Loeve Expansion Research Articles

A Distributed Non-Intrusive Load Monitoring Method Using Karhunen–Loeve Feature Extraction and an Improved Deep Dictionary

In recent years, the non-invasive load monitoring (NILM) method based on sparse coding has shown promising research prospects. This type of method learns a sparse dictionary for each monitoring target device, and it expresses load decomposition as a problem of signal reconstruction using dictionaries and sparse vectors. The existing NILM methods based on sparse coding have problems such as inability to be applied to multi-state and time-varying devices, single-load characteristics, and poor recognition ability for similar devices in distributed manners. Using the analysis above, this paper focuses on devices with similar features in households and proposes a distributed non-invasive load monitoring method using Karhunen–Loeve (KL) feature extraction and an improved deep dictionary. Firstly, Karhunen–Loeve expansion (KLE) is used to perform subspace expansion on the power waveform of the target device, and a new load feature is extracted by combining singular value decomposition (SVD) dimensionality reduction. Afterwards, the states of all the target devices are modeled as super states, and an improved deep dictionary based on the distance separability measure function (DSM-DDL) is learned for each super state. Among them, the state transition probability matrix and observation probability matrix in the hidden Markov model (HMM) are introduced as the basis for selecting the dictionary order during load decomposition. The KL feature matrix of power observation values and improved depth dictionary are used to discriminate the current super state based on the minimum reconstruction error criterion. The test results based on the UK-DALE dataset show that the KL feature matrix can effectively reduce the load similarity of devices. Combined with DSM-DDL, KL has a certain information acquisition ability and acceptable computational complexity, which can effectively improve the load decomposition accuracy of similar devices, quickly and accurately estimating the working status and power demand of household appliances.

Random Responses of Shield Tunnel to New Tunnel Undercrossing Considering Spatial Variability of Soil Elastic Modulus

This paper investigates the effect of spatial variability of soil elastic modulus on the longitudinal responses of the existing shield tunnel to the new tunnel undercrossing using a random two-stage analysis method (RTSAM). The Timoshenko–Winkler-based deterministic method considering longitudinal variation in the subgrade reaction coefficient and the random field of the soil elastic modulus discretized by the Karhunen–Loeve expansion method are combined to establish the RTSAM. Then, the proposed RTSAM is applied to carry out a random analysis based on an actual engineering case. Results show that the increases in the scale of fluctuation and the coefficient of variation of the soil elastic modulus lead to higher variabilities of tunnel responses. A decreasing pillar depth and mean value of the soil elastic modulus and an increasing skew angle strengthen the effect of the spatial variability of the soil elastic modulus on tunnel responses. The variabilities of tunnel responses under the random field of the soil elastic modulus are overestimated by the Euler–Bernoulli beam model. The results of this study provide references for the uncertainty analysis of the new tunneling-induced responses of the existing tunnel under the random field of soil properties.

Band Gap Properties in Metamaterial Beam with Spatially Varying Interval Uncertainties

First, this study proposed a metamaterial beam model with spatially varying interval density. The interval dynamic equation of this model could be established by incorporating the decomposition results of the interval field based on Karhunen–Loeve expansion into the finite element method. An interval perturbation finite element method was developed to evaluate the bounds of the dynamic response interval vector. Then, an interval vibration transmission analysis could be performed, and the frequency range of the safe band gap could be determined. Meanwhile, Monte Carlo simulations and the vertex method are also presented to provide reference solutions. By comparison, it was found that the calculation accuracy of the interval perturbation finite element method was acceptable. The numerical results also showed that the safe band gap range was significantly smaller than that of the deterministic band gap.

Modeling the biological growth with a random logistic differential equation

We modeled biological growth using a random differential equation (RDE), where the initial condition is a random variable, and the growth rate is a suitable stochastic process. These assumptions let us obtain a model that represents well the random growth process observed in nature, where only a few individuals of the population reach the maximal size of the species, and the growth curve for every individual behaves randomly. Since we assumed that the initial condition is a random variable, we assigned a priori density, and we performed Bayesian inference to update the initial condition’s density of the RDE. The Karhunen–Loeve expansion was then used to approximate the random coefficient of the RDE. Then, using the RDE’s approximations, we estimated the density f(p, t). Finally, we fitted this model to the biological growth of the giant electric ray (or Cortez electric ray) Narcine entemedor. Simulations of the solution of the random logistic equation were performed to construct a curve that describes the solutions’ mean for each time. As a result, we estimated confidence intervals for the mean growth that described reasonably well the observed data. We fit the proposed model with a training dataset, and the model is tested with a different dataset. The model selection is performed with the square of the errors.

Sparse polynomial chaos expansion for nonlinear finite element simulations with random material properties

AbstractThis contribution deals with the uncertainty quantification for applied nonlinear structural engineering problems, including high stochastic dimensions. A finite element problem with different material models is investigated. The efficiency, accuracy and convergence of sparse PCE are studied numerically and compared with Monte‐Carlo Simulation (MCS) for non‐linear structural analysis including elasto‐plastic and damage models. In both models, the Young's modulus is considered as random fields discretised by Karhunen Loeve Expansion (KLE). In the provided studies, sparse PCE converges fast and is highly efficient for linear elastic and elasto‐plastic material models. However, sparse PCE loses its effectiveness and exhibits lower accuracy for the damage material model.

Bayesian inversion of log-normal eikonal equations

We study the Bayesian inverse problem for inferring the log-normal slowness function of the eikonal equation, given noisy observation data on its solution at a set of spatial points. We contribute rigorous proofs on the existence and well-posedness of the problem. We then study approximation of the posterior probability measure by solving the truncated eikonal equation, which contains only a finite number of terms in the Karhunen–Loeve expansion of the slowness function, by the fast marching method (FMM). The error of this approximation in the Hellinger metric is deduced in terms of the truncation level of the slowness and the grid size in the FMM resolution. It is well known that the plain Markov chain Monte Carlo (MCMC) procedure for sampling the posterior probability is highly expensive. We develop and justify the convergence of a multilevel MCMC method. Using the heap sort procedure in solving the forward eikonal equation by the FMM, our multilevel MCMC method achieves a prescribed level of accuracy for approximating the posterior expectation of quantities of interest, requiring only an essentially optimal level of complexity. Numerical examples confirm the theoretical results.

Uncertain Dynamic Characteristic Analysis for Structures with Spatially Dependent Random System Parameters.

This work presents a robust non-deterministic free vibration analysis for engineering structures with random field parameters in the frame of stochastic finite element method. For this, considering the randomness and spatial correlation of structural physical parameters, a parameter setting model based on random field theory is proposed to represent the random uncertainty of parameters, and the stochastic dynamic characteristics of different structural systems are then analyzed by incorporating the presented parameter setting model with finite element method. First, Gauss random field theory is used to describe the uncertainty of structural material parameters, the random parameters are then characterized as the standard deviation and correlation length of the random field, and the random field parameters are then discretized with the Karhunen-Loeve expansion method. Moreover, based on the discretized random parameters and finite element method, structural dynamic characteristics analysis is addressed, and the probability distribution density function of the random natural frequency is estimated based on multi-dimensional kernel density estimation method. Finally, the random field parameters of the structures are quantified by using the maximum likelihood estimation method to verify the effectiveness of the proposed method and the applicability of the constructed model. The results indicate that (1) for the perspective of maximum likelihood estimation, the parameter setting at the maximum value point is highly similar to the input parameters; (2) the random field considering more parameters reflects a more realistic structure.

Homotopy based stochastic finite element model updating with correlated static measurement data

This paper focuses on studying the impaction of the correlation between the measurement data on structural model updating. A new homotopy based stochastic finite element model updating frame is constructed to cope with the correlated static measurement data. To judge whether considering the correlation of the measurement data, the sensitivity analysis of structural responses about structural parameters is implemented firstly. Then the discrete Karhunen–Loeve expansion is utilized to transform the correlated measurement data into a linear combination of multiple independent random variables. Furthermore, a novel stochastic model updating equation about the correlated static measurement data is set up, and is solved by the homotopy stochastic finite element method. It is significant to check the consistence of the correlation coefficients of the responses of the updated model with those of the measured data when the correlation actually exists. In this case, the proposed approach can effectively update the structural model.

Influence of Spatially Distributed Out-of-Plane CFRP Fiber Waviness on the Estimation of Knock-Down Factors Based on Stochastic Numerical Analysis

The presence of waviness defects in CFRP materials due to fiber undulation affects the structural performance of composite structures. Hence, without a reliable assessment of the resulting material properties, the full weight-saving potential cannot be exploited. Within the paper, a probabilistic numerical approach for improved estimation of material properties based on spatially distributed fiber waviness is presented. It makes use of a homogenization approach to derive viable knock-down factors for the different plies on the laminate level for reference material and is demonstrated for a representative tension loadcase. For the stochastic analysis, a random field is selected which describes the complex inner geometry of the plies in the laminate model and is numerically discretized by the Karhunen–Loeve expansion methods to fit into an FE model for the strength analysis. Conducted analysis studies reveal a substantial influence of randomly distributed waviness defects on the derived knock-down factors. Based on a topological analysis of the waviness fields, the reduction of the material properties was found to be weakly negatively correlated related to simple geometrical properties such as maximum amplitudes of the waviness field, which justifies the need for further subsequent sensitivity studies.

Fast Bayesian inference for inverse heat conduction problem using polynomial chaos and Karhunen–Loeve expansions

We present a fast Bayesian inference framework for solving inverse heat conduction problems (IHCP) for the estimation of boundary heat flux. The framework leverages the polynomial chaos expansions (PCEs) for generating computationally efficient and statistically equivalent surrogate model of the computationally expensive forward model and dimensionality reduction based on Karhunen–Loeve (K–L) expansion. We demonstrate the potential of this approach using three model problems to estimate heat flux in inverse heat conduction models. We represent the heat flux using K–L expansion to reduce the high dimensionality of the heat flux and compute the posterior probability distribution of the heat flux using the temperature measurements. The first inverse problem involves the estimation of time–varying heat flux in a one–dimensional slab using transient temperature measurements. The second problem involves the estimation of the unknown time–dependent heat flux of the disc in a realistic axisymmetric disc brake system. The third problem involves the estimation of the surface heat flux on a sounding rocket having a realistic three–dimensional geometry of the rocket module used in actual flight tests. We further obtain the inverse solution by computing the expectation of the distribution and demonstrate that this approach becomes orders of magnitude more efficient when we use the PCE surrogate model instead of the direct forward model for comparable accuracy. The numerical results show that this framework overcomes the drawback of high computational cost of Markov chain Monte Carlo (MCMC) without compromising on solution accuracy. The combination of PC–based surrogate modeling and K–L based dimensionality reduction leads to over 2000–fold speed up of numerical solution of IHCP.

Nonlinear Random Vibrations Using Second-Order Adjoint and Projected Differentiation Methods

Abstract This paper proposes a new computationally efficient methodology for random vibrations of nonlinear vibratory systems using a time-dependent second-order adjoint variable (AV2) method and a second-order projected differentiation (PD2) method. The proposed approach is called AV2–PD2. The vibratory system can be excited by stationary Gaussian or non-Gaussian random processes following the traditional translation process model. A Karhunen–Loeve (KL) expansion expresses each input random process in terms of standard normal random variables. A second-order adjoint approach is used to obtain the required first- and second-order output derivatives accurately by solving as many sets of equations of motion (EOMs) as the number of KL random variables. These derivatives are used to compute the marginal cumulative distribution function (CDF) of the output process with second-order accuracy. Then, a second-order projected differentiation method calculates the autocorrelation function of each output process with second-order accuracy, at an additional cost of solving as many sets of EOMs as the number of outputs of interest, independently of the time horizon (simulation time). The total number of solutions of the EOM scales linearly with the number of input KL random variables and the number of output processes. The efficiency and accuracy of the proposed approach are demonstrated using a nonlinear Duffing oscillator problem under a quadratic random excitation and a nonlinear half-car suspension example.

Stall-induced fatigue damage in nonlinear aeroelastic systems under stochastic inflow: Numerical and experimental analyses

This study focuses on characterizing the fatigue damage accumulated in nonlinear aeroelastic systems subjected to stochastic inflows through both numerical simulations and wind tunnel experiments. In the mathematical model, nonlinearities are assumed to exist either in the structure (via a cubic hardening nonlinearity in the pitch stiffness), or in the flow (via dynamic stall condition), or simultaneously in both the structural and aerodynamic counterparts. The aerodynamic loads in the attached flow and dynamic stall conditions are estimated using Wagner’s formulation and semi-empirical Leishman–Beddoes model, respectively. To augment the findings to in-field flow conditions, the oncoming wind flow is considered to be randomly time-varying in nature. The stochastic input flow fluctuations are modeled using a Karhunen–Loeve Expansion formulation. The response dynamics and the associated fatigue damage of the aeroelastic system, possessing different sources of nonlinearities, are systematically investigated under isolated cases of deterministic and stochastic input flows. Specifically, the pertinent role of stochasticity in the input flow is brought out by presenting the response dynamics and the associated fatigue damage accumulation for different values of noise intensity and time scale of the input flow fluctuation. It is demonstrated that under fluctuating flow conditions, the dynamics intermittently switch between attached flow and the dynamic stall regimes even at low mean flow speeds. The intermittent nature of the response varies as the time scale and intensity of the oncoming flow are varied. The role of torsional stresses as the predominant component dictating the fatigue damage accumulation irrespective of the source of nonlinearity is illustrated. Using the rainflow counting method and Miner’s linear damage accumulation theory, it is shown that the accumulated fatigue damage is substantially higher under stochastic flow conditions as compared to deterministic input flows. Importantly, it is observed that different time scales and intensities of the oncoming flow fluctuation play a pivotal role in dictating the fatigue damage in aeroelastic systems. Finally, fatigue damage is observed to be significantly higher for torsionally dominant oscillations in the dynamical stall regime compared to the oscillations at the attached flow regime. The numerical findings are strengthened by drawing comparisons with the preliminary results obtained from wind tunnel experiments performed on a NACA 0012 airfoil undergoing dynamic stall. To the best of our knowledge, this is the first study that systematically bridges the dichotomy between the stall-induced dynamical signatures in stochastic aeroelastic systems and maps the same to the corresponding structural damage.

Behavior of Shallow Circular Tunnels—Impact of the Soil Spatial Variability

Spatial variability is unavoidable for soils and it is important to consider such a feature in the design of geotechnical engineering as it may lead to some structure behaviors which cannot be predicted by a calculation assuming homogenous soils. This paper attempts to evaluate the performance of a shallow circular tunnel, in a context of the service limit state, considering the soil spatial variability. The Log-normal distributed random fields, generated by the Karhunen–Loeve expansion method, are used for the spatial modeling. A two-dimensional numerical model, based on the finite difference method, is constructed to deterministically estimate two quantities of interest (i.e., tunnel lining bending moment and surface settlement). The model is combined with the random fields and is implemented into the Monte Carlo simulation to investigate the effects of the soil spatial variability on the tunnel responses. The autocorrelation distance, an important parameter for random fields, is varied within multiple probabilistic analyses. For both of the two tunnel responses, their variabilities are increased with increasing the autocorrelation distance, while a minimum mean value can be observed with this parameter being approximately the tunnel radius. Such finding is very useful for practical designs. A sensitivity analysis is also conducted to show the importance of each random parameter.

Stochastic dynamics of veering modes in a symmetric coupled system

ABSTRACT A conceptual model of a coupled symmetric system consisting of three beams coupled using a common base beam was developed. The purpose was to analyse the behaviour of close modes of the repeated system. A study on variation in the dynamic behaviour due to parametric uncertainty was carried out experimentally and theoretically. Experimentally the mass of the system was perturbed and the frequency response function (FRF) was analysed. Theoretically the mean and standard deviation of the frequency response function was determined using stochastic finite element (SFEM). Stochastic finite element analysis was carried out considering uncertainty in stiffness and the system response. Karhunen–Loeve (KL) expansion and polynomial chaos (PC) were carried out for Young's modulus uncertainty and the response uncertainty respectively. Optimal number of KL terms and PC order were determined. Using the optimal SFEM model, the influence of uncertainty in the individual subsystem on the dynamic response was determined. The results from the study indicated that a relaxed manufacturing tolerance could reduce the possibility of shock amplification. The frequency range over which high system response was observed in the FRF increased with lower tolerance, however, the maximum amplitude in the mean response was reduced.

Quadratic-exponential functionals of Gaussian quantum processes

This paper is concerned with exponential moments of integral-of-quadratic functions of quantum processes with canonical commutation relations of position-momentum type. Such quadratic-exponential functionals (QEFs) arise as robust performance criteria in control problems for open quantum harmonic oscillators (OQHOs) driven by bosonic fields. We develop a randomised representation for the QEF using a Karhunen–Loeve expansion of the quantum process on a bounded time interval over the eigenbasis of its two-point commutator kernel, with noncommuting position-momentum pairs as coefficients. This representation holds regardless of a particular quantum state and employs averaging over an auxiliary classical Gaussian random process whose covariance operator is specified by the commutator kernel. This allows the QEF to be related to the moment-generating functional of the quantum process and computed for multipoint Gaussian states. For stationary Gaussian quantum processes, we establish a frequency-domain formula for the QEF rate in terms of the Fourier transform of the quantum covariance kernel in composition with trigonometric functions. A differential equation is obtained for the QEF rate with respect to the risk sensitivity parameter for its approximation and numerical computation. The QEF is also applied to large deviations and worst-case mean square cost bounds for OQHOs in the presence of statistical uncertainty with a quantum relative entropy description.

Evaluating the Dynamic Response of the Bridge-Vehicle System considering Random Road Roughness Based on the Moment Method

The bridge-vehicle interaction (BVI) system vibration is caused by the vehicles passing through the bridge. The road roughness has a great impact on the system vibration. In this regard, poor road roughness is known to affect the comfort of the vehicle crossing the bridge and aggravate the fatigue damage of the bridge. Road roughness is usually regarded as a random process in numerical calculation. To fully consider the influence of road roughness randomness on the response of the BVI system, a random BVI model was established. Thereafter, the random process of road roughness was expressed by Karhunen–Loeve expansion (KLE), after which the moment method was used to calculate the maximum probability value of the BVI system response. The proposed method has higher accuracy and efficiency than the Monte Carlo simulation (MCS) calculation method. Subsequently, the influences of vehicle speed, roughness grade, and bridge span on the impact factor (IMF) were analyzed. The results show that the road roughness grade has a greater impact on the bridge IMF than the bridge span and vehicle speed.

Efficient Propagation of Epistemic Uncertainty for Probabilistic Seismic Hazard Analyses (PSHAs) Including Partial Correlation of Magnitude–Distance Scaling

ABSTRACT Probabilistic seismic hazard analysis (PSHA) is moving from ergodic ground-motion models (GMMs) to nonergodic GMMs that account for site-source-specific source, path, and site effects and which require a much larger number of GMM branches on the logic tree to capture the full epistemic uncertainty. An efficient method for computing PSHA with a large number of GMM branches was developed by Lacour and Abrahamson (2019) using polynomial chaos (PC) expansion with the key assumption that the epistemic uncertainty in the median ground motion is fully correlated. In the current study, we remove the assumption of full correlation using a multivariate PC expansion. The correlation structure of the available median GMMs across scenarios is computed empirically. The median ground motion is modeled as a Gaussian random process with the correlation structure of the GMMs across the range of relevant earthquake scenarios. This Gaussian random process is discretized using the Karhunen–Loeve expansion, which leads to multivariate PC expansions of uncertain hazard curves. The hazard fractiles can be reconstructed during an efficient postprocessing phase that includes the effects of partial correlation between the GMMs. Multivariate PC expansions require significantly more terms than for the fully correlated case, which increases the calculation time by about a factor of 5, but it is still much more efficient than direct sampling of the branches of the GMM logic for a large number of branches. An example hazard calculation shows that the effect of using partial correlation in place of full correlation of the GMMs is small for the Next Generation Attenuation-West2 (NGA-West2) set of GMMs, indicating that the fully correlated assumption may be adequate for many applications. The multivariate PC method can be used to evaluate the effects of the partial correlation of the GMMs for sets of GMMs that are different from the NGA-West2 GMMs.

Common Mode Component and Its Potential Effect on GPS-Inferred Crustal Deformations in Greenland

The effective extraction and elimination of the Global Positioning System's (GPS) common mode error (CME) is of great significance to improving the signal-to-noise ratio of the coordinate time series and accurately estimating the characteristics of the crustal deformation. In this study, the principal component analysis (PCA), Karhunen–Loeve expansion (KLE) and stacking filtering methods are used to extract the CMEs of Greenland GPS station coordinates in the East-North-Up (ENU) directions. Their influence on GPS coordinate residuals and the relationship amongst the observations are also analyzed. The results show that the PCA, KLE and stacking filtering methods can effectively eliminate the common mode noise in the GPS station coordinate time series and reduce their uncertainty. The PCA method is better than KLE and stacking methods, and the filtering effect in the up (U) direction is better than those in the east (E) and north (N) directions. Then, as the CME can be detected periodically in the U direction, its physical mechanism is studied. On the one hand, the period of CME maximum power is 1 year but is not constant. On the other hand, the correlation between the GPS CME and GRACE ice loading deformation is 0.5. When the ice loading deformation is deduced by the GRACE method, the CME of the GPS network can be eliminated by 25.6%, which indicates that the ice loading deformation is part of the source of GPS CME in Greenland. Finally, the influence of the U direction CME on the station noise characteristics is analyzed using maximum likelihood estimation (MLE). The results show that the CME noise type is mainly WN + FN. Moreover, the estimated coordinate velocities and most of the annual amplitudes of the stations have an increasing trend after CME elimination, whereas the uncertainty has a decreasing trend. These findings are consistent with the trend of noise component change. The above results indicate that the CME elimination of the GPS stations cannot be ignored, as they can help to improve the estimation of station velocity and annual amplitude and reduce the uncertainty.

Sequential Sampling Framework for Metamodeling Uncertainty Reduction in Multilevel Optimization of Hierarchical Systems

Abstract Metamodels instead of computer simulations are often adopted to reduce the computational cost in the uncertainty-based multilevel optimization. However, metamodel techniques may bring prediction discrepancy, which is defined as metamodeling uncertainty, due to the limited training data. An unreliable solution will be obtained when the metamodeling uncertainty is ignored, while an overly conservative solution, which contradicts the original intension of the design, may be got when both parametric and metamodeling uncertainty are treated concurrently. Hence, an adaptive sequential sampling framework is developed for the metamodeling uncertainty reduction of multilevel systems to obtain a solution that approximates the true solution. Based on the Kriging model for the probabilistic analytical target cascading (ATC), the proposed framework establishes a revised objective-oriented sampling criterion and sub-model selection criterion, which can realize the location of additional samples and the selection of subsystem requiring sequential samples. Within the sampling criterion, the metamodeling uncertainty is decomposed by the Karhunen–Loeve expansion into a set of stochastic variables, and then polynomial chaos expansion (PCE) is used for uncertainty quantification (UQ). The polynomial coefficients are encoded and integrated in the selection criterion to obtain subset sensitivity indices for the sub-model selection. The effectiveness of the developed framework for metamodeling uncertainty reduction is demonstrated on a mathematical example and an application.

An adaptive scheme for random field discretization using KL expansion

With the increase in computational facilities, interest in probabilistic analysis of engineering structures are observed to grow for a realistic assessment of physical systems. While the probabilistic analyses are generally carried out considering random variable models of physical parameters, a spatially varying random field model is more realistic. A discretization method converts the continuous parameter random field to a set of random variables. Truncated Karhunen–Loeve (KL) expansion is one of the popular methods for the discretization of a random field. An accurate discretization can be achieved by considering a fine mesh along with a large number of terms. However, an increase in the number of elements and/or inclusion of more terms leads to an increase in computational cost. It is also found that only an increase in the number of terms in the expansion or number of elements alone does not provide an accurate representation of the random field. An adaptive discretization strategy is presented, which will suitably discretize the concerned domain while ensuring that both global and local level errors are kept within the prescribed limit.