arXiv subjects

James L. Beck

Publications and source records attributed to James L. Beck.

At least 19 recordsLinked to original sources

arXiv2026

Adaptive Meta-Learning Stochastic Gradient Hamiltonian Monte Carlo Simulation for Bayesian Updating of Structural Dynamic Models

In the last few decades, Markov chain Monte Carlo (MCMC) methods have been widely applied to Bayesian updating of structural dynamic models in the field of structural health monitoring. Recently, several MCMC algorithms have been developed that incorporate neural networks to enhance their performance for specific Bayesian model updating problems. However, a common challenge with these approaches lies in the fact that the embedded neural networks often necessitate retraining when faced with new tasks, a process that is time-consuming and significantly undermines the competitiveness of these methods. This paper introduces a newly developed adaptive meta-learning stochastic gradient Hamiltonian Monte Carlo (AM-SGHMC) algorithm. The idea behind AM-SGHMC is to optimize the sampling strategy by training adaptive neural networks, and due to the adaptive design of the network inputs and outputs, the trained sampler can be directly applied to various Bayesian updating problems of the same type of structure without further training, thereby achieving meta-learning. Additionally, practical issues for the feasibility of the AM-SGHMC algorithm for structural dynamic model updating are addressed, and two examples involving Bayesian updating of multi-story building models with different model fidelity are used to demonstrate the effectiveness and generalization ability of the proposed method.

stat.AP↗

arXiv2026

Probabilistic Graphical Model using Graph Neural Networks for Bayesian Inversion of Discrete Structural Component States

The health condition of components in civil infrastructures can be described by various discrete states according to their performance degradation. Inferring these states from measurable responses is typically an ill-posed inverse problem. Although Bayesian methods are well-suited to tackle such problems, computing the posterior probability density function (PDF) presents challenges. The likelihood function cannot be analytically formulated due to the unclear relationship between discrete states and structural responses, and the high-dimensional state parameters resulting from numerous components severely complicates the computation of the marginal likelihood function. To address these challenges, this study proposes a novel Bayesian inversion paradigm for discrete variables based on Probabilistic Graphical Models (PGMs). The Markov networks are employed as modeling tools, with model parameters learned from data and structural topology prior. It has been proved that inferring this PGM produces the same probabilistic estimation as the posterior PDF derived from Bayesian inference, which effectively solves the above challenges. The inference is accomplished by Graph Neural Networks (GNNs), and a graph property-based GNN training strategy is developed to enable accurate inference across varying graph scales, thereby significantly reducing the computational overhead in high-dimensional problems. Both synthetic and experimental data are used to validate the proposed framework

stat.ML↗

arXiv2026

MCMC with Adaptive Principal-Component Transformation: Rotation-Invariant Universal Samplers for Bayesian Structural System Identification

Over decades, Markov chain Monte Carlo (MCMC) methods have been widely studied, with a typical application being the quantification of posterior uncertainties in Bayesian system identification of structural dynamic models. To address the issue of excessively low sampling efficiency in generic MCMC methods when applied to specific problems, researchers developed several MCMC algorithms that integrate trainable neural networks to replace and enhance their critical components. Later, meta-learning MCMC methods emerged to reduce training time. However, they require considerable similarity between test and training tasks, while their sampling efficiency is constrained by trade-off-simplified network designs. This paper proposes the Adaptive Principal-Component (PC) Meta-learning Stochastic Gradient Hamiltonian Monte Carlo (APM-SGHMC) algorithm. It adaptively rotates coordinate axes in the parameter space to align with the PC directions of the current posterior samples, ensuring rotation-invariance of sampling performance with respect to the posterior distribution. By incorporating translation-invariance, scale-invariance, and rotation-invariance in a unified framework, APM-SGHMC enables universal samplers to acquire generalizable knowledge across diverse Bayesian system identification tasks using minimalistic tasks while eliminating the constraints imposed by network design trade-offs on sampling efficiency. Practical feasibility issues are also addressed. Two Bayesian system identification case studies demonstrate its effectiveness and universality: our method overcomes the case-by-case limitations of traditional data-driven approaches, achieving zero-shot generalization across structurally distinct models without retraining and maintaining consistent superior performance across all scenarios.

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arXiv2025

Dynamic Analysis of Flexible Stepping Frames for Earthquakes

This paper investigates the nonlinear dynamics of stepping flexible frames under seismic excitation. The conventional iterative method of solution of peak quasi-dynamic displacement of stepping frames is not guaranteed to converge. To address this limitation, we present closed-form solutions and stability criteria for displacement response of stepping flexible frames. Bifurcation of displacements in response of such systems is next studied through the extension of dynamics of stepping rigid bodies. An approximate analytical expression is presented to account for the effects of moving resonance under earthquake ground motions. The closed-form solutions for displacement demand can be readily adjusted to incorporate the influence of moving resonance on the quasi-dynamic response of stepping oscillators. While the quasi-dynamic method of analysis may be useful in the early stages of design, numerical integration of the nonlinear system of differential equations of motion is recommended for the solution of dynamic response in such applications. Implications for formal limit-state analysis of stepping response are discussed, accompanied by several examples demonstrating the procedures.

physics.geo-ph↗

arXiv2024

Relativistic Mechanics Theory for Electrons that Exhibits Spin, Zitterbewegung, Superposition and Produces Dirac's Wave Equation

A neo-classical relativistic mechanics theory is presented where the spin of an electron is a natural part of its space-time path as a point particle. The fourth-order equation of motion corresponds to the same Lagrangian function in proper time as in special relativity except for an additional spin energy term. The total motion can be decomposed into a sum of a local spin motion about a point and a global motion of this point, called the spin center. The global motion is sub-luminal and obeys Newton's second law in proper time, the time for a clock fixed at the spin center, while the total motion occurs at the speed of light c, consistent with the eigenvalues of Dirac's velocity operators having magnitude c. The local spin motion corresponds to Schrödinger's zitterbewegung and is a perpetual motion, which for a free electron has a circular path in the spin-center frame. In an electro-magnetic field, this spin motion generates magnetic and electric dipole energies through the Lorentz force on the electron's point charge. The corresponding electric dipole energy is consistent with the spin-orbit coupling term in the corrected Pauli non-relativistic Hamiltonian but the magnetic dipole energy is one half of that in Dirac's theory. By defining a spin tensor as the angular momentum of the electron's total motion about its spin center, the fundamental equations of motion can be re-written in an identical form to those of the Barut-Zanghi electron theory. These equations of motion can then be expressed using operators applied to a state function of proper time satisfying a Dirac-Schrödinger spinor equation. The operators produce dynamic variables without any probability implications. For the free electron, the state function satisfies Dirac's relativistic wave equation when the Lorentz transformation is applied to express proper time in terms of an observer's space-time coordinates.

quant-ph↗

arXiv2019

Exploiting Convexification for Bayesian Optimal Sensor Placement by Maximization of Mutual Information

Bayesian optimal sensor placement, in its full generality, seeks to maximize the mutual information between uncertain model parameters and the predicted data to be collected from the sensors for the purpose of performing Bayesian inference. Equivalently, the expected information entropy of the posterior of the model parameters is minimized over all possible sensor configurations for a given sensor budget. In the context of structural dynamical systems, this minimization is computationally expensive because of the large number of possible sensor configurations. Here, a very efficient convex relaxation scheme is presented to determine informative and possibly-optimal solutions to the problem, thereby bypassing the necessity for an exhaustive, and often infeasible, combinatorial search. The key idea is to relax the binary sensor location vector so that its components corresponding to all possible sensor locations lie in the unit interval. Then, the optimization over this vector is a convex problem that can be efficiently solved. This method always yields a unique solution for the relaxed problem, which is often binary and therefore the optimal solution to the original problem. When not binary, the relaxed solution is often suggestive of what the optimal solution for the original problem is. An illustrative example using a fifty-story shear building model subject to sinusoidal ground motion is presented, including a case where there are over 47 trillion possible sensor configurations. The solutions and computational effort are compared to greedy and heuristic methods.

stat.AP↗

arXiv2018

Bayesian Updating and Uncertainty Quantification using Sequential Tempered MCMC with the Rank-One Modified Metropolis Algorithm

Bayesian methods are critical for quantifying the behaviors of systems. They capture our uncertainty about a system's behavior using probability distributions and update this understanding as new information becomes available. Probabilistic predictions that incorporate this uncertainty can then be made to evaluate system performance and make decisions. While Bayesian methods are very useful, they are often computationally intensive. This necessitates the development of more efficient algorithms. Here, we discuss a group of population Markov Chain Monte Carlo (MCMC) methods for Bayesian updating and system reliability assessment that we call Sequential Tempered MCMC (ST-MCMC) algorithms. These algorithms combine 1) a notion of tempering to gradually transform a population of samples from the prior to the posterior through a series of intermediate distributions, 2) importance resampling, and 3) MCMC. They are a form of Sequential Monte Carlo and include algorithms like Transitional Markov Chain Monte Carlo and Subset Simulation. We also introduce a new sampling algorithm called the Rank-One Modified Metropolis Algorithm (ROMMA), which builds upon the Modified Metropolis Algorithm used within Subset Simulation to improve performance in high dimensions. Finally, we formulate a single algorithm to solve combined Bayesian updating and reliability assessment problems to make posterior assessments of system reliability. The algorithms are then illustrated by performing prior and posterior reliability assessment of a water distribution system with unknown leaks and demands.

stat.CO↗

arXiv2018

Contrasting Implications of the Frequentist and Bayesian Interpretations of Probability when Applied to Quantum Mechanics Theory

An examination is made of the differing implications from applying the two mainstream interpretations of probability, frequentist and Bayesian, to QM (quantum mechanics) theory for the Bohm-EPR experiment. The joint probability distribution for the possible spin outcomes for two particles with coupled spins is examined. Contrasting conclusions are made: (i) Under the frequentist interpretation, the QM spin distribution implies a widely-discussed non-locality because probabilistic conditioning on a spin measurement is viewed as corresponding to a causal influence. Under the Bayesian interpretation, this conditioning is viewed as providing information relevant to the spin probabilities and the argument for non-locality loses its force. (ii) The frequentist interpretation leads to the locality condition used by John Bell in 1964 to establish conditions for the existence of hidden variables behind the spin probability distribution but this condition is not consistent with the product (Bayes) rule of probability theory. Under the Bayesian interpretation, there is no motivation for this locality condition and a new stochastic hidden-variable model is given that reproduces the spin distribution. (iii) Bell's original 1964 inequality provides a necessary condition for the existence of a third-order joint probability distribution for three spin variables that is compatible through marginalization with the second-order joint distributions for the three possible spin pairs. Bell's inequality is logically equivalent to the 1969 CHSH inequality that involves expectations of four pairs of four spin variables; the CHSH inequality must be satisfied in order for a fourth-order joint probability distribution for the four spin variables to exist. When any of these inequalities are violated, a joint probability distribution for three spin outcomes fails to be valid because some probabilities are negative.

quant-ph↗

arXiv2017

Bayesian System Identification based on Hierarchical Sparse Bayesian Learning and Gibbs Sampling with Application to Structural Damage Assessment

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 serve 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.

stat.AP↗

arXiv2016

Hierarchical Stochastic Model in Bayesian Inference: Theoretical Implications and Efficient Approximation

We classify two types of Hierarchical Bayesian Model found in the literature as Hierarchical Prior Model (HPM) and Hierarchical Stochastic Model (HSM). Then, we focus on studying the theoretical implications of the HSM. Using examples of polynomial functions, we show that the HSM is capable of separating different types of uncertainties in a system and quantifying uncertainty of reduced order models under the Bayesian model class selection framework. To tackle the huge computational cost for analyzing HSM, we propose an efficient approximation scheme based on Importance Sampling and Empirical Interpolation Method. We illustrate our method using two examples - a Molecular Dynamics simulation for Krypton and a pharmacokinetic/pharmacodynamic model for cancer drug.

stat.AP↗

arXiv2016

Using Approximate Bayesian Computation by Subset Simulation for Efficient Posterior Assessment of Dynamic State-Space Model Classes

Approximate Bayesian Computation (ABC) methods have gained in their popularity over the last decade because they expand the horizon of Bayesian parameter inference methods to the range of models for which only forward simulation is available. The majority of the ABC methods rely on the choice of a set of summary statistics to reduce the dimension of the data. However, as has been noted in the ABC literature, the lack of convergence guarantees that is induced by the absence of a vector of sufficient summary statistics that assures inter-model sufficiency over the set of competing models, hinders the use of the usual ABC methods when applied to Bayesian model selection or assessment. In this paper, we present a novel ABC model selection procedure for dynamical systems based on a newly appeared multi-level Markov chain Monte Carlo method, self-regulating ABC-SubSim, and a hierarchical state-space formulation of dynamic models. We show that this formulation makes it possible to independently approximate the model evidence required for assessing the posterior probability of each of the competing models. We also show that ABC-SubSim not only provides an estimate of the model evidence as a simple by-product but also it gives the posterior probability of each model as a function of the tolerance level, which allows the ABC model choices made in previous studies to be understood. We illustrate the performance of the proposed framework for ABC model updating and model class selection by applying it to two problems in Bayesian system identification: a single degree-of-freedom bilinear hysteretic oscillator and a three-story shear building with Masing hysteresis, both of which are subject to a seismic excitation.

stat.CO↗

arXiv2015

Rare Event Simulation

Rare events are events that are expected to occur infrequently, or more technically, those that have low probabilities (say, order of $10^{-3}$ or less) of occurring according to a probability model. In the context of uncertainty quantification, the rare events often correspond to failure of systems designed for high reliability, meaning that the system performance fails to meet some design or operation specifications. As reviewed in this section, computation of such rare-event probabilities is challenging. Analytical solutions are usually not available for non-trivial problems and standard Monte Carlo simulation is computationally inefficient. Therefore, much research effort has focused on developing advanced stochastic simulation methods that are more efficient. In this section, we address the problem of estimating rare-event probabilities by Monte Carlo simulation, Importance Sampling and Subset Simulation for highly reliable dynamic systems.

stat.CO↗

arXiv2015

Robust Bayesian compressive sensing with data loss recovery for structural health monitoring signals

The application of compressive sensing (CS) to structural health monitoring is an emerging research topic. The basic idea in CS is to use a specially-designed wireless sensor to sample signals that are sparse in some basis (e.g. wavelet basis) directly in a compressed form, and then to reconstruct (decompress) these signals accurately using some inversion algorithm after transmission to a central processing unit. However, most signals in structural health monitoring are only approximately sparse, i.e. only a relatively small number of the signal coefficients in some basis are significant, but the other coefficients are usually not exactly zero. In this case, perfect reconstruction from compressed measurements is not expected. A new Bayesian CS algorithm is proposed in which robust treatment of the uncertain parameters is explored, including integration over the prediction-error precision parameter to remove it as a "nuisance" parameter. The performance of the new CS algorithm is investigated using compressed data from accelerometers installed on a space-frame structure and on a cable-stayed bridge. Compared with other state-of-the-art CS methods including our previously-published Bayesian method which uses MAP (maximum a posteriori) estimation of the prediction-error precision parameter, the new algorithm shows superior performance in reconstruction robustness and posterior uncertainty quantification. Furthermore, our method can be utilized for recovery of lost data during wireless transmission, regardless of the level of sparseness in the signal.

stat.AP↗

arXiv2015

Hierarchical sparse Bayesian learning: theory and application for inferring structural damage from incomplete modal data

Structural damage due to excessive loading or environmental degradation typically occurs in localized areas in the absence of collapse. This prior information about the spatial sparseness of structural damage is exploited here by a hierarchical sparse Bayesian learning framework with the goal of reducing the source of ill-conditioning in the stiffness loss inversion problem for damage detection. Sparse Bayesian learning methodologies automatically prune away irrelevant or inactive features from a set of potential candidates, and so they are effective probabilistic tools for producing sparse explanatory subsets. We have previously proposed such an approach to establish the probability of localized stiffness reductions that serve as a proxy for damage by using noisy incomplete modal data from before and after possible damage. The core idea centers on a specific hierarchical Bayesian model that promotes spatial sparseness in the inferred stiffness reductions in a way that is consistent with the Bayesian Ockham razor. In this paper, we improve the theory of our previously proposed sparse Bayesian learning approach by eliminating an approximation and, more importantly, incorporating a constraint on stiffness increases. Our approach has many appealing features that are summarized at the end of the paper. We validate the approach by applying it to the Phase II simulated and experimental benchmark studies sponsored by the IASC-ASCE Task Group on Structural Health Monitoring. The results show that it can reliably detect, locate and assess damage by inferring substructure stiffness losses from the identified modal parameters. The occurrence of missed and false damage alerts is effectively suppressed.

stat.AP↗

arXiv2014

Robust Bayesian compressive sensing for signals in structural health monitoring

In structural health monitoring (SHM) systems, massive amounts of data are often generated that need data compression techniques to reduce the cost of signal transfer and storage. Compressive sensing (CS) is a novel data acquisition method whereby the compression is done in a sensor simultaneously with the sampling. If the original sensed signal is sufficiently sparse in terms of some basis, the decompression can be done essentially perfectly up to some critical compression ratio. In this article, a Bayesian compressive sensing (BCS) method is investigated that uses sparse Bayesian learning to reconstruct signals from a compressive sensor. By explicitly quantifying the uncertainty in the reconstructed signal, the BCS technique exhibits an obvious benefit over existing regularized norm-minimization CS methods that provide a single signal estimate. However, current BCS algorithms suffer from a robustness problem: sometimes the reconstruction errors are very large when the number of measurements is a lot less than the number of signal degrees of freedom that are needed to capture the signal accurately in a directly sampled form. In this paper, we present improvements to the BCS reconstruction method to enhance its robustness so that even higher compression ratios can be used and we examine the tradeoff between efficiently compressing data and accurately decompressing it. Synthetic data and actual acceleration data collected from a bridge SHM system are used as examples. Compared with the state-of-the-art BCS reconstruction algorithms, the improved BCS algorithm demonstrates superior performance. With the same reconstruction error, the proposed BCS algorithm works with relatively large compression ratios and it can achieve perfect lossless compression performance with quite high compression ratios. Furthermore, the error bars for the signal reconstruction are also quantified effectively.

stat.AP↗

arXiv2014

Hierarchical sparse Bayesian learning for structural health monitoring with incomplete modal data

For civil structures, structural damage due to severe loading events such as earthquakes, or due to long-term environmental degradation, usually occurs in localized areas of a structure. A new sparse Bayesian probabilistic framework for computing the probability of localized stiffness reductions induced by damage is presented that uses noisy incomplete modal data from before and after possible damage. This new approach employs system modal parameters of the structure as extra variables for Bayesian model updating with incomplete modal data. A specific hierarchical Bayesian model is constructed that promotes spatial sparseness in the inferred stiffness reductions in a way that is consistent with the Bayesian Ockham razor. To obtain the most plausible model of sparse stiffness reductions together with its uncertainty within a specified class of models, the method employs an optimization scheme that iterates among all uncertain parameters, including the hierarchical hyper-parameters. The approach has four important benefits: (1) it infers spatially-sparse stiffness changes based on the identified modal parameters; (2) the uncertainty in the inferred stiffness reductions is quantified; (3) no matching of model and experimental modes is needed, and (4) solving the nonlinear eigenvalue problem of a structural model is not required. The proposed method is applied to two previously-studied examples using simulated data: a ten-story shear-building and the three-dimensional braced-frame model from the Phase II Simulated Benchmark problem sponsored by the IASC-ASCE Task Group on Structural Health Monitoring. The results show that the occurrence of false-positive and false-negative damage detection is clearly reduced in the presence of modeling error. Furthermore, the identified most probable stiffness loss ratios are close to their actual values.

stat.AP↗

arXiv2014

Approximate Bayesian Computation by Subset Simulation

A new Approximate Bayesian Computation (ABC) algorithm for Bayesian updating of model parameters is proposed in this paper, which combines the ABC principles with the technique of Subset Simulation for efficient rare-event simulation, first developed in S.K. Au and J.L. Beck [1]. It has been named ABC- SubSim. The idea is to choose the nested decreasing sequence of regions in Subset Simulation as the regions that correspond to increasingly closer approximations of the actual data vector in observation space. The efficiency of the algorithm is demonstrated in two examples that illustrate some of the challenges faced in real-world applications of ABC. We show that the proposed algorithm outperforms other recent sequential ABC algorithms in terms of computational efficiency while achieving the same, or better, measure of ac- curacy in the posterior distribution. We also show that ABC-SubSim readily provides an estimate of the evidence (marginal likelihood) for posterior model class assessment, as a by-product.

stat.CO↗

arXiv2011

Bayesian Post-Processor and other Enhancements of Subset Simulation for Estimating Failure Probabilities in High Dimensions

Estimation of small failure probabilities is one of the most important and challenging computational problems in reliability engineering. The failure probability is usually given by an integral over a high-dimensional uncertain parameter space that is difficult to evaluate numerically. This paper focuses on enhancements to Subset Simulation (SS), proposed by Au and Beck, which provides an efficient algorithm based on MCMC (Markov chain Monte Carlo) simulation for computing small failure probabilities for general high-dimensional reliability problems. First, we analyze the Modified Metropolis algorithm (MMA), an MCMC technique, which is used in SS for sampling from high-dimensional conditional distributions. We present some observations on the optimal scaling of MMA, and develop an optimal scaling strategy for this algorithm when it is employed within SS. Next, we provide a theoretical basis for the optimal value of the conditional failure probability $p_0$, an important parameter one has to choose when using SS. Finally, a Bayesian post-processor SS+ for the original SS method is developed where the uncertain failure probability that one is estimating is modeled as a stochastic variable whose possible values belong to the unit interval. Simulated samples from SS are viewed as informative data relevant to the system's reliability. Instead of a single real number as an estimate, SS+ produces the posterior PDF of the failure probability, which takes into account both prior information and the information in the sampled data. This PDF quantifies the uncertainty in the value of the failure probability and it may be further used in risk analyses to incorporate this uncertainty. The relationship between the original SS and SS+ is also discussed

stat.CO↗