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1. An Integral Equation Model for PET Imaging

   Tang, Xinhuang; Schmidtlein, Charles Ross; Li, Si; Xu, Yuesheng (January 2021, International journal of numerical analysis and modeling)

   Positron emission tomography (PET) is traditionally modeled as discrete systems. Such models may be viewed as piecewise constant approximations of the underlying continuous model for the physical processes and geometry of the PET imaging. Due to the low accuracy of piecewise constant approximations, discrete models introduce an irreducible modeling error which fundamentally limits the quality of reconstructed images. To address this bottleneck, we propose an integral equation model for the PET imaging based on the physical and geometrical considerations, which describes accurately the true coincidences. We show that the proposed integral equation model is equivalent to the existing idealized model in terms of line integrals which is accurate but not suitable for numerical approximation. The proposed model allows us to discretize it using higher accuracy approximation methods. In particular, we discretize the integral equation by using the collocation principle with piecewise linear polynomials. The discretization leads to new ill-conditioned discrete systems for the PET reconstruction, which are further regularized by a novel wavelet-based regularizer. The resulting non-smooth optimization problem is then solved by a preconditioned proximity fixed-point algorithm. Convergence of the algorithm is established for a range of parameters involved in the algorithm. The proposed integral equation model combined with the discretization, regularization, and optimization algorithm provides a new PET image reconstruction method. Numerical results reveal that the proposed model substantially outperforms the conventional discrete model in terms of the consistency to simulated projection data and reconstructed image quality. This indicates that the proposed integral equation model with appropriate discretization and regularizer can significantly reduce modeling errors and suppress noise, which leads to improved image quality and projection data estimation. 

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2. Dynamically regularized Lagrange multiplier schemes with energy dissipation for the incompressible Navier-Stokes equations

   https://doi.org/10.1016/j.jcp.2024.113550  

   Doan, Cao-Kha; Hoang, Thi-Thao-Phuong; Ju, Lili; Lan, Rihui (January 2025, Journal of Computational Physics)

   In this paper, we present efficient numerical schemes based on the Lagrange multiplier approach for the Navier-Stokes equations. By introducing a dynamic equation (involving the kinetic energy, the Lagrange multiplier, and a regularization parameter), we form a new system which incorporates the energy evolution process but is still equivalent to the original equations. Such nonlinear system is then discretized in time based on the backward differentiation formulas, resulting in a dynamically regularized Lagrange multiplier (DRLM) method. First- and second-order DRLM schemes are derived and shown to be unconditionally energy stable with respect to the original variables. The proposed schemes require only the solutions of two linear Stokes systems and a scalar quadratic equation at each time step. Moreover, with the introduction of the regularization parameter, the Lagrange multiplier can be uniquely determined from the quadratic equation, even with large time step sizes, without affecting accuracy and stability of the numerical solutions. Fully discrete energy stability is also proved with the Marker-and-Cell (MAC) discretization in space. Various numerical experiments in two and three dimensions verify the convergence and energy dissipation as well as demonstrate the accuracy and robustness of the proposed DRLM schemes. 

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3. State-space modeling for degrading systems with stochastic neural networks and dynamic Bayesian layers

   https://doi.org/10.1080/24725854.2023.2185323  

   Farhat, Md Tanzin; Moghaddass, Ramin (February 2023, IISE Transactions)

   To monitor the dynamic behavior of degrading systems over time, a flexible hierarchical discrete-time state-space model (SSM) is introduced that can mathematically characterize the stochastic evolution of the latent states (discrete, continuous, or hybrid) of degrading systems, dynamic measurements collected from condition monitoring sources (e.g., sensors with mixed-type out-puts), and the failure process. This flexible SSM is inspired by Bayesian hierarchical modeling and recurrent neural networks without imposing prior knowledge regarding the stochastic structure of the system dynamics and its variables. The temporal behavior of degrading systems and the relationship between variables of the corresponding system dynamics are fully characterized by stochastic neural networks without having to define parametric relationships/distributions between deterministic and stochastic variables. A Bayesian filtering-based learning method is introduced to train the structure of the proposed framework with historical data. Also, the steps to utilize the proposed framework for inference and prediction of the latent states and sensor outputs are dis-cussed. Numerical experiments are provided to demonstrate the application of the proposed framework for degradation system modeling and monitoring. 

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4. Data assimilation for online model calibration in discrete event simulation

   https://doi.org/10.1177/00375497231221578  

   Hu, Xiaolin; Yan, Mingxi (June 2024, SIMULATION)

   The increasing availability of real-time data collected from dynamic systems brings opportunities for simulation models to be calibrated online for improving the accuracy of simulation-based studies. Systematical methods are needed for assimilating real-time measurement data into simulation models. This paper presents a particle filter-based data assimilation method to support online model calibration in discrete event simulation. A joint state-parameter estimation problem is defined, and a particle filter-based data assimilation algorithm is presented. The developed method is applied to a discrete event simulation of a one-way traffic control system. Experiments results demonstrate the effectiveness of the developed method for calibrating simulation models’ parameters in real time and for improving data assimilation results. 

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5. Dissipative timescales from coarse-graining irreversibility

   https://doi.org/10.1088/1742-5468/acdce6  

   Cisneros, Freddy A; Fakhri, Nikta; Horowitz, Jordan M (July 2023, Journal of Statistical Mechanics: Theory and Experiment)

   Abstract We propose and investigate a method for identifying timescales of dissipation in nonequilibrium steady states modeled as discrete-state Markov jump processes. The method is based on how the irreversibility—measured by the statistical breaking of time-reversal symmetry—varies under temporal coarse-graining. We observe a sigmoidal-like shape of the irreversibility as a function of the coarse-graining time whose functional form we derive for systems with a fast driven transition. This theoretical prediction allows us to develop a method for estimating the dissipative time scale from time-series data by fitting estimates of the irreversibility to our predicted functional form. We further analyze the accuracy and statistical fluctuations of this estimate. 

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