State and Parameter Estimation in Three-Dimensional Subsurface Contaminant Transport Modeling using Kalman Filter Coupled with Monte Carlo Sampling

Error measures for fuzzy linear regression: Monte Carlo simulation approach

Fractional-order models (FOMs) have been recognized as superior tools for capturing the complex electrochemical dynamics of lithium-ion batteries, outperforming integer-order models in accuracy, robustness, and adaptability. Parameter identification (PI) is essential for FOMs, as its accuracy directly affects the model’s ability to predict battery behavior and estimate critical states such as state of charge (SOC) and state of health (SOH). In this study, a hybrid deep learning approach has been introduced for FOM PI, representing the first application of deep learning in this domain. A simulation platform was developed to generate datasets using Sobol and Monte Carlo sampling methods. Five deep learning models were constructed: long short-term memory (LSTM), gated recurrent unit (GRU), one-dimensional convolutional neural network (1DCNN), and hybrid models combining 1DCNN with LSTM and GRU. Hyperparameters were optimized using Optuna, and enhancements such as Huber loss for robustness to outliers, stochastic weight averaging, and exponential moving average for training stability were incorporated. The primary contribution lies in the hybrid architectures, which integrate convolutional feature extraction with recurrent temporal modeling, outperforming standalone models. On a test set of 1000 samples, the improved 1DCNN + GRU model achieved an overall root mean square error (RMSE) of 0.2223 and a mean absolute percentage error (MAPE) of 0.27%, representing nearly a 50% reduction in RMSE compared to its baseline. This performance surpasses that of the improved LSTM model, which yielded a MAPE of 9.50%, as evidenced by tighter scatter plot alignments along the diagonal and reduced error dispersion in box plots. Terminal voltage prediction was validated with an average RMSE of 0.002059 and mean absolute error (MAE) of 0.001387, demonstrating high-fidelity dynamic reconstruction. By advancing data-driven PI, this framework is well-positioned to enable real-time integration into battery management systems.

Take a typical cross-section of rockfill embankment slope in Yaan-Luku highway as the research object, reliability analysis is studied under the condition of water table of 840.85m, 851.50m, and loading condition of natural state and horizontal seismic acceleration of 0.2g, respectively. Raw data use Kolmogorov-Smirnov test (K-S test) to determine the distribution type of parametric variation. And the parameters were sampling with Latin hypercube sampling (LHS) method and Monte Carlo (MC) method, respectively, to obtain state function and determine safety factors and reliability indexes. A conclusion is drawn that the times of simulation based on LHS method were less than Monte Carlo method. Also the convergence of failure probability is better than the Monte Carlo method. The safety factor is greater than one and the failure probability has reached to 35.45% in condition of earthquake, which indicating that the instability of rockfill embankment slope is still possible.

Objective. Monte Carlo (MC) simulations are the benchmark for accurate radiotherapy dose calculations, notably in patient-specific high dose rate brachytherapy (HDR BT), in cases where considering tissue heterogeneities is critical. However, the lengthy computational time limits the practical application of MC simulations. Prior research used deep learning (DL) for dose prediction as an alternative to MC simulations. While accurate dose predictions akin to MC were attained, graphics processing unit limitations constrained these predictions to large voxels of 3 mm × 3 mm × 3 mm. This study aimed to enable dose predictions as accurate as MC simulations in 1 mm × 1 mm × 1 mm voxels within a clinically acceptable timeframe. Approach. Computed tomography scans of 98 breast cancer patients treated with Iridium-192-based HDR BT were used: 70 for training, 14 for validation, and 14 for testing. A new cropping strategy based on the distance to the seed was devised to reduce the volume size, enabling efficient training of 3D DL models using 1 mm × 1 mm × 1 mm dose grids. Additionally, novel DL architecture with layer-level fusion were proposed to predict MC simulated dose to medium-in-medium (D m,m ). These architectures fuse information from TG-43 dose to water-in-water (D w,w ) with patient tissue composition at the layer-level. Different inputs describing patient body composition were investigated. Main results. The proposed approach demonstrated state-of-the-art performance, on par with the MC D m,m maps, but 300 times faster. The mean absolute percent error for dosimetric indices between the MC and DL-predicted complete treatment plans was 0.17% ± 0.15% for the planning target volume V 100, 0.30% ± 0.32% for the skin D 2cc , 0.82% ± 0.79% for the lung D 2cc , 0.34% ± 0.29% for the chest wall D 2cc and 1.08% ± 0.98% for the heart D 2cc . Significance. Unlike the time-consuming MC simulations, the proposed novel strategy efficiently converts TG-43 D w,w maps into precise D m,m maps at high resolution, enabling clinical integration.

Bayesian inference is usually divided into two categories: parameter estimation and model selection. Parameter estimation is mostly performed using Markov Chain Monte Carlo (MCMC) sampling methods, most often based on the standard Metropolis-Hastings algorithm or its variants, such as slice, Gibbs' and Hamiltonian sampling. These methods can be highly inefficient in exploring multi-modal distributions or ones exhibiting degeneracies between parameters. Moreover, in order to perform Bayesian model selection, estimation of Bayesian evidence is needed which is by definition, a (possibly high dimensional) integration over the entire parameter space. Unfortunately, the computational expense involved in the evaluation of Bayesian evidence is typically an order of magnitude higher than doing the parameter estimation. In this paper, we discuss some techniques, built around the nested sampling framework, that have been developed to efficiently calculate the Bayesian evidence. We also discuss applications of Bayesian evidence in astrophysics and particle physics, in problems involving model selection, source detection, consistency checks between different data-sets, multi-model inference and in determining the constraining power of different observables.

With the continuous development of social economy, China's port construction scale has become saturated. As an important part of the harbor, the breakwater plays a crucial role in the safety of the working environment in the harbor, and the reliability of the breakwater is an assessment of its safety. Build on the previous studies, this paper puts forward a semi-submersible breakwater using waste tires. The reliability of this breakwater is analyzed by Monte Carlo method of artificial neural network based on Matlab, and the results are compared to those of Direct sampling Monte Carlo method and Important sampling Monte Carlo method. The results show that the Monte Carlo method is able to analyze the reliability of overturning failure of semi-submersible breakwater of waste tire. Compared with the other two consequences, the result is more accurate. The Monte Carlo method of artificial neural network based on Matlab has the obvious advantage in being devoted to the problem of complex structure and variable. Safety of the breakwater meets the relevant requirements and can be applied to the actual engineering. It can be seen that waste tires has a high degree of reliability, daptability, and a wide range of applications.

Special Issue on Toxics and Pathogens in the Environment

Thermal-pulse method is a powerful tool for measuring space charge distributions in polymer films. The data analysis for thermal-pulse method involves the Fredholm integral equation of the first kind, which requires an appropriate numerical procedure to obtain a solution. Various numerical techniques, including scale transformation and regulation method, are proposed. Of those numerical methods, the scale transformation (ST) is the simplest and the most widely used method. However, it presents a high spatial resolution only near the sample surface. Monte Carlo (MC) method is one of the recently proposed ways to solve the equation numerically and has been successfully applied to the analysis of laser intensity modulation method data, which also involves the Fredholm integral equation of the first kind. In this paper we attempt to analyze thermal-pulse data in frequency domain with the MC method and discuss its effectiveness based on some numerical simulations. The simulation results indicate that the electric field profiles can be effectively extracted by the MC method. The computed profiles by the MC method consist well with the supposed distributions in the entire thickness of the sample, while the profiles reconstructed by the ST method fit very well to the supposed one at the vicinity of the target surface and distort sharply along the direction of the thermal pulse propagation in the sample bulk. On the other hand, the oscillations in the computed results by the MC method could deteriorate its accuracy in this study. The influence of noise level on the analysis based on the MC method is also tested by the use of the simulated data. The results show that the computed profiles would become more fluctuant as the noise level increases. This problem can be solved by selecting a larger value of tolerance during the singular value decomposition procedure. Thus, the value of tolerance is considered to be one of the key parameters in this algorithm, which is actually hard to determine. Additionally, the experimental data obtained from a polypropylene film under applied electric field are analyzed to illustrate the feasibility of MC method to be applied to the thermal-pulse experimental data. The results also show that the spatial accuracy by the MC method in the entire sample thickness is higher than by the ST method, which verifies that the MC method is more suitable for detecting the electric field distribution in the deep bulk of the sample. Owing to noise and error, the accuracy of MC calculation depends on the chosen tolerance value, which is now considered to be an obstacle in applying this method to the practical thermal-pulse measurement.

This paper develops a parallel implementation of the ensemble Kalman filter (EnKF) based on shrinkage covariancematrix estimation. The EnKF is a sequential Monte Carlo method for parameter and state estimation of highly nonlinearmodels. In the context of ensemble based methods, background error correlations are estimated via anensemble of model realizations. In practice, the model dimension (number of model components) is several timeslarger than the ensemble size owing to the computational effort involved in one single model propagation. Thisconstraint yields to spurious correlations (correlation between distant model components in space) in the backgrounderror estimations. Commonly, localization methods are utilized in order to reduce the impact of spuriouscorrelations. For instance, correlations of model components are reduced based on some function of their physicaldistances. Some EnKF implementations are formulated in such manner that, localization methods are implicitlyinvolved in the assimilation of observations. In this context, one of the best ensemble Kalman filter implementationsis the local ensemble transform Kalman filter (LETKF). In the LETKF, each model component is surrounded by alocal box of size r and then, the information contained within the local domain (i.e., observed components) is used toperform the assimilation. However, for sparse observational networks, r can become sufficiently large with respectto the ensemble size and therefore, the local analysis corrections can be impacted by spurious correlations. Wepropose an ensemble Kalman filter implementation in which the background error covariance matrix is estimated viathe Rao-Blackwell Ledoit and Wolf (RBLW) estimator. This estimator has proven to be better conditioned than thetrue (background) error covariance matrix. In addition, the explicit computation of the RBLW estimator is notrequired in our formulation. This implies enormous savings in terms of memory considering the typical order ofbackground error covariance matrices O(10e8) Furthermore, the proposed EnKF implementation can be efficientlyperformed in parallel: each model component is surrounded by a local box, a local background error covariancematrix is estimated based on the RBLW estimator and then, the ensemble Kalman filter equations are locally used inorder to compute the local analysis corrections. After the local assimilation, each model component is mapped backonto the full model domain from where the global analysis solution is obtained. In operational data assimilation, EnKF implementations are attractive owing to their potential for parallelization. Experiments are performed making use of the Atmospheric General Circulation Model SPEEDY. The proposed method is coded in FORTRAN and MPI is used for the local assimilation of observations. The predicted variables are the zonal and meridional wind components, the temperature and the specific humidity. For each model variable, 8 layers with the T-63 resolution (192 x 96 grid points) are considered which provide a total number of 589,824 model components. The ensemble size is 94. The number of processors utilized are 96, 256, 512, 768, 1024 and 1536. The results reveal that, the use of the RBLW estimator can mitigate the impact of spurious correlations under sparse observational networks and even more, for different number of processors, the computational effort of the proposed EnKF implementation is comparable to that of the LETKF where no covariance estimation is performed. In terms of accuracy, the proposed implementation performs better than the LETKF formulation in the Root Mean Square Error sense.

An important feature of Bayesian statistics is the opportunity to do sequential inference: the posterior distribution obtained after seeing a dataset can be used as prior for a second inference. However, when Monte Carlo sampling methods are used for inference, we only have a set of samples from the posterior distribution. To do sequential inference, we then either have to evaluate the second posterior at only these locations and reweight the samples accordingly, or we can estimate a functional description of the posterior probability distribution from the samples and use that as prior for the second inference. Here, we investigated to what extent we can obtain an accurate joint posterior from two datasets if the inference is done sequentially rather than jointly, under the condition that each inference step is done using Monte Carlo sampling. To test this, we evaluated the accuracy of kernel density estimates, Gaussian mixtures, mixtures of factor analyzers, vine copulas and Gaussian processes in approximating posterior distributions, and then tested whether these approximations can be used in sequential inference. In low dimensionality, Gaussian processes are more accurate, whereas in higher dimensionality Gaussian mixtures, mixtures of factor analyzers or vine copulas perform better. In our test cases of sequential inference, using posterior approximations gives more accurate results than direct sample reweighting, but joint inference is still preferable over sequential inference whenever possible. Since the performance is case-specific, we provide an R package mvdens with a unified interface for the density approximation methods.

Parameter estimation when the measurement information may be incomplete is discussed. An ARX model is used as a basic system representation. The presentation covers both missing output and missing input. First reconstruction of the missing values is discussed. The reconstruction is based on a state-space formulation of the system, and is performed using Kalman filtering or fixed-interval smoothing formulas. Several approaches to the identification problem are presented, including a new method based on the EM (expectation maximization) algorithm. The different approaches are tested and compared using Monte Carlo simulations. The choice of method is always a tradeoff between estimation accuracy and computational complexity. According to the simulations the gain in accuracy using the EM method can be considerable if many data are missing.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The Power Function distribution is one of the most suitable distributions for failure times modeling, especially of electric components and devices. The article proposes some new modified estimators for parameter estimation of the Power Function distribution. The proposed modified estimators are based on the mean and geometric mean of the distribution. The performance of these estimators is compared with existing traditional and modified estimators by means of Monte Carlo simulation and two real‐life examples. Total mean square error, total relative deviation, root mean square error and mean absolute error are used as performance criteria. The performance of the estimators is also compared on two real‐life data sets repressing the device failures times. From both, Monte Carlo simulation and real‐life applications, the results indicated that the proposed modified estimators perform better than the competing estimators and hence their use is recommended for parameter estimation of the Power Function probability model. The findings will help to get more precise parameter estimates for the distribution which is very much applied in different fields particularly in modelling failure times of electronic devices and components.

Buys-Ballot methodology was developed to, among other things, obtain estimates of trend parameters and seasonal indices in descriptive time series analysis (Iwueze et al, 2011). In developing this method, Iwueze and Nwogu (2004, 2005), Iwueze and Ohakwe (2005), proposed these estimates (of trend parameters and seasonal indices) were derived from row, column and overall averages and standard deviations of the Buys-Ballot table. These averages and standard deviations were known to be affected by outliers. In their proposed methods, Iwueze and Nwogu (2004, 2005), Iwueze and Ohakwe (2005), made no provision for the effects of these outliers on the proposed estimates. Hence, the purpose of this study was to assess the effects of outliers on Buys-Ballot estimates of trend parameters and seasonal indices. Simulated series were used in this study in the absence and presence of outliers. To capture the presence of outliers, 5, 10, and 15 outliers were introduced in the simulations. Comparison of the estimates (in the absence and presence of outliers) was done using the Accuracy Measures (Mean Square Error (MSE), the Mean Absolute Error (MAE), and the Mean Absolute Percentage Error (MAPE)). The accuracy measures were based on the deviations of parameters estimated from the corresponding parameters. The results of the analysis showed MSE is 0.024 in the absence of outliers. The values of the MSE increased as the number of outliers increased to 15.37, 21.23, and 31.23 for 5,10 and 15 outliers respectively. Thus in the presence of outlier, the Buys-Ballot Estimates (BBE) is not effective in estimating the trend parameters and seasonal indices when trend-cycle component is linear for additive model. Hence, it has been recommended that in estimating the trend parameters and seasonal indices using a Buys-Ballot procedure of the Buys-Ballot table, the effect of outlier should be considered and handled before further analysis.

In recent years, cone-beam computed tomography (CBCT) has played an important role in medical imaging. However, the applications of CBCT are limited due to the severe scatter contamination. Conventional Monte Carlo (MC) simulation can provide accurate scatter estimation for scatter correction, but the expensive computational cost has always been the bottleneck of MC method in clinical application. In this work, an MC simulation method combined with a variance reduction technique called correlated sampling is proposed for fast iterative scatter correction. Correlated sampling exploits correlation between similar simulation systems to reduce the variance of interest quantities. Specifically, conventional MC simulation is first performed on the scatter-contaminated CBCT to generate the initial scatter signal. In the subsequent correction iterations, scatter estimation is then updated by applying correlated MC sampling to the latest corrected CBCT images by reusing the random number sequences of the task-related photons in conventional MC. Afterward, the corrected projections obtained by subtracting the scatter estimation from raw projections are utilized for FDK reconstruction. These steps are repeated until an adequate scatter correction is obtained. The performance of the proposed framework is evaluated by the accuracy of the scatter estimation, the quality of corrected CBCT images and efficiency. Overall, the difference in mean absolute percentage error between scatter estimation with and without correlated sampling is 0.25% for full-fan case and 0.34% for half-fan case, respectively. In simulation studies, scatter artifacts are substantially eliminated, where the mean absolute error value is reduced from 15 to 2HU in full-fan case and from 53 to 13HU in half-fan case. Scatter-to-primary ratio is reduced to 0.02 for full-fan and 0.04 for half-fan, respectively. In phantom study, the contrast-to-noise ratio (CNR) is increased by a factor of 1.63, and the contrast is increased by a factor of 1.77. As for clinical studies, the CNR is improved by 11% and 14% for half-fan and full-fan, respectively. The contrast after correction is increased by 19% for half-fan and 44% for full-fan. Furthermore, root mean square error is also effectively reduced, especially from 78 to 4HU for full-fan. Experimental results demonstrate that the figure of merit is improved between 23 and 43 folds when using correlated sampling. The proposed method takes less than 25s for the whole iterative scatter correction process. The proposed correlated sampling-based MC simulation method can achieve fast and accurate scatter correction for CBCT, making it suitable for real-time clinical use.

IntroductionEpidemiological models benefit from incorporating detailed time-to-event data to understand how disease risk evolves. For example, decompensation risk in liver cirrhosis depends on sojourn time spent with cirrhosis. Semi-Markov and related models capture these details by modeling time-to-event distributions based on published survival data. However, implementations of semi-Markov processes rely on Monte Carlo sampling methods, which increase computational requirements and introduce stochastic variability. Explicitly calculating the evolving transition likelihood can avoid these issues and provide fast, reliable estimates.MethodsWe present the sojourn time density framework for computing semi-Markov and related models by calculating the evolving sojourn time probability density as a system of partial differential equations. The framework is parametrized by commonly used hazard and models the distribution of current disease state and sojourn time. We describe the mathematical background, a numerical method for computation, and an example model of liver disease.ResultsModels developed with the sojourn time density framework can directly incorporate time-to-event data and serial events in a deterministic system. This increases the level of potential model detail over Markov-type models, improves parameter identifiability, and reduces computational burden and stochastic uncertainty compared with Monte Carlo methods. The example model of liver disease was able to accurately reproduce targets without extensive calibration or fitting and required minimal computational burden.ConclusionsExplicitly modeling sojourn time distribution allows us to represent semi-Markov systems using detailed survival data from epidemiological studies without requiring sampling, avoiding the need for calibration, reducing computational time, and allowing for more robust probabilistic sensitivity analyses.HighlightsTime-inhomogeneous semi-Markov models and other time-to-event–based modeling approaches can capture risks that evolve over time spent with a disease.We describe an approach to computing these models that represents them as partial differential equations representing the evolution of the sojourn time probability density.This sojourn time density framework incorporates complex data sources on competing risks and serial events while minimizing computational complexity.