Karhunen Loeve Research Articles

Differential evolution-based topology optimization using Karhunen-Loève expansion

Topology optimization is a widely adopted structural optimization approach that aims to maximize desired performance based on mathematical and physical principles. Gradient-based methods are common for obtaining optimal configurations; however, they struggle with problems involving numerous local optima and non-differentiable problems, which prevent the derivation of design sensitivity. To address these issues, this study explores the application of the Success History-Based Differential Evolution with Linear Population Size Reduction and Semi-Parameter Adaptation, which is regarded as LSHADE-SPA algorithm, known for its strong search capabilities. Nevertheless, its computational cost makes it impractical for topology optimization, especially as the number of design variables increases. To overcome this challenge, a novel structure representation method utilizing the Karhunen-Loève (KL) expansion is constructed. Applying truncated KL expansion instead of conventional density methods to represent structural configurations reduces the number of design variables. This article introduces a differential evolution-based topology optimization method utilizing truncated KL expansion and validates its effectiveness through applications in various structural and fluid problems. The method is particularly suited to highly nonlinear problems, such as negative Poisson's ratio structures and micro-mixer designs. The findings here indicate that differential evolution with truncated KL expansion can serve as an efficient and robust topology optimization method, proving particularly effective for solving optimization problems that are challenging for conventional gradient-based approaches. Notably, the proposed methodology not only delivers superior results but also alleviates computational burdens, thereby advancing the prospects of topology optimization in various engineering applications.

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.

A combined Markov Chain Monte Carlo and Levenberg–Marquardt inversion method for heterogeneous subsurface reservoir modeling

In this study, we systematically investigate an inverse method for heterogeneous porous media to obtain porosity and permeability fields considering only production well data. The forward modeling of the input data, namely pressure and saturation fields, is based on the motion equations of a coupled Darcy flow system involving two phases of isothermal fluid flow. We discretize these equations using a multiscale finite volume simulation technique in the spatial domain, and the backward Euler method in time domain. In the inversion procedure, we combine a global optimization method, the Markov Chain Monte Carlo (MCMC) method, with a local optimization method, the Levenberg–Marquardt (LM) method. The MCMC was implemented as the Random Walk algorithm, and to generate the samples of the porosity and permeability fields, we employed Karhunen–Loève (KL) expansion of second-order stationary fields with Gaussian covariance. The coefficients of the KL expansion are estimated by minimizing the norm of the residual dependent on these fields. At the end of the MCMC iterations, we refine the KL coefficients associated with the porosity and permeability fields using the LM method. We verify in the numerical experiments that the accuracy of the porosity and permeability fields is improved by the LM refinement step.

An interval finite element method based on bilevel Kriging model

This study introduces a new approach utilizing an interval finite element method combined with a bilevel Kriging model to determine the bounds of structural responses in the presence of spatial uncertainties. A notable benefit of this approach is its ability to determine the response bounds across all degrees of freedom with a small sample size, which means that it has high efficiency. Firstly, the spatially varying uncertain parameters are quantified using an interval field model, which is described by a series of standard interval variables within a truncated interval Karhunen-Loève (K-L) series expansion. Secondly, considering that the bound of structural response is a function of spatial position with the property of continuity, a surrogate model for the response bound is constructed, namely the first-level Kriging model. The training samples required for this surrogate model are obtained by establishing the second-level Kriging model. The second-level Kriging model is established to describe the structural responses at particular locations relative to the interval variables so as to facilitate the upper and lower bounds of the node response required by the first-level Kriging model. Finally, the accuracy and effectiveness of the method are verified through examples.

Forward variable selection enables fast and accurate dynamic system identification with Karhunen-Loève decomposed Gaussian processes

A promising approach for scalable Gaussian processes (GPs) is the Karhunen-Loève (KL) decomposition, in which the GP kernel is represented by a set of basis functions which are the eigenfunctions of the kernel operator. Such decomposed kernels have the potential to be very fast, and do not depend on the selection of a reduced set of inducing points. However KL decompositions lead to high dimensionality, and variable selection thus becomes paramount. This paper reports a new method of forward variable selection, enabled by the ordered nature of the basis functions in the KL expansion of the Bayesian Smoothing Spline ANOVA kernel (BSS-ANOVA), coupled with fast Gibbs sampling in a fully Bayesian approach. It quickly and effectively limits the number of terms, yielding a method with competitive accuracies, training and inference times for tabular datasets of low feature set dimensionality. Theoretical computational complexities are O ( N P 2 ) in training and O ( P ) per point in inference, where N is the number of instances and P the number of expansion terms. The inference speed and accuracy makes the method especially useful for dynamic systems identification, by modeling the dynamics in the tangent space as a static problem, then integrating the learned dynamics using a high-order scheme. The methods are demonstrated on two dynamic datasets: a ‘Susceptible, Infected, Recovered’ (SIR) toy problem, along with the experimental ‘Cascaded Tanks’ benchmark dataset. Comparisons on the static prediction of time derivatives are made with a random forest (RF), a residual neural network (ResNet), and the Orthogonal Additive Kernel (OAK) inducing points scalable GP, while for the timeseries prediction comparisons are made with LSTM and GRU recurrent neural networks (RNNs) along with the SINDy package.

Bayesian Inference and Condition Assessment Based on the Deflection of Aging Reinforced Concrete Hollow Slab Bridges

This paper presents a Bayesian inference framework for updating the structural rigidity ratio of aging hollow slab RC bridges using deflection measurements. The framework models the structural rigidity ratio as a stochastic field along the hollow RC slabs, using the Karhunen–Loeve (KL) transform to capture spatial correlation and variation. Bayesian inference is then applied using deflection data from static loading tests, supported by a finite element model (FEM) and a Kriging surrogate model to enhance computational efficiency. The posterior distribution of the structural rigidity ratio is derived using a Markov chain Monte Carlo (MCMC) sampler. The proposed method was tested on an RC bridge with hollow slabs, using deflection measurements taken before and after reinforcement. The Bayesian updates indicated increased structural rigidity ratios after reinforcement, validating the effectiveness of the reinforcement. The deflection predictions from the updated models closely matched the measurements, with the 95% confidence bounds encompassing most of the data. This demonstrates the method’s validity and robustness in capturing the structural improvements post-reinforcement.

Stochastic Modeling of Two-Phase Transport in Fractured Porous Media Under Geological Uncertainty Using an Improved Probabilistic Collocation Method

Summary Simulation of multiphase transport through fractured porous media is highly affected by the uncertainty in fracture distribution and matrix block size that arises from inherent heterogeneity. To quantify the effect of such uncertainties on displacement performance in porous media, the probabilistic collocation method (PCM) has been applied as a feasible and accurate approach. However, propagation of uncertainty during the simulation of unsteady-state transport through porous media could not be computed by this method or even by the direct-sampling Monte Carlo (MC) approach. Therefore, with this research, we implement a novel numerical modeling workflow that improves PCM on sparse grids and combines it with the Smolyak algorithm for selection of collocation points sets, Karhunen-Loeve (KL) decomposition, and polynomial chaos expansion (PCE) to compute the uncertainty propagation in oil-gas flow through fractured porous media in which gravity drainage force is enabled. The effect of uncertainty in the vertical dimension of matrix blocks, which are frequently an uncertain and history-matching parameter, on simulation results of randomly synthetic 3D fractured media is explored. The developed numerical model is innovatively coupled with solving governing deterministic partial differential equations (PDEs) to compute uncertainty propagation from the first timestep to the last timestep of the simulation. The uncertainty interval and aggregation of uncertainty in ultimate recovery are quantified, and statistical moments for simulation outputs are presented at each timestep. The results reveal that the model properly quantifies uncertainty and extremely reduces central processing unit (or CPU) time in comparison with MC simulation.

Random field of homogeneous and multi-material structures by the smoothed finite element method and Karhunen–Loève expansion

A random field of homogeneous and multi-material structures with uncertain scenarios is addressed by constructed the generalized stochastic cell-based smoothed finite element model. Smoothed finite element method (S-FEM) is “Jacobian-free”, which is not only overcomes the limitations of “overly-stiff” and low accuracy of the standard FEM, but also demonstrates strong resistance to mesh distortion. This paper proposes an efficient non-intrusive stochastic smoothed finite element method (SS-FEM) based on cell-based smoothing domains (SD) for the stochastic analysis of elastic problems, which describe much more realistically real physical systems under uncertain scenarios. This SS-FEM starts from the Gaussian random field based on representation related to the spatial variability in material parameters based on the Karhunen-Loève (KL) expansion, and then establishes the basic formulation that considers the effects of uncertainties. A generalized solving framework based on numerical integration is further developed. The non-intrusive dimension reduction method is also adopted to obtain the statistical moments and probability density function. The proposed SS-FEM can capture the structural stochastic responses under uncertainty condition by combining gradient smoothing technology and uncertainty quantification. A number of engineering examples are compared with the solution of Monte Carlo simulation to demonstrate both the accuracy and the efficiency of the proposed method.

A nonlinear interval finite element method for elastic–plastic problems with spatially uncertain parameters

This paper proposes a nonlinear interval finite element method for elastic–plastic analysis of structures with spatially uncertain parameters. The spatially uncertain parameters are described by the interval field, and the variation bounds of the elastic–plastic structural responses can be calculated effectively. Quantified by the interval field, the spatially uncertain parameters are represented by the interval Karhunen–Loève (K-L) expansion, based on which the nonlinear interval finite element equilibrium equation is formulated. An interval iterative method is then presented to solve the equilibrium equation and obtain an outer solution of the variation bounds of structural responses such as displacement. In this method, the Newton-Raphson iterative method is used to transform the nonlinear problem into a linear one, and then the interval iterative method is introduced to solve the interval linear equations. Three numerical examples are employed to illustrate the feasibility and accuracy of the proposed method.

Analysis of sensitive clay landslide mechanism and impact pipeline bearing characteristics considering spatial variability

Landslides caused by the progressive failure of sensitive clay slopes may cause substantial economic losses and environmental pollution to pipelines. This study utilizes the Karhunen-Loève (K-L) expansion method, the coupled Eulerian-Lagrangian (CEL) method, and the Monte Carlo method to model the large deformation behaviour of slopes with spatial variability. The proposed RCEL-MC framework for joint analysis of extensive deformation-probability statistics of landslide impact on pipelines confirms that the spatial variability clay slope instability failure mechanism, considering the soil softening effect under earthquake action, aligns more closely with real-world scenarios. Furthermore, based on the statistical failure probability of pipelines impacted by landslides, a sensitive clay slope pipeline safety design method is suggested. The research highlights that the random field parameters significantly influence the load-bearing characteristics of pipelines affected by spatially variable slope progressive failure landslides. The uncertainty surrounding the maximum impact force on pipelines notably increases with the rise of the horizontal correlation length and variability coefficient. When accounting for spatial variability, the maximum impact force on landslide pipelines surpasses the deterministic model analysis results by 28-69 %. It is emphasized that the deterministic model analysis, which neglects the spatial variability of soil and the soil softening effect, may seriously underestimate landslide risk. Based on the criteria for pipeline buckling failure and yield failure, it is advised that the safe design parameters for pipelines in practical projects should fall within the intersection of the secure regions defined by the two failure criteria.

A neural network approach for the reliability analysis on failure of shallow foundations on cohesive soils

A collection of feed forward neural networks (FNN) for estimating the limit pressure load and the according displacements at limit state of a footing settlement is presented. The training procedure is through supervised learning with error loss function the mean squared error norm. The input dataset is originated from Monte Carlo simulations for a variety of loadings and stochastic uncertainty of the material of the clayey soil domain. The material yield function is the Modified Cam Clay model. The accuracy of the FNN’s is in terms of relative error no more than 10-5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^{-5}$$\\end{document} and this applies to all output variables. Furthermore, the epochs of the training of the FNN’s required for construction are found to be small in amount, in the order of magnitude of 90,000, leading to an alleviated data cost and computational expense. The input uncertainty of Karhunen Loeve random field sum appears to provide the most detrimental values for the displacement field of the soil domain. The most unfavorable situation for the displacement field result to limit displacements in the order of magnitude of 0.05 m, that may result to structural collapse if they appear to the founded structure. These series can provide an easy and reliable estimation for the failure of shallow foundation and therefore it can be a useful implement for geotechnical engineering analysis and design.

Research on uncertainties in fuel centrifugal pump based on prediction and reconstruction of internal flow field

Geometric machining errors in the blade profile and variable operating conditions in the extreme operating environment are primary factors leading to the uncertainties in pump performance. This paper presents an analysis of uncertainties of fuel centrifugal pumps by modeling the geometry uncertainty in blade machining based on the Karhunen–Loève (KL) expansion and using a polynomial chaos expansion (PCE) model. First, the geometric uncertainty in the blade machining is described by the KL expansion in three sections and a stochastic simulation of the blade geometry is performed. Then, a PCE surrogate model is trained based on the least angle regression method and validated by the bootstrap method to quantify the uncertainties of performance indices. Finally, the influence mechanism and relative importance of each input uncertainty parameter are investigated using a quasi-Monte Carlo simulation method. The results show that the KL expansion of the blade profile uses the random vector perturbation superposition of three stream surface, achieving the dimensional reduction in the blade machining error. The PCE surrogate model, trained with a dataset of 3 × 106 sample points, exhibits excellent fit, and the R-squared and adjusted R-squared for head coefficient and efficiency are both above 80%. The variance of parameter control points of the reconstructed flow field is less than 0.002. The uncertainties in both operating conditions and parameters have an influence on the distribution of the global flow field, while the influence of the uncertainty in machining error on the global flow field mainly concentrates on the power-generating positions of the blade.

High-Resolution Estimation of Soil Saturated Hydraulic Conductivity via Upscaling and Karhunen–Loève Expansion within DREAM(ZS)

Quantification of the soil hydraulic conductivity is key to the study of water flow and solute transport in unsaturated soils. Rapid advances in measurement technology have provided a large number of observations at different scales, offering unprecedented opportunities and challenges for the estimation of hydraulic parameters. This paper proposes an inverse estimation method for downscaling of observations on coarse scales to estimate hydraulic parameters on high-resolution scales. Due to the significant spatial heterogeneity, the inversion faces the problems of dynamics-based integration of data at different scales, model uncertainty due to hundreds and thousands of parameters, and computational consumption due to the large number of forward simulations. To overcome these problems, this paper uses an efficient Bayesian optimization DREAM(ZS) as an inverse framework, and incorporates an analytical upscaling method and Karhunen–Loève (KL) expansion to infer finer-scale saturated hydraulic conductivity distribution conditioned on coarse-scale measurements. The efficient upscaling method is used to link measurements and hydraulic parameters at different scales, and Karhunen–Loève (KL) expansion is incorporated to greatly reduce the dimension of the parameter to be estimated. To further improve the efficiency of the inversion, a locally one-dimensional (LOD) algorithm is used to solve the multidimensional water flow model at coarse scales. The proposed inverse model is applied in a series of numerical experiments to demonstrate its applicability and effectiveness under different flow boundary conditions, different levels of ratio between coarse- and fine-scale grids, different densities of observation points, and different degrees of statistic heterogeneity of soil mediums.

Self-Adaptive Incremental PCA-Based DBSCAN of Acoustic Features for Anomalous Sound Detection

In modern industry, maintaining continuous machine operations is important for improving production efficiency and reducing costs. Therefore, the smart technology of acoustic monitoring to detect anomalous machine conditions earlier before breakdowns works as part of predictive maintenance and is applied not only in industry fault detection but also in safety monitoring and surveillance systems. This paper proposes a self-adaptive unsupervised machine learning algorithm with dimension-reduction technology to detect anomalous sounds after extracting acoustic machine features. Technically, the automatic EPS calculation algorithm-based genetic algorithm optimizes the automatic clustering algorithm’s configuration for incremental principal component analysis and density-based spatial clustering algorithms with noise. IPCA is enhanced by the sequential Karhunen–Loeve (SKL) algorithm, and the condensation algorithm works as the second layer of the algorithm to reduce the number of effective components. This architecture could select an optimized set of parameters based on different test environments and keeps performance quality with fewer computational requirements. In the experiments, 228 sets of normal sounds and 100 sets of anomaly sounds are used. The sound files are collected from the same machine type (stepper motors) at a real plant site. We compare the proposed algorithm with K-means++, one-class SVM, agglomerative clustering, DCGAN and DCNN-Autoencoder, and this new algorithm performs best, with an AUC of 0.84 and the shortest execution time. The algorithm is generic and can be applied to detect anomalies in machines to provide early warning to people to avoid serious accidents or disasters.

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.

Algorithm for Surface Wave Suppression on 2D Seismic Data Using the Slant Karhunen–Loeve Transform in a Time-Frequency Domain

Abstract —Surface waves are the main source of coherent noise in land seismic survey, and their suppression is one of the main stages of common depth point data processing designed to improve the quality of tracking primary reflections on time sections. In practice, noise reduction is carried out using procedures from modern software based on numerical modeling of waveforms. However, they are too resource-intensive and have a large number of subjectively customizable parameters. The known algorithms have a common drawback: either the energy of reflected waves is distorted in an interference zone with a noise wave or the noise suppression quality is unsatisfactory. The current research is aimed at improving the filtering algorithm in a time-frequency domain using the slant Karhunen–Loeve transform in order to overcome these limitations, to increase the accuracy and rate of its software implementation, and also to test it when processing profile field data from land-based 2D seismic surveys. The algorithm is modified by developing a new method for determining static corrections for surface wave hodograph rectification in a time-frequency domain and by the application of preprocessing in which the reflected wave signal is removed preliminarily. These and other modifications ensure faster calculations and improve the quality of surface wave interference suppression. In addition, the slant Karhunen–Loeve transform is accelerated by parallelizing calculations across logical processor cores. In this paper, the algorithm is described in detail, its significant advantage over the standard methods of bandpass filtering and F–K filtering is shown, and the results of processing the field data obtained by the SWANA procedure (Geovation 2.0) and by the slant Karhunen–Loeve transform. The result obtained by the slant Karhunen–Loeve transform is superior to the SWANA procedure in terms of the surface wave filtering quality and has only four adjustable parameters (SWANA has 20 parameters)

Domain-decomposed Bayesian inversion based on local Karhunen-Loève expansions

In many Bayesian inverse problems the goal is to recover a spatially varying random field. Such problems are often computationally challenging especially when the forward model is governed by complex partial differential equations (PDEs). The challenge is particularly severe when the spatial domain is large and the unknown random field needs to be represented by a high-dimensional parameter. In this paper, we present a domain-decomposed method to attack the dimensionality issue and the method decomposes the spatial domain and the parameter domain simultaneously. On each subdomain, a local Karhunen-Loève (KL) expansion is constructed, and a local inversion problem is solved independently in a parallel manner, and more importantly, in a lower-dimensional space. After local posterior samples are generated through conducting Markov chain Monte Carlo (MCMC) simulations on subdomains, a novel projection procedure is developed to effectively reconstruct the global field. In addition, the domain decomposition interface conditions are dealt with an adaptive Gaussian process-based fitting strategy. Numerical examples are provided to demonstrate the performance of the proposed method.

A novel general method for simulating a one dimensional random field based on the active learning Kriging model

Random fields are widely used to represent the uncertainty of some parameters in engineering, and numerous simulation approaches have been developed for Gaussian and non-Gaussian random fields. However, the unified methods among them suffer from low computational accuracy and efficiency or discontinuities in the simulated random fields. Therefore, an easy-to-implement general simulation method based on the active learning Kriging model is proposed for a one dimensional(1D) Gaussian or non-Gaussian random field in this paper. In the proposed method, there are two stages. One stage, called the inner loop, is to construct the Kriging approximation of a random field sample with enough accuracy by some samples of the random variables at some discretized locations, in which an active learning strategy based on the error estimation for the Kriging model is introduced to select adaptively the added locations, and a fast sampling method is presented to determine efficiently the samples at the added locations. In the other stage, called the outer loop, some random field samples are represented accurately by their corresponding Kriging approximations through training iteratively. Furthermore, several numerical examples are presented to show the accuracy, effectiveness and generality of the proposed method for 1D Gaussian and non-Gaussian random fields by comparing with the Karhunen–Loève(KL) expansion method. Meanwhile, the effects of the types of correlation function and the scales of fluctuation on the simulation results are analyzed.

Groundwater Responses of Foundation Subjected to Water Level Fluctuation of Reservoir Considering Variability of Layered Structure

The effect of the variability in a layered structure, characterized by the spatial variability of the saturated hydraulic conductivity, on the distribution of a pressure head p in a foundation subjected to water level fluctuation in a reservoir is investigated with the aid of the random field theory, Karhunen–Loève (K-L) expansion, first-order moment approach, and cross-correlation analysis. The results show that the variability in the foundation structure has significant impacts on the groundwater response to the reservoir’s water level fluctuations. Regions with relatively large uncertainties of the p and σp values in the foundation are those around the initial water level at the reservoir side, and those at the distal end away from the reservoir. In addition, there is a larger variance of Ks, denoted as σlnKs2, a larger correlation scale in the horizontal direction λh, a larger correlation scale in the vertical direction λv, and a larger one-way time consumption of fluctuations T to a larger uncertainty in p. Moreover, the four factors (σlnKs2, λh, λv, and T) all have positive correlations with σp. σlnKs2 has the largest impact on σp in the foundation, λv has the second largest impact, and λh has the smallest impact. A foundation with small Ks values around the initial water level at the reservoir side and large Ks values around the highest water level at the reservoir side may produce larger p values in the foundation. These results yield useful insight into the effect of the variability in a layered structure on the distribution of the pressure head in a foundation subjected to water level fluctuation in a reservoir.

Application of the improved particle swarm optimization method in slope probability analysis

To improve the probability analysis efficiency of slope, this paper proposes an improved particle swarm optimization (IPSO) method to determine the critical sliding surface (CSS) and its corresponding minimum safety factor, which is used to calculate the failure probability of slope. Based on the random fields generated by using Karhunen-Loève (KL) expansion method, the simplified Bishop’s method combined with the entry and exit method, the particle swarm optimization (PSO) method, and the IPSO method is used to determine the CSS and its corresponding minimum factor of safety. Then, Monte Carlo simulation is used to estimate the failure probability of slope. The application potential of the IPSO method is demonstrated by re-analyzing two examples. Meaningful comparisons are made to demonstrate the calculating accuracy and calculating efficiency of the IPSO method in searching for the minimum safety factor of slope. Results show that the IPSO can accurately and efficiently determine the minimum safety factor in slope probability analysis considering the spatially variable soils. The IPSO method provides a promising tool for an efficient slope probability analysis.