Posterior Posterior Probability Density Function Research Articles

Large-scale Gravitational Lens Modeling with Bayesian Neural Networks for Accurate and Precise Inference of the Hubble Constant

Abstract We investigate the use of approximate Bayesian neural networks (BNNs) in modeling hundreds of time delay gravitational lenses for Hubble constant (H 0) determination. Our BNN was trained on synthetic Hubble Space Telescope quality images of strongly lensed active galactic nuclei with lens galaxy light included. The BNN can accurately characterize the posterior probability density functions (PDFs) of model parameters governing the elliptical power-law mass profile in an external shear field. We then propagate the BNN-inferred posterior PDFs into an ensemble H 0 inference, using simulated time delay measurements from a plausible dedicated monitoring campaign. Assuming well-measured time delays and a reasonable set of priors on the environment of the lens, we achieve a median precision of 9.3% per lens in the inferred H 0. A simple combination of a set of 200 test lenses results in a precision of 0.5 km s−1 Mpc−1 (0.7%), with no detectable bias in this H 0 recovery test. The computation time for the entire pipeline—including the generation of the training set, BNN training and H 0 inference—translates to 9 minutes per lens on average for 200 lenses and converges to 6 minutes per lens as the sample size is increased. Being fully automated and efficient, our pipeline is a promising tool for exploring ensemble-level systematics in lens modeling for H 0 inference.

A Bayesian inference framework for fault slip distributions based on ensemble modelling of the uncertainty of underground structure: with a focus on uncertain fault dip

SUMMARYThe model prediction errors that originate from the uncertainty of underground structure are often a major contributor of the errors between the data and the model predictions in fault slip estimation using geodetic or seismic waveform data. However, most studies on slip inversions either neglect the model prediction errors or do not distinguish them from observation errors. Several methods that explicitly incorporated the model prediction errors in slip estimation, which has been proposed in the past decade, commonly assumed a Gaussian distribution for the stochastic property of the model prediction errors to simplify the formulation. Moreover, the information on both the slip distribution and the underground structure is expected to be successfully extracted from the data by incorporating the stochastic property of the model prediction errors. In this study, we propose a novel flexible Bayesian inference framework for estimating fault slips that can accurately incorporate non-Gaussian model prediction errors. This method considers the uncertainty of the underground structure, including fault geometry, based on the ensemble modelling of the uncertainty of Green’s functions. Furthermore, the framework allows the estimation of the posterior probability density function (PDF) of the parameters of the underground structure by calculating the likelihood of each sample in the ensemble. We performed numerical experiments for estimating the slip deficit rate (SDR) distribution on a 2-D thrust fault using synthetic data of surface displacement rates, which is the simplest problem setting that can essentially demonstrate the fundamental idea and validate the advantage of the proposed method. In the experiments, the dip angle of the fault plane was the parameter used to characterize the underground structure. The proposed method succeeded in estimating a posterior PDF of SDR that is consistent with the true one, despite the uncertain and inaccurate information of the dip angle. In addition, the method could estimate a posterior PDF of the dip angle that has a strong peak near the true angle. In contrast, the estimation results obtained using a conventional approach, which introduces regularization based on smoothing constraints and does not explicitly distinguish the prediction and observation errors, included a significant amount of bias, which was not noted in the results obtained using the proposed method. The estimation results obtained using different settings of the parameters suggested that inaccurate prior information of the underground structure with a small variance possibly results in significant bias in the estimated PDFs, particularly the posterior PDFs for SDR, those for the underground structure, and the posterior predicted PDF of the displacement rates. The distribution shapes of the model prediction errors for the representative model parameters in certain observation points are significantly asymmetric with large absolute values of the sample skewness, suggesting that they would not be well-modelled by Gaussian approximations.

Bayesian model updating of a twin-tower masonry structure through subset simulation optimization using ambient vibration data

A Bayesian model updating method using subset simulation optimization given ambient vibration data is proposed to improve the efficiency in sampling and avoid local optimums, which is applied to a full-scale high-rise structure. In the proposed method, the Bayesian fast Fourier transform method is used to extract the most probable values (MPVs) and coefficients of variation (COVs) of modal parameters of the structure, which are considered as the weighting factors in the likelihood function. The posterior probability density function (PDF) of the updating parameters is then derived considering the prior PDF as the regularization term. The subset simulation optimization algorithm is extended to find the global optimal solution of the posterior PDF. The MPVs and COVs of the updating parameters are finally presented. The proposed method is first verified by updating the 15 story stiffness scaling factors of a shear building model. Then, the proposed method is investigated through a numerical application of a four-story frame structure, in which accurate parameter estimations are observed independent of the prior PDF. Finally, the proposed method is applied to a real-world 13-story twin-tower masonry structure which is a modern heritage building. Ambient vibrations were measured through a series of accelerometers instrumented in the structure. The updating parameters are the moduli of elasticity of selected substructures with two model classes studied: three parameters and six parameters. The model-predicted modal parameters from the updated model are in good agreement with their identified counterparts. It is also shown that the case using three parameters provides larger evidence based on the Bayesian model class selection method. The updated refined finite element model provides a basis for the long-term structural health monitoring.

A Gaussian Mixture Model as a Proposal Distribution for Efficient Markov-Chain Monte Carlo Characterization of Uncertainty in Reservoir Description and Forecasting

SummaryGenerating an estimate of uncertainty in production forecasts has become nearly standard in the oil industry, but is often performed with procedures that yield at best a highly approximate uncertainty quantification. Formally, the uncertainty quantification of a production forecast can be achieved by generating a correct characterization of the posterior probability-density function (PDF) of reservoir-model parameters conditional to dynamic data and then sampling this PDF correctly. Although Markov-chain Monte Carlo (MCMC) provides a theoretically rigorous method for sampling any target PDF that is known up to a normalizing constant, in reservoir-engineering applications, researchers have found that it might require extraordinarily long chains containing millions to hundreds of millions of states to obtain a correct characterization of the target PDF. When the target PDF has a single mode or has multiple modes concentrated in a small region, it might be possible to implement a proposal distribution dependent on a random walk so that the resulting MCMC algorithm derived from the Metropolis-Hastings acceptance probability can yield a good characterization of the posterior PDF with a computationally feasible chain length. However, for a high-dimensional multimodal PDF with modes separated by large regions of low or zero probability, characterizing the PDF with MCMC using a random walk is not computationally feasible. Although methods such as population MCMC exist for characterizing a multimodal PDF, their computational cost generally makes the application of these algorithms far too costly for field application. In this paper, we design a new proposal distribution using a Gaussian mixture PDF for use in MCMC where the posterior PDF can be multimodal with the modes spread far apart. Simply put, the method generates modes using a gradient-based optimization method and constructs a Gaussian mixture model (GMM) to use as the basic proposal distribution. Tests on three simple problems are presented to establish the validity of the method. The performance of the new MCMC algorithm is compared with that of random-walk MCMC and is also compared with that of population MCMC for a target PDF that is multimodal.

Gaussian mixture model fitting method for uncertainty quantification by conditioning to production data

For most history matching problems, the posterior probability density function (PDF) may have multiple local maxima, and it is extremely challenging to quantify uncertainty of model parameters and production forecasts by conditioning to production data. In this paper, a novel method is proposed to improve the accuracy of Gaussian mixture model (GMM) approximation of the complex posterior PDF by adding more Gaussian components. Simulation results of all reservoir models generated during the history matching process, e.g., using the distributed Gauss-Newton (DGN) optimizer, are used as training data points for GMM fitting. The distance between the GMM approximation and the actual posterior PDF is estimated by summing up the errors calculated at all training data points. The distance is an analytical function of unknown GMM parameters such as covariance matrix and weighting factor for each Gaussian component. These unknown GMM parameters are determined by minimizing the distance function. A GMM is accepted if the distance is reasonably small. Otherwise, new Gaussian components will be added iteratively to further reduce the distance until convergence. Finally, high-quality conditional realizations are generated by sampling from each Gaussian component in the mixture, with the appropriate relative probability. The proposed method is first validated using nonlinear toy problems and then applied to a history-matching example. GMM generates better samples with a computational cost comparable to or less than other methods we tested. GMM samples yield production forecasts that match production data reasonably well in the history-matching period and are consistent with production data observed in the blind test period.

Markov chain Monte Carlo-based Bayesian model updating of a sailboat-shaped building using a parallel technique

In unidentifiable model updating problems, the posterior probability density function (PDF) of uncertain model parameters cannot be well approximated by a multivariate Gaussian distribution. An alternative solution is to estimate the posterior PDF using samples from a multilevel Markov chain Monte Carlo (MCMC) simulation. In general, the accuracy of the approximated posterior PDF highly depends on the number of MCMC samples, which, in turn, depends on the available computational power. For model updating using field test data, a large number of samples are required to ensure the accuracy of the updated results. Inevitably, the computational power needed will be largely increased. To increase the efficiency of the MCMC method, this paper puts forward the parallel MCMC method, which generates several Markov chains (instead of a single chain) using multiple CPUs. As a result, more samples are available for the approximation of the posterior PDF. With the fast development of multicore processors in desktop or even laptop computing, parallel MCMC provides an efficient way to approximate the posterior PDF in model updating accurately, even if the problem is unidentifiable. To demonstrate the algorithm, an ambient vibration test of a 20-story office building was carried out. Owing to the limited number of sensors, the vibration test was divided into multiple setups. This paper not only reports the field test and the operational modal analysis but also the model class selection and the updating of the finite element model of the office building following the parallel MCMC method. The proposed algorithm together with the case studies using field test data presented in this study contributes to the development of structural model updating and structural health monitoring (SHM) on civil engineering structures.

Robust Uncertainty Quantification Through Integration of Distributed-Gauss-Newton Optimization With a Gaussian Mixture Model and Parallelized Sampling Algorithms

Summary Uncertainty quantification of production forecasts is crucially important for business planning of hydrocarbon–field developments. This is still a very challenging task, especially when subsurface uncertainties must be conditioned to production data. Many different approaches have been proposed, each with their strengths and weaknesses. In this work, we develop a robust uncertainty–quantification work flow by seamless integration of a distributed–Gauss–Newton (GN) (DGN) optimization method with a Gaussian mixture model (GMM) and parallelized sampling algorithms. Results are compared with those obtained from other approaches. Multiple local maximum–a–posteriori (MAP) estimates are determined with the local–search DGN optimization method. A GMM is constructed to approximate the posterior probability–density function (PDF) by reusing simulation results generated during the DGN minimization process. The traditional acceptance/rejection (AR) algorithm is parallelized and applied to improve the quality of GMM samples by rejecting unqualified samples. AR–GMM samples are independent, identically distributed samples that can be directly used for uncertainty quantification of model parameters and production forecasts. The proposed method is first validated with 1D nonlinear synthetic problems with multiple MAP points. The AR–GMM samples are better than the original GMM samples. The method is then tested with a synthetic history–matching problem using the SPE01 reservoir model (Odeh 1981; Islam and Sepehrnoori 2013) with eight uncertain parameters. The proposed method generates conditional samples that are better than or equivalent to those generated by other methods, such as Markov–chain Monte Carlo (MCMC) and global–search DGN combined with the randomized–maximum–likelihood (RML) approach, but have a much lower computational cost (by a factor of five to 100). Finally, it is applied to a real–field reservoir model with synthetic data, with 235 uncertain parameters. A GMM with 27 Gaussian components is constructed to approximate the actual posterior PDF. There are 105 AR–GMM samples accepted from the 1,000 original GMM samples, and they are used to quantify the uncertainty of production forecasts. The proposed method is further validated by the fact that production forecasts for all AR–GMM samples are quite consistent with the production data observed after the history–matching period. The newly proposed approach for history matching and uncertainty quantification is quite efficient and robust. The DGN optimization method can efficiently identify multiple local MAP points in parallel. The GMM yields proposal candidates with sufficiently high acceptance ratios for the AR algorithm. Parallelization makes the AR algorithm much more efficient, which further enhances the efficiency of the integrated work flow.

Covariance control joint integrated probabilistic data association filter for multi‐target tracking

The joint integrated probabilistic data association (JIPDA) filter is effective for automatic multi-target tracking in cluttered environment. At each time step, the posterior probability density function (PDF) is approximated by a Gaussian PDF to estimate track states. However, JIPDA suffers from the track coalescence problem. When targets are closely spaced, their tracking gates may overlap, so that the track state estimates become biased and tend to coalesce. The authors note that the covariance can influence both the size of the tracking gate and the accuracy of the Gaussian approximation. Therefore, in this study, the posterior PDF is optimised by controlling the covariance to improve the accuracy of the track state estimates. A cost function measuring the track estimate error covariance is developed as the optimisation criterion, and an iterative strategy is adopted for minimising it. Finally, the posterior PDF with the minimal cost function is derived as the desired one. The theoretical analysis and example show that the proposed approach can reduce the extent of tracking gate overlap and improve the accuracy of the Gaussian approximation. Simulation results show that the new approach can handle track coalescence and performs better than the traditional methods.

Bayes-Mode-ID: A Bayesian modal-component-sampling method for operational modal analysis

A Bayesian modal-component-sampling system identification (Bayes-Mode-ID) method is developed in this paper. This method can efficiently identify the modal parameters of civil engineering structures under operational conditions even when the number of measured degrees of freedom (DOFs) is large. The mathematical model of the dynamic system is constructed with the modal parameters being the system parameters and the posterior probability density function (PDF) of these modal parameters is formulated using Bayes theorem. Bayesian modal analysis is conducted through generating samples of the modal parameters in the important regions of the posterior PDF. The proposed method can identify the most probable (maximum posterior) values (MPVs) of the modal parameters, together with the corresponding posterior uncertainties based on the generated samples, without assuming an approximate form for the posterior PDF. There are two main difficulties in sampling modal parameters from the posterior PDF. Firstly, it is not possible to analytically normalize the posterior PDF. Secondly, the number of the modal parameters is usually large so the samples cannot be efficiently generated in the important region of the posterior PDF. The proposed component sampling algorithm is tailor made to handle these two problems. This paper covers the theoretical development of the Bayes-Mode-ID for operational modal analysis together with two experimental case studies under laboratory conditions.

An efficient adaptive sequential Monte Carlo method for Bayesian model updating and damage detection

This paper reports the development of an efficient adaptive sequential Monte Carlo (ASMC) method for Bayesian model updating and damage detection of a structural system using measured vibration data. The proposed method can efficiently tackle two challenging problems commonly encountered in Bayesian inference, namely, identifying the posterior probability density function (PDF) in a complicated parameter space and evaluating the high-dimensional integral. The posterior PDF is identified through sampling from a series of bridge PDFs. A new formulation based on the idea of a backward kernel is proposed. This formulation makes use of the process of sampling at multiple levels and the optimal situation in which the importance density equals the bridge PDF. A new adaptive sampling scheme using importance weights is proposed to generate samples in the important region of the posterior PDF. Rather than directly controlling the uncertainty measure of the bridge PDF in each level, the ASMC method allows the important regions of these PDFs to change adaptively. The model updating methodology was experimentally verified using a four-floor shear-building model. The effects of different amounts of measured information on the uncertainty of the model updating results were studied. The application of the proposed methodology in structural damage detection was experimentally investigated using a scaled transmission tower model. The probability of damage was calculated using the posterior PDF constructed by the ASMC method. The structural damage was clearly identified from the probability of damage in the case studies.

Inversion of Surface Deformation Data for Rapid Estimates of Source Parameters and Uncertainties: A Bayesian Approach

AbstractNew satellite missions (e.g., the European Space Agency's Sentinel‐1 constellation), advances in data downlinking, and rapid product generation now provide us with the ability to access space‐geodetic data within hours of their acquisition. To truly take advantage of this opportunity, we need to be able to interpret geodetic data in a prompt and robust manner. Here we present a Bayesian approach for the inversion of multiple geodetic data sets that allows a rapid characterization of posterior probability density functions (PDFs) of source model parameters. The inversion algorithm efficiently samples posterior PDFs through a Markov chain Monte Carlo method, incorporating the Metropolis‐Hastings algorithm, with automatic step size selection. We apply our approach to synthetic geodetic data simulating deformation of magmatic origin and demonstrate its ability to retrieve known source parameters. We also apply the inversion algorithm to interferometric synthetic aperture radar data measuring co‐seismic displacements for a thrust‐faulting earthquake (2015 Mw 6.4 Pishan earthquake, China) and retrieve optimal source parameters and associated uncertainties. Given its robustness and rapidity in estimating deformation source parameters and uncertainties, our Bayesian framework is capable of taking advantage of real‐time geodetic measurements. Thus, our approach can be applied to geodetic data to study magmatic, tectonic, and other geophysical processes, especially in rapid‐response operational settings (e.g., volcano observatories). Our algorithm is fully implemented in a MATLAB®‐based software package (Geodetic Bayesian Inversion Software) that we make freely available to the scientific community.

Automated reliability assessment for spectroscopic redshift measurements

Context.Future large-scale surveys, such as the ESAEuclidmission, will produce a large set of galaxy redshifts (≥106) that will require fully automated data-processing pipelines to analyze the data, extract crucial information and ensure that all requirements are met. A fundamental element in these pipelines is to associate to each galaxy redshift measurement a quality, or reliability, estimate.Aim.In this work, we introduce a new approach to automate the spectroscopic redshift reliability assessment based on machine learning (ML) and characteristics of the redshift probability density function.Methods.We propose to rephrase the spectroscopic redshift estimation into a Bayesian framework, in order to incorporate all sources of information and uncertainties related to the redshift estimation process and produce a redshift posterior probability density function (PDF). To automate the assessment of a reliability flag, we exploit key features in the redshift posterior PDF and machine learning algorithms.Results.As a working example, public data from the VIMOS VLT Deep Survey is exploited to present and test this new methodology. We first tried to reproduce the existing reliability flags using supervised classification in order to describe different types of redshift PDFs, but due to the subjective definition of these flags (classification accuracy ~58%), we soon opted for a new homogeneous partitioning of the data into distinct clusters via unsupervised classification. After assessing the accuracy of the new clusters via resubstitution and test predictions (classification accuracy ~98%), we projected unlabeled data from preliminary mock simulations for theEuclidspace mission into this mapping to predict their redshift reliability labels.Conclusions.Through the development of a methodology in which a system can build its own experience to assess the quality of a parameter, we are able to set a preliminary basis of an automated reliability assessment for spectroscopic redshift measurements. This newly-defined method is very promising for next-generation large spectroscopic surveys from the ground and in space, such asEuclidand WFIRST.

Probabilistic calibration of discrete element simulations using the sequential quasi-Monte Carlo filter

The calibration of discrete element method (DEM) simulations is typically accomplished in a trial-and-error manner. It generally lacks objectivity and is filled with uncertainties. To deal with these issues, the sequential quasi-Monte Carlo (SQMC) filter is employed as a novel approach to calibrating the DEM models of granular materials. Within the sequential Bayesian framework, the posterior probability density functions (PDFs) of micromechanical parameters, conditioned to the experimentally obtained stress–strain behavior of granular soils, are approximated by independent model trajectories. In this work, two different contact laws are employed in DEM simulations and a granular soil specimen is modeled as polydisperse packing using various numbers of spherical grains. Knowing the evolution of physical states of the material, the proposed probabilistic calibration method can recursively update the posterior PDFs in a five-dimensional parameter space based on the Bayes’ rule. Both the identified parameters and posterior PDFs are analyzed to understand the effect of grain configuration and loading conditions. Numerical predictions using parameter sets with the highest posterior probabilities agree well with the experimental results. The advantage of the SQMC filter lies in the estimation of posterior PDFs, from which the robustness of the selected contact laws, the uncertainties of the micromechanical parameters and their interactions are all analyzed. The micro–macro correlations, which are byproducts of the probabilistic calibration, are extracted to provide insights into the multiscale mechanics of dense granular materials.

A new formulation of vector weights in localized particle filters

Particle filters (PFs) constitute a sequential data assimilation method based on the Monte Carlo approximation of Bayesian estimation theory. Standard PFs use scalar weights derived from the likelihood of the approximate posterior probability density functions (PDFs) of observations and use resampling schemes to generate posterior particles. However, the scalar weights approach interferes with the localization algorithm and often results in filter degeneracy.Recently, a localized particle filter (LPF) was developed by extending the scalar weights of PFs to vector weights, which produces various (local) posterior PDFs for different model grids and variables. With a sampling and merging approach in the resampling, a LPF can effectively solve the filter degeneracy problem and offer a practical, efficient algorithm for localization. However, this algorithm assumes the variations in the weights of a state variable of neighbouring grids to be continuous and uses a spatially linear interpolation of PF weights to determine the local weights.In this paper, we first analyse the possible concerns associated with the linear continuity of PF weights. This assumption is found to challenge the theoretical properties of nonlinear and non‐Gaussian variations in weights and alleviate the intrinsic spatial variations of PF weights. On this basis, we propose a new algorithm to produce vector weights for PFs for neighbouring grids. Numerical experiments using the Lorenz 96 model show that our new LPF performs better than the existing LPF algorithm, indicating the advantages and potential applications of this new algorithm of vector weights in the field of data assimilation.

Efficient joint probabilistic data association filter based on Kullback–Leibler divergence for multi‐target tracking

To deal with the track coalescence problem of the joint probabilistic data association (JPDA) filter, a novel approach based on the Kullback-Leibler divergence (KLD) is developed in this study. In JPDA, the posterior probability density function (PDF) is approximated by a single Gaussian PDF at each time step. The authors propose a novel method of optimising the posterior PDF to obtain a single Gaussian PDF that minimises the KLD from the posterior PDF. However, the KLD is intractable because the posterior PDF is a Gaussian mixture model. Hence, an approximation of the KLD is introduced as the cost function to simplify the problem. The cost function is a linear combination of multiple objective functions which are not conflicting. Therefore, the minimisation of the cost function is easier to operate, because all objective functions can be optimised simultaneously. In addition, an iterative method is adopted for minimising the cost function. In the iteration process, the tracking accuracy is improved with the monotonic decrease of the cost function. Theoretical analysis and example show the feasibility of the proposed approach. Simulation results demonstrate the advantages of the new approach over others when tracking closely spaced targets with contaminated sensor measurements.

Objectively combining AR5 instrumental period and paleoclimate climate sensitivity evidence

Combining instrumental period evidence regarding equilibrium climate sensitivity with largely independent paleoclimate proxy evidence should enable a more constrained sensitivity estimate to be obtained. Previous, subjective Bayesian approaches involved selection of a prior probability distribution reflecting the investigators’ beliefs about climate sensitivity. Here a recently developed approach employing two different statistical methods—objective Bayesian and frequentist likelihood-ratio—is used to combine instrumental period and paleoclimate evidence based on data presented and assessments made in the IPCC Fifth Assessment Report. Probabilistic estimates from each source of evidence are represented by posterior probability density functions (PDFs) of physically-appropriate form that can be uniquely factored into a likelihood function and a noninformative prior distribution. The three-parameter form is shown accurately to fit a wide range of estimated climate sensitivity PDFs. The likelihood functions relating to the probabilistic estimates from the two sources are multiplicatively combined and a prior is derived that is noninformative for inference from the combined evidence. A posterior PDF that incorporates the evidence from both sources is produced using a single-step approach, which avoids the order-dependency that would arise if Bayesian updating were used. Results are compared with an alternative approach using the frequentist signed root likelihood ratio method. Results from these two methods are effectively identical, and provide a 5–95% range for climate sensitivity of 1.1–4.05 K (median 1.87 K).

Bayesian system identification based on hierarchical sparse Bayesian learning and Gibbs sampling with application to structural damage assessment

Bayesian system identification has attracted substantial interest in recent years for inferring structural models based on measured dynamic response from a structural dynamical system. The focus in this paper is Bayesian system identification based on noisy incomplete modal data where we can impose spatially-sparse stiffness changes when updating a structural model. To this end, based on a similar hierarchical sparse Bayesian learning model from our previous work, we propose two Gibbs sampling algorithms. The algorithms differ in their strategies to deal with the posterior uncertainty of the equation-error precision parameter, but both sample from the conditional posterior probability density functions (PDFs) for the structural stiffness parameters and system modal parameters. The effective dimension for the Gibbs sampling is low because iterative sampling is done from only three conditional posterior PDFs that correspond to three parameter groups, along with sampling of the equation-error precision parameter from another conditional posterior PDF in one of the algorithms where it is not integrated out as a “nuisance” parameter. A nice feature from a computational perspective is that it is not necessary to solve a nonlinear eigenvalue problem of a structural model. The effectiveness and robustness of the proposed algorithms are illustrated by applying them to the IASE-ASCE Phase II simulated and experimental benchmark studies. The goal is to use incomplete modal data identified before and after possible damage to detect and assess spatially-sparse stiffness reductions induced by any damage. Our past and current focus on meeting challenges arising from Bayesian inference of structural stiffness serves to strengthen the capability of vibration-based structural system identification but our methods also have much broader applicability for inverse problems in science and technology where system matrices are to be inferred from noisy partial information about their eigenquantities.

Highly Efficient Probabilistic Finite Element Model Updating Using Intelligent Inference With Incomplete Modal Information

A highly efficient probabilistic framework of finite element model updating in the presence of measurement noise/uncertainty using intelligent inference is presented. This framework uses incomplete modal measurement information as input and is built upon the Bayesian inference approach. To alleviate the computational cost, Metropolis–Hastings Markov chain Monte Carlo (MH MCMC) is adopted to reduce the size of samples required for repeated finite element modal analyses. Since adopting such a sampling technique in Bayesian model updating usually yields a sparse posterior probability density function (PDF) over the reduced parametric space, Gaussian process (GP) is then incorporated in order to enrich analysis results that can lead to a comprehensive posterior PDF. The PDF obtained with densely distributed data points allows us to find the most optimal model parameters with high fidelity. To facilitate the entire model updating process with automation, the algorithm is implemented under ansys Parametric Design Language (apdl) in ansys environment. The effectiveness of the new framework is demonstrated via systematic case studies.

Inferring and Propagating Kinetic Parameter Uncertainty for Condensed Phase Burning Models

Kinetic parameters for serial pyrolysis reactions were calibrated from thermogravimetric analysis (TGA) data using Bayesian inference via Markov Chain Monte Carlo (MCMC) simulations assuming a serial reaction mechanism. Calibrations were performed for high-impact polystyrene (HIPS), bisphenol-A polycarbonate (PC), and poly(vinyl chloride) (PVC) at heating rates of 3 K/min and 10 K/min. The resulting parameter inferences are probabilistic as opposed to the point estimates calibrated in previous studies and are visualized using posterior probability density functions (PDFs) generated by kernel density estimation (KDE). Correlations between the parameters are identified and discussed. In particular, it is clear that pre-exponential constants and activation energies for a given reaction have a strong positive correlation. It is hypothesized that the degree of overlap in the posterior PDFs might be a measure of model adequacy. Point-estimates of the kinetic parameters were made by finding the mode of the posterior PDFs. For HIPS, it was determined that a one-reaction pyrolysis model is most appropriate, and that the posterior modes for \(\log \left( A_1\right) \) and \(E_1\) are \(19.5\,\log (1/s)\) and 292 kJ/mol, respectively, for the 3 K/min data. To evaluate the effect of kinetic parameter uncertainty on predictions of burning rate, samples from the posterior PDF were used to simulate gasification and cone calorimetry experiments using the fire dynamics simulator (FDS). In some cases, it was found that models with fewer parameters provided better predictions due to over-fitting associated with greater model complexity. Another important observation is that for the predictions of PVC cone calorimetry, the time to peak heat release rate can range from around 40 s to 180 s for a number of different kinetic parameter combinations that all fit the TGA data fairly well. It is argued that the proposed methodology is necessary for progress in modeling of condensed phase physics for fire problems as it supports both model validation and engineering predictions.

The Effect of the Equivalent-Weights Particle Filter on Dynamical Balance in a Primitive Equation Model

Abstract The disadvantage of the majority of data assimilation schemes is the assumption that the conditional probability density function of the state of the system given the observations [posterior probability density function (PDF)] is distributed either locally or globally as a Gaussian. The advantage, however, is that through various different mechanisms they ensure initial conditions that are predominantly in linear balance and therefore spurious gravity wave generation is suppressed. The equivalent-weights particle filter is a data assimilation scheme that allows for a representation of a potentially multimodal posterior PDF. It does this via proposal densities that lead to extra terms being added to the model equations and means the advantage of the traditional data assimilation schemes, in generating predominantly balanced initial conditions, is no longer guaranteed. This paper looks in detail at the impact the equivalent-weights particle filter has on dynamical balance and gravity wave generation in a primitive equation model. The primary conclusions are that (i) provided the model error covariance matrix imposes geostrophic balance, then each additional term required by the equivalent-weights particle filter is also geostrophically balanced; (ii) the relaxation term required to ensure the particles are in the locality of the observations has little effect on gravity waves and actually induces a reduction in gravity wave energy if sufficiently large; and (iii) the equivalent-weights term, which leads to the particles having equivalent significance in the posterior PDF, produces a change in gravity wave energy comparable to the stochastic model error. Thus, the scheme does not produce significant spurious gravity wave energy and so has potential for application in real high-dimensional geophysical applications.