IPM framework for using the L1 norm indifferently on the two terms of an inverse problem. We use electrical impedance tomography as an example inverse problem to demonstrate the implementation of the algorithms we derive, and the effect of choosing the L2 or the L1 norm on the two terms of the inverse problem. Pseudo-codes for the algorithms and a public domain implementation are provided.

Phylogeographic Ancestral Inference Using the Coalescent Model on Haplotype Trees

Phylogeographic ancestral inference is issue frequently arising in population ecology that aims to understand the geographical roots and structure of species. Here, we specifically address relatively small scale mtDNA datasets (typically less than 500 sequences with fewer than 1000 nucleotides), focusing on ancestral location inference. Our approach uses a coalescent modelling framework projected onto haplotype trees in order to reduce computational complexity, at the same time adhering to complex evolutionary processes. Statistical innovations of the last few years have allowed for computationally feasible yet accurate inferences in phylogenetic frameworks. We implement our methods on a set of synthetic datasets and show how, despite high uncertainty in terms of identifying the root haplotype, estimation of the ancestral location naturally encompasses lower uncertainty, allowing us to pinpoint the Maximum A Posteriori estimates for ancestral locations. We exemplify our methods on a set of synthetic datasets and then combine our inference methods with the phylogeographic clustering approach presented in Manolopoulou et al. (2011) on a real dataset from weevils in the Iberian peninsula in order to infer ancestral locations as well as population substructure.

A Bayesian generalized multiple group IRT model with model-fit assessment tools

The multiple group IRT model (MGM) proposed by Bock and Zimowski (1997) provides a useful framework for analyzing item response data from clustered respondents. In the MGM, the selected groups of respondents are of specific interest such that group-specific population distributions need to be defined. The main goal is to explore the potentials of an MCMC estimation procedure and Bayesian model-fit tools for the MGM. We develop a full Gibbs sampling algorithm (FGSA) for estimation as well as a Metropolis-Hastings within Gibss sampling algorithm (MHWGS) in order to use non-conjugate priors. The FGSA is compared with Bilog–MG, which uses marginal maximum likelihood (MML) and marginal maximum a posteriori (MMAP) methods. That is; Bilog–MG provides maximum likelihood (ML) and expected a posteriori (EAP) estimates for both item and population parameters, and maximum a posteriori (MAP) estimates for the latent traits. We conclude that, in general, the results from our approach are slightly better than Bilog–MG. Besides a simultaneous MCMC estimation procedure, model-fit assessment tools are developed. Furthermore, the prior sensitivity is investigated with respect to the parameters of the latent population distributions. It will be shown that the FGSA provides a wide set of model-fit tools. The proposed model assessment tools can evaluate important model assumptions of (1) the item response function (IRF) and (2) the latent trait distributions. The utility of the proposed estimation and model-fit assessment methods will be shown using data from a longitudinal data study concerning first to fourth graders of sampled Brazilian public schools.

A posteriori error estimates of higher-order finite elements for frictional contact problems

In this paper, a posteriori estimates are derived for higher-order finite element methods and frictional contact problems. The discretization is based on a mixed approach where the geometrical and frictional constraints are captured by Lagrange multipliers. The use of higher-order polynomials leads to a certain non-conformity in the discretization which requires special attention in the error analysis. As a main result an error estimation is proposed which consists of the dual norm of a residual plus some computable remainder terms. The residual is estimated by well-known a posteriori error estimates for variational equations. The remainder terms represent typical sources resulting from the non-conforming mixed discretization. Numerical experiments confirm the applicability of the a posteriori estimates to adaptive mesh refinements.

Características reprodutivas e suas associações com outras características de importância econômica na raça Nelore

Objetivou-se com este trabalho estimar as herdabilidades (h²) e as correlações genéticas (r g) entre idade ao primeiro parto (IPP) e primeiro intervalo de partos (PIEP) e outras características como peso (PS) ao ano (A) e ao sobreano (S), altura do posterior (ALT) e perímetro escrotal (PE450) em animais da raça Nelore. Os parâmetros genéticos foram estimados em uma análise multicaracterística por modelo animal, utilizando-se a inferência bayesiana via algoritmo de "Gibbs Sampling". Os parâmetros genéticos estimados sugerem a existência de variabilidade genética para IPP (h² = 0,26), sendo que a seleção para a diminuição da IPP de fêmeas Nelore deve responder à seleção individual, sem causar antagonismo do valor genético dos animais para PS (r g = -0,22 (A) e -0,44 (S)) e PE450 (r g = 0,02). A seleção para a diminuição da IPP, no longo prazo, pode levar a um aumento da ALT dos animais, embora essa associação seja relativamente baixa (-0,35). A estimativa de herdabilidade a posteriori para a característica PIEP foi baixa, 0,11±0,03. As r g entre PIEP e as demais características estudadas indicam que a seleção para essas características de crescimento não afetará o PIEP.

An Improved Semi-blind Joint Data Detection and Channel Estimation Algorithm for MIMO-OFDM System

In this paper, a new Semi-blind joint Grover’s Quantum Search (GS) based data detection and Space-Alternating Generalized Expectation-maximization (SAGE) channel estimation algorithm for Multiple-Input Multiple-Output (MIMO)-Orthogonal frequency division multiplexing (OFDM) system is proposed. According to the training symbols inserted in the head of sub-frame, we get the initial estimation of joint algorithm with the Maximum a posteriori (MAP) estimator. Then, an iterative optimization loop which joints the SAGE’s E&M steps and GS based data detection is employed. In the iteration process, we apply channel estimation of the previous transmitted symbols to initial estimation in the current transmitted symbols with GS based data detection, update channel estimate and data detection until converge. All transmitted symbols can be updated in turn. The simulation results show that our proposed joint algorithm for MIMO-OFDM system have low Bit Error Rate (BER), and also have lower complexity than joint Maximum-Likelihood (ML) based data detection and SAGE based channel estimation algorithm.

An A Posteriori Error Analysis of Mixed Finite Element Galerkin Approximations to Second Order Linear Parabolic Problems

In this article, a posteriori error estimates are derived for mixed finite element Galerkin approximations to second order linear parabolic initial and boundary value problems. Using mixed elliptic reconstructions, a posteriori error estimates in $L^{\infty}(L^2)$- and $L^2(L^2)$-norms for the solution as well as its flux are proved for the semidiscrete scheme. Finally, based on a backward Euler method, a completely discrete scheme is analyzed and a posteriori error bounds are derived, which improves upon earlier results on a posteriori estimates of mixed finite element approximations to parabolic problems. Results of numerical experiments verifying the efficiency of the estimators have also been provided.

Certified Reduced Basis Method for the Electric Field Integral Equation

In [B. Fares et al., J. Comput. Phys., 230 (2011), pp. 5532--5555], a reduced basis method (RBM) for the electric field integral equation (EFIE) using the boundary element method (BEM) is developed, based on a simplified a posteriori error estimator for the greedy-based snapshot selection. In this paper, we extend this work and propose a certified RBM for the EFIE based on a mathematically rigorous a posteriori estimator. A central difficulty of the certified method is that the intrinsic solution space of the EFIE is ${\boldsymbol{H}^{-1/2}_{{\rm div}}(\Gamma)}$, inducing a relatively complicated norm. Since the measured error consists of the difference between the reduced basis solution and the boundary element solution, which is a member of the discrete boundary element space, we clarify that the intrinsic norm can be replaced by an alternative norm and in this work use the $\boldsymbol{H}({\rm div})$-norm, which is computable and demonstrated to not degrade the quality of the error estimator. A successive constraint method (SCM) for complex matrices is discussed in detail, and numerical tests for the SCM and then the certified RBM confirm the analysis.

A posteriori accuracy estimations of solutions to ill-posed inverse problems and extra-optimal regularizing algorithms for their solution

A new scheme of a posteriori accuracy estimation for approximate solutions to ill-posed inverse problems is presented along with an algorithm of calculating this estimation. A new notion of extra-optimal regularizing algorithmis introduced as a method for solving ill-posed inverse problems having optimal in order a posteriori accuracy estimation. Sufficient conditions of extra-optimality are formulated and an example of extra-optimal regularizing algorithm is given. The developed theory is illustrated by numerical experiments.

Research on Satellite Platform Vibration Compensation Method Based on Liquid Crystal Steering System

This paper presents the disturbance compensation scheme based on liquid crystal steering system according to the problem that satellite platform vibration affects the tracking and pointing accuracy in space optical communication. The mathematical model of liquid crystal steering system is built. For improving convergence speed, the adaptive controller based on posteriori estimation errors RLS lattice filter is designed, which not only achieves fast adaptation, but also reaches optimal steady-state disturbance compensation. Finally, the simulation results show that this control scheme is valid and feasible.