Posteriori Estimates Research Articles

A retinex inspired deep image prior model for despeckling and deblurring of aerial and satellite images using proximal gradient method

ABSTRACT Unsupervised learning models, particularly in the remote sensing domain, have gained significant attention in recent years. Various degradations in the satellite images, primarily occurring during acquisition, pose a substantial hurdle in obtaining reliable ground truth and extensive training data. The Deep Image Prior model (DIP) addresses these issues by performing restoration tasks using a single image, relying on the implicit regularization inherent in the network architecture. In this paper, we propose a novel approach, integrating the DIP model within the retinex framework to restore aerial and satellite images from the Gamma distributed speckles and linear shift-invariant Gaussian blur along with contrast enhancement using the alternating proximal gradient descent ascent (PGDA) method. Our proposed methodology combines implicit regularization with explicit total variational (TV) regularization, incorporating automated estimation of local regularization parameters. The data-fidelity component in the optimization function is formulated using the Bayesian Maximum A posteriori (MAP) estimate, assuming the speckles follow the Gamma distribution. Demonstration of despeckling and deblurring alone and in addition as a combined task is carried out on aerial and Synthetic Aperture Radar (SAR) images with different resolutions and polarization from various sources. Results obtained are compared with various state-of-the-art despeckling and deblurring models using distinct image quality metrics such as Equivalent Number of Looks (ENL), Contrast to Noise Ratio (CNR), Edge Preserving Index (EPI), Entropy, Global Contrast Factor (GCF), Natural Image Quality Evaluator (NIQE), Blind/Referenceless Image Spatial Quality Evaluator (BRISQUE) and Bradley-Terry (B-T) score based on the various factors. The quality of restored images depicted superior performance of the proposed method over the existing models under study.

Multi-state Markovian-random walk adaptive filter for time-varying block sparse system identification

This paper presents a higher order multi-state Markov model for block sparsity model of a system impulse response. To capture the time-varying behavior of the system, a random walk model for temporal evolution is incorporated; resulting in a two-dimensional multi-state Markov-Random walk model across spatial and temporal domains. Additionally, a block batch data structure is utilized for the adaptive filter design. The proposed adaptive filter is aimed to find the Maximum A Posteriori (MAP) estimator of the batch time-varying impulse response. The associated optimization problem is demonstrated to be convex and efficiently solvable through the Steepest-Descent (SD) approach. Furthermore, the final recursion of the proposed adaptive filter is derived and a mean convergence analysis of the algorithm is provided. Simulation results show the effectiveness of the proposed algorithm under severe time-varying conditions of the system impulse response, where alternative algorithms may exhibit divergence or poor convergence. Synthetic experiments and real-world experiments using acoustic channel estimation further validate the superiority of the proposed algorithm.

A posteriori-based, local multilevel mesh refinement for the Darcy porous media flow problem

In this work we develop an a posteriori-based adaptive local multilevel mesh refinement strategy for the Darcy porous media flow problem discretized with the finite volume method. Our approach is based on coupling the FIC “flux interface correction” method [1] and the equilibrated flux a posteriori error estimate [2]. The choice of the FIC method is motivated by the fact that it is well-adapted to problems discretized in conservative form since that the FIC provides corrections by balancing fluxes computed from both coarse and fin grids across the interface. The equilibrated flux a posteriori error estimate provides a guaranteed upper bound on the total error in the fluxes and takes advantage of the conservative scheme in such a way that the estimate can be easily coded and cheaply evaluated. We use this a posteriori estimate to detect automatically the zone of interest where the error is dominant in order to defined the local sub-grids. Numerous numerical experiments on practical problems illustrate the performance of our methodology. A comparison with a classical h-adaptive method is also carried out in this work.

State updating in Xin’anjiang model by Asynchronous Ensemble Kalman filtering with enhanced error models

For flood simulation in humid catchments, utilizing discharge observations to update the states of hydrological models may enhance performance. Asynchronous Ensemble Kalman Filter (AEnKF), an asynchronous variant of the Ensemble Kalman Filter (EnKF), holds substantial application potential in hydrological assimilation due to its ability to utilize more observations with almost no additional computational time. This study employs AEnKF to update the state variables of the Xin’anjiang model, necessitating the use of error models to perturb both model states and observations to generate ensemble spread. The Bias-corrected Gaussian Error Model (BGEM) is used to mitigate the systematic bias brought by perturbating soil moisture, and the Maximum a Posteriori Estimation Method (MAP) is employed for the estimation of hyperparameters of error models. Through synthetic and real-world data testing, it has been validated that the rectification of soil moisture perturbations using the BGEM significantly reduces the systematic bias induced by Gaussian perturbations. Moreover, the assimilation scheme introduced in this study, based on AEnKF with enhanced error models, outperforms the EnKF with those models. It substantially reduces the accumulation of past errors in the initial conditions at the start of the forecast, thereby aiding in elevating the accuracy of flood forecasting.

Modal–based uncertainty quantification for deterministically estimated structural parameters in low-fidelity model updating of complex connections

Modeling complex joints in structures entails significant time and effort, necessitating simplifications. Epistemic uncertainties arising from low-fidelity modeling can be quantified through probabilistic model updating. However, finding a surrogate physical model to represent simplified joint configurations poses challenges. Additionally, establishing a Bayesian formulation capable of incorporating structural parameters of connections is necessary. This study employs a validated simplifying parameterization approach for surrogate modeling of complex semi-rigid connections in a benchmark laboratory steel grid. It proposes a modal probabilistic Bayesian methodology to quantify uncertainties in the structure's joints. Three modal-based objective functions are utilized for finite element model updating. The modal properties of the structure are extracted by experimental modal analysis during an impact test, which will be utilized in the model updating process. Deterministic and probabilistic structural parameter estimations are integrated to enhance the robustness of the Bayesian technique. Furthermore, a guideline for selecting optimal hyperparameters is provided. Results demonstrate that utilizing deterministically estimated parameters as prior knowledge can facilitate and improve modal probabilistic model updating for structures with complex joints. Also, it is found that despite significant simplifications of joints, structural parameter tolerance around the maximum a posteriori estimate in surrogate models remains low.

Tikhonov regularization as a nonparametric method for uncertainty quantification in aggregate data problems

Dynamical systems are widely used in mathematical models in engineering and in the applied sciences. However, the parameters of these systems are usually unknown and they must be estimated from measured data. Performing parameter estimation and quantifying the uncertainty in these estimates becomes critical to realize the predictive power of dynamical systems. In some applications, only measurements about the states of an ensemble of systems are available, where each one evolves according to the same model but for different parameter values, leading to aggregate data problems. In this case, an approach proposed in the literature is to estimate a density that is consistent with some estimates of the expected value of some quantities of interest. To solve the problem in practice, the density is discretized and, to solve the resulting ill-posed problem, Tikhonov regularization is used. In this work, we propose a Bayesian model that shows this approach can be interpreted as a maximum a posteriori estimate for the density. We show that the infinite-dimensional problem defining the MAP can be reformulated as a finite-dimensional problem that does not require discretizing the density. In several cases of interest, this problem is convex, unconstrained, and the objective function is smooth. Thus, it can be solved using algorithms with optimal convergence rates. The trade-off is that the objective is defined by an integral. However, our results characterize the regularity of the integrand, allowing the use of tailored numerical schemes to approximate it. Furthermore, our theoretical results characterize the form of the optimal density, whereas our numerical results illustrate the performance of our method and confirm our theoretical findings.

Parameters correction of an elliptical equation using an a posteriori estimate

In nature several phenomena touch humanity to act and control these phenomena we try to model their evolution. To simplify the study of the equations obtained, most of the time we try to use linear forms and in this way modelling errors are made which can influence the correct analysis of these phenomena. In this work we focus on the Richards' equation, where the coefficients changes their forms from a given value hs: This value is unknown then the coefficients are approximated. We prove an a priori and an a posteriori estimates on the modelling error. This estimates allows us, using local indicators, to build an adaptive algorithm to control the modelling error and automatically determine the "best" approximation of hs. Numerical results confirm theconvergence of this procedure and the interest of this approach.

A priori and a posteriori estimates for solving one evolutionary inverse problem

This article considers an initial-boundary value problem for a system of parabolic equations, which arises when studying the flow of a binary mixture in a horizontal channel with walls heated non-uniformly. The problem was reduced to a sequence of initial-boundary value problems with Dirichlet or Neumann conditions. Among them, an inverse problem with a non-local overdetermination condition was distinguished. The solution was constructed using the Fourier method and validated as classical. The behavior of the non-stationary solution at large times was discussed. It was shown that certain functions within the solution tend to their stationary analogs exponentially at large times. For some functions, only boundedness was proved. The problem and its solution are relevant for modeling the thermal modes associated with the separation of liquid mixtures.

Learning Representations on the Unit Sphere: Investigating Angular Gaussian and Von Mises-Fisher Distributions for Online Continual Learning

We use the maximum a posteriori estimation principle for learning representations distributed on the unit sphere. We propose to use the angular Gaussian distribution, which corresponds to a Gaussian projected on the unit-sphere and derive the associated loss function. We also consider the von Mises-Fisher distribution, which is the conditional of a Gaussian in the unit-sphere. The learned representations are pushed toward fixed directions, which are the prior means of the Gaussians; allowing for a learning strategy that is resilient to data drift. This makes it suitable for online continual learning, which is the problem of training neural networks on a continuous data stream, where multiple classification tasks are presented sequentially so that data from past tasks are no longer accessible, and data from the current task can be seen only once. To address this challenging scenario, we propose a memory-based representation learning technique equipped with our new loss functions. Our approach does not require negative data or knowledge of task boundaries and performs well with smaller batch sizes while being computationally efficient. We demonstrate with extensive experiments that the proposed method outperforms the current state-of-the-art methods on both standard evaluation scenarios and realistic scenarios with blurry task boundaries. For reproducibility, we use the same training pipeline for every compared method and share the code at https://github.com/Nicolas1203/ocl-fd.

Exact, Fast and Expressive Poisson Point Processes via Squared Neural Families

We introduce squared neural Poisson point processes (SNEPPPs) by parameterising the intensity function by the squared norm of a two layer neural network. When the hidden layer is fixed and the second layer has a single neuron, our approach resembles previous uses of squared Gaussian process or kernel methods, but allowing the hidden layer to be learnt allows for additional flexibility. In many cases of interest, the integrated intensity function admits a closed form and can be computed in quadratic time in the number of hidden neurons. We enumerate a far more extensive number of such cases than has previously been discussed. Our approach is more memory and time efficient than naive implementations of squared or exponentiated kernel methods or Gaussian processes. Maximum likelihood and maximum a posteriori estimates in a reparameterisation of the final layer of the intensity function can be obtained by solving a (strongly) convex optimisation problem using projected gradient descent. We demonstrate SNEPPPs on real, and synthetic benchmarks, and provide a software implementation.

Spatial air quality prediction in urban areas via message passing

Air pollution in urban areas poses a significant and pressing challenge for modern society. Unfortunately, the existing network of pollution detectors in many cities is limited in scope and fails to adequately cover the entire geographical area. Consequently, the implementation of spatial prediction algorithms becomes essential to generate high-resolution data. In this paper, we introduce two significant contributions: 1) We formalize the air pollution prediction problem as a Maximum A Posteriori (MAP) estimate within the framework of a Markov Random Field and 2) we propose a message-passing algorithm, which stands out as an efficient solution that surpasses the current state of the art. The experimental procedure has been carried out using the case study of the city of Barcelona, based on a dataset extracted from the BCN Open Data portal.

Uncertainty cues amplify late positive potential responses to aversive emotional stimuli

ABSTRACT Uncertainty is unavoidable, and maladaptive responses to uncertainty may underlie the etiology and maintenance of psychopathology. A general tendency to associate uncertainty with aversive consequences, a type of covariation bias, can amplify aversive emotional experiences. To address questions about uncertainty during emotion regulation, we examined the Late Positive Potential (LPP) – an electrocortical marker of attention to and appraisal of motivationally relevant emotional stimuli – during a task designed to measure the effect of covariation bias and its emotional response consequences. Event-related potentials (ERPs) were recorded while participants (N = 52) were presented with a pre-stimulus cue that either conveyed information about the valence of an upcoming emotional image, or left them in ambiguity. We replicated findings that demonstrate expectancy biases in a priori and online expectancies of emotion-eliciting images, as well as in a posteriori estimates for concurrence of uncertainty cues and aversive images. Moreover, we demonstrate a novel finding that uncertainty cues amplify the LPP in response to subsequent aversive emotional stimuli. These findings advance research by conjoining existing emotion regulation research on the LPP with study of the effects of uncertainty on emotional appraisal and highlight the importance of accounting for stimulus uncertainty in emotion regulation research.

The Minimally-Invasive Oral Glucose Minimal Model: Estimation of Gastric Retention, Glucose Rate of Appearance, and Insulin Sensitivity From Type 1 Diabetes Data Collected in Real-Life Conditions.

Modeling the effect of meal composition on glucose excursion would help in designing decision support systems (DSS) for type 1 diabetes (T1D) management. In fact, macronutrients differently affect post-prandial gastric retention (GR), rate of appearance (R[Formula: see text]), and insulin sensitivity (S[Formula: see text]). Such variables can be estimated, in inpatient settings, from plasma glucose (G) and insulin (I) data using the Oral glucose Minimal Model (OMM) coupled with a physiological model of glucose transit through the gastrointestinal tract (reference OMM, R-OMM). Here, we present a model able to estimate those quantities in daily-life conditions, using minimally-invasive (MI) technologies, and validate it against the R-OMM. Forty-seven individuals with T1D (weight =78±13 kg, age =42±10 yr) underwent three 23-hour visits, during which G and I were frequently sampled while wearing continuous glucose monitoring (CGM) and insulin pump (IP). Using a Bayesian Maximum A Posteriori estimator, R-OMM was identified from plasma G and I measurements, and MI-OMM was identified from CGM and IP data. The MI-OMM fitted the CGM data well and provided precise parameter estimates. GR and R[Formula: see text] model parameters were not significantly different using the MI-OMM and R-OMM (p 0.05) and the correlation between the two S[Formula: see text] was satisfactory ( ρ =0.77). The MI-OMM is usable to estimate GR, R[Formula: see text], and S[Formula: see text] from data collected in real-life conditions with minimally-invasive technologies. Applying MI-OMM to datasets where meal compositions are available will allow modeling the effect of each macronutrient on GR, R[Formula: see text], and S[Formula: see text]. DSS could finally exploit this information to improve diabetes management.

Error analysis for a spectral element method for solving two-parameter singularly perturbed diffusion equation

In this paper, we study the two-parameter spectral element method based on weighted shifted orthogonal polynomials for solving singularly perturbed diffusion equation on an interval [0, 1] which are modeled with singular parameters. We continue our study to estimate the lower bound of the weighted orthogonal polynomial coefficient and the upper bound of a posteriori error estimates of the method through different weighted norms to minimize the computational cost. Numerical examples are implemented to study the applicability and efficiency of the technique. The obtained error bounds for the coefficient of orthogonal polynomials and the posteriori estimates fall within the bounds derived in the theoretical section. It is also observed that the two weighted norms decreases when the values of [Formula: see text] and [Formula: see text] increases for the three choices of [Formula: see text] and for different values of x and y. The quality and accuracy of the solution can be realized through figures and tables.

A probabilistic algorithm for optimising the steady-state diffusional flux into a partially absorbing body

Cells in an aqueous environment absorb diffusing nutrient molecules through nanoscale protein channels in their outer membranes. Assuming that there are constraints on the number of such channels a cell can produce, we ask the question: given a nondepleting source of nutrients, what is the optimal distribution of these channels over the cell surface? We coarse-grain this problem, phrasing it as a diffusion problem with position-dependent Robin boundary conditions on the surface. The aim is to maximize the steady-state total flux through the partially absorbing surface under an integral constraint on the local reactivities. We develop an algorithm to tackle this problem that uses the stored and processed results of a particle-based simulation with reflective boundary conditions to a posteriori estimate absorption flux at essentially negligible additional computational cost. We validate the algorithm against a few cases for which analytical or semi-analytical results are available. We apply it to two examples: a spherical cell in the presence of a point source and a spheroidal cell with an isotropic source at infinity. In the former case, there is a significant gain relative to the homogeneous case, while in the latter case the gain is only 1%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\%$$\\end{document}.

Optimal nonlinear dynamic sparse model selection and Bayesian parameter estimation for nonlinear systems

This work develops algorithms for the estimation of sparse interpretable data-driven models. Linear combination of linear and nonlinear basis functions is considered for candidate models, where the optimal basis functions are selected by using the branch and bound algorithm. Model parameters are optimally estimated by employing Bayesian inferencing to seek the maximum a posteriori estimates conditioned on the available data. This is implemented in a two-step expectation-maximization algorithm enabling simultaneous estimation of noise characteristics along with the model parameters. A priori analysis is undertaken based on the ranking of the basis functions in the library. In presence of correlated noise, special properties of the Jacobian matrix are exploited for efficient pruning. Model selection is done by using modified Akaike information criterion that rewards model fitness while penalizing model size and complexity. The algorithm, called Bayesian identification of Dynamic Sparse Algebraic Models (BIDSAM), is tested for three case studies.

A DIA method based on maximum a posteriori estimate for multiple outliers

A DIA method based on maximum a posteriori estimate for multiple outliers

Automated component analysis in DOSY NMR using information criteria

This study introduces a model selection technique based on Bayesian information criteria for estimating the number of components in a mixture during Diffusion-Ordered Spectroscopy (DOSY) Nuclear Magnetic Resonance (NMR) data analysis. As the accuracy of this technique is dependent on the efficiency of parameter estimators, we further investigate the performance of the Weighted Least Squares (WLS) and Maximum a Posteriori (MAP) estimators. The WLS method, enhanced with meticulously tuned L2-regularization, effectively detects components when the difference in self-diffusion coefficients is more than two-fold, especially when the component with the smaller coefficient has a larger weight ratio. The MAP method, strengthened by a substantial database of prior information, exhibits outstanding precision, decreasing this threshold to 1.5 times. Both estimators provide weight ratio estimates with standard deviations of approximately around 1 percentage point, although the MAP method tends to overestimate the component with a larger self-diffusion coefficient. Deviations from the expected values can exceed 10 percentage points, often due to inaccuracies in component detection. The error estimates are determined using data resampling techniques derived from a large-scale 1000-point experiment and an additional five measurements from a single-component mixture. This approach allowed us to thoroughly examine data distribution characteristics, thereby laying a robust groundwork for future refinement efforts.

Adaptive nonlinear optimization of district heating networks based on model and discretization catalogs

We propose an adaptive optimization algorithm for operating district heating networks in a stationary regime. The behavior of hot water flow in the pipe network is modeled using the incompressible Euler equations and a suitably chosen energy equation. By applying different simplifications to these equations, we derive a catalog of models. Our algorithm is based on this catalog and adaptively controls where in the network which model is used. Moreover, the granularity of the applied discretization is controlled in a similar adaptive manner. By doing so, we are able to obtain optimal solutions at low computational costs that satisfy a prescribed tolerance w.r.t. the most accurate modeling level. To adaptively control the switching between different levels and the adaptation of the discretization grids, we derive error measure formulas and a posteriori error measure estimators. Under reasonable assumptions we prove that the adaptive algorithm terminates after finitely many iterations. Our numerical results show that the algorithm is able to produce solutions for problem instances that have not been solvable before.

Analysis of weakly symmetric mixed finite elements for elasticity

We consider mixed finite element methods for linear elasticity where the symmetry of the stress tensor is weakly enforced. Both an a priori and a posteriori error analysis are given for several known families of methods that are uniformly valid in the incompressible limit. A posteriori estimates are derived for both the compressible and incompressible cases. The results are verified by numerical examples.