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Journal articles on the topic 'Karhunen-Loeve expansion'

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Published: 4 June 2021

Last updated: 28 January 2023

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1

Paff, W. G., and G. Ahmadi. "On the Convergence of Karhunen-Loeve Series Expansion for a Brownian Particle." Journal of Applied Mechanics 60, no. 3 (September 1, 1993): 783–84. http://dx.doi.org/10.1115/1.2900876.

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A linear Langevin equation for the velocity of a Brownian particle is considered. The equation of motion is solved and the Karhunen-Loeve expansion for the particle velocity is derived. The mean-square velocity as obtained by the truncated Karhunen-Loeve expansion is compared with the exact solution. It is shown, as the number of terms in the series increases, the result approaches that of the exact solution asymptotically.

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2

BABUŠKA, IVO, and KANG-MAN LIU. "ON SOLVING STOCHASTIC INITIAL-VALUE DIFFERENTIAL EQUATIONS." Mathematical Models and Methods in Applied Sciences 13, no. 05 (May 2003): 715–45. http://dx.doi.org/10.1142/s0218202503002696.

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This paper addresses the issues involved in solving systems of linear ODE's with stochastic coefficients and loadings described by the Karhunen–Loeve expansion. The Karhunen–Loeve expansion is used to discretize random functions into a denumerable set of uncorrelated random variables, thus providing us for transforming this problem into an equivalent deterministic one. Perturbation error estimates and a priori error estimates between the exact solution and the finite element solution in the framework of Sobolev space are given. The method of successive approximations for finite element solutions is analyzed.

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3

Courmontagne, Ph. "A new formulation for the Karhunen–Loeve expansion." Signal Processing 79, no. 3 (December 1999): 235–49. http://dx.doi.org/10.1016/s0165-1684(99)00099-7.

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4

Jaimez, Ramón Gutiérrez, and Mariano J. Valderrama Bonnet. "On the Karhunen-Loeve expansion for transformed processes." Trabajos de Estadistica 2, no. 2 (September 1987): 81–90. http://dx.doi.org/10.1007/bf02863594.

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5

Li, Heng. "Conditional Simulation of Flow in Heterogeneous Porous Media with the Probabilistic Collocation Method." Communications in Computational Physics 16, no. 4 (October 2014): 1010–30. http://dx.doi.org/10.4208/cicp.090513.040414a.

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AbstractA stochastic approach to conditional simulation of flow in randomly heterogeneous media is proposed with the combination of the Karhunen-Loeve expansion and the probabilistic collocation method (PCM). The conditional log hydraulic conductivity field is represented with the Karhunen-Loeve expansion, in terms of some deterministic functions and a set of independent Gaussian random variables. The propagation of uncertainty in the flow simulations is carried out through the PCM, which relies on the efficient polynomial chaos expansion used to represent the flow responses such as the hydraulic head. With the PCM, existing flow simulators can be employed for uncertainty quantification of flow in heterogeneous porous media when direct measurements of hydraulic conductivity are taken into consideration. With illustration of several numerical examples of groundwater flow, this study reveals that the proposed approach is able to accurately quantify uncertainty of the flow responses conditioning on hydraulic conductivity data, while the computational efforts are significantly reduced in comparison to the Monte Carlo simulations.

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6

Phoon, K. K., S. P. Huang, and S. T. Quek. "Simulation of second-order processes using Karhunen–Loeve expansion." Computers & Structures 80, no. 12 (May 2002): 1049–60. http://dx.doi.org/10.1016/s0045-7949(02)00064-0.

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7

Hilai, Ran, and Jacob Rubinstein. "Recognition of rotated images by invariant Karhunen–Loeve expansion." Journal of the Optical Society of America A 11, no. 5 (May 1, 1994): 1610. http://dx.doi.org/10.1364/josaa.11.001610.

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8

Smallwood, David. "Characterization and Simulation of Gunfire with Karhunen-Loeve Expansion." Journal of the IEST 47, no. 1 (September 14, 2004): 47–50. http://dx.doi.org/10.17764/jiet.47.1.3476326r02g27247.

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Gunfire is used as an example to illustrate how the Karhunen-Loeve (K-L) expansion can be used to characterize and simulate nonstationary random events. This paper will develop a method to describe the nonstationary random process in terms of a K-L expansion. The gunfire record is broken up into a sequence of transient waveforms, each representing the response to the firing of a single round. First, the mean is estimated and subtracted from each waveform. The mean is an estimate of the deterministic part of the gunfire. The autocovariance matrix is estimated from the matrix of these single-round gunfire records. Each column is a realization of the firing of a single round. The gunfire is characterized with the K-L expansion of the autocovariance matrix. The gunfire is simulated by generating realizations of records of a single-round firing from the expansion and the mean waveform. The individual realizations are then assembled into a realization of a time history of many rounds firing. The method is straightforward and easy to implement, and produces a simulated record very much like the original measured gunfire record.

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9

Ai, Xiaohui. "Karhunen–Loeve expansion for the additive detrended Brownian motion." Communications in Statistics - Theory and Methods 46, no. 16 (August 2, 2016): 8210–16. http://dx.doi.org/10.1080/03610926.2016.1177079.

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10

Lenz, Reiner, and Mats Österberg. "Computing the Karhunen-Loeve Expansion with a Parallel, Unsupervised Filter System." Neural Computation 4, no. 3 (May 1992): 382–92. http://dx.doi.org/10.1162/neco.1992.4.3.382.

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We use the invariance principle and the principles of maximum information extraction and maximum signal concentration to design a parallel, linear filter system that learns the Karhunen-Loeve expansion of a process from examples. In this paper we prove that the learning rule based on these principles forces the system into stable states that are pure eigenfunctions of the input process.

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11

Wan, Chengbiao, Mengchun Pan, Qi Zhang, Dixiang Chen, Hongfeng Pang, and Xuejun Zhu. "Performance improvement of magnetic anomaly detector using Karhunen–Loeve expansion." IET Science, Measurement & Technology 11, no. 5 (August 1, 2017): 600–606. http://dx.doi.org/10.1049/iet-smt.2016.0392.

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12

Phoon, K. K., H. W. Huang, and S. T. Quek. "Simulation of strongly non-Gaussian processes using Karhunen–Loeve expansion." Probabilistic Engineering Mechanics 20, no. 2 (April 2005): 188–98. http://dx.doi.org/10.1016/j.probengmech.2005.05.007.

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13

Ai, Xiaohui, and Yang Sun. "Karhunen–Loeve expansion for the additive two-sided Brownian motion." Communications in Statistics - Theory and Methods 47, no. 13 (October 30, 2017): 3085–91. http://dx.doi.org/10.1080/03610926.2017.1346809.

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14

Wornell, G. W. "A Karhunen-Loeve-like expansion for 1/f processes via wavelets." IEEE Transactions on Information Theory 36, no. 4 (July 1990): 859–61. http://dx.doi.org/10.1109/18.53745.

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15

Liu, Zhangjun, Zixin Liu, and Yongbo Peng. "Dimension reduction of Karhunen-Loeve expansion for simulation of stochastic processes." Journal of Sound and Vibration 408 (November 2017): 168–89. http://dx.doi.org/10.1016/j.jsv.2017.07.016.

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16

Barat, P., and S. K. Bandyopadhyay. "Expansion of ultrasonic signals in Karhunen–Loeve basis for data compression." Computational Materials Science 12, no. 1 (August 1998): 57–63. http://dx.doi.org/10.1016/s0927-0256(98)00018-4.

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17

Alvarez-Borrego, Josue´. "Karhunen-Loeve expansion of stationary random signals with exponentially oscillating covariance function." Optical Engineering 42, no. 4 (April 1, 2003): 1018. http://dx.doi.org/10.1117/1.1558089.

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18

Phoon, K. K., S. P. Huang, and S. T. Quek. "Implementation of Karhunen–Loeve expansion for simulation using a wavelet-Galerkin scheme." Probabilistic Engineering Mechanics 17, no. 3 (July 2002): 293–303. http://dx.doi.org/10.1016/s0266-8920(02)00013-9.

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19

Urbina, Angel, and Thomas Paez. "Probabilistic Numerical Analysis of Large, Complex, Structural Dynamic System Models." Journal of the IEST 46, no. 1 (September 14, 2003): 119–27. http://dx.doi.org/10.17764/jiet.46.1.p3k33743858u56hx.

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In recent years, great progress has been made in the construction and solution of large finite element models of complex structural dynamic systems. For example, structural models with millions of degrees of freedom are being built and used to approximate responses of structural systems. Further, great progress is being made in stochastic system analysis. Techniques for the construction of stochastic system models have been developed and solution techniques proposed. However, the two areas have not been combined, on a large scale, because stochastic finite element approaches appear very intrusive in their pure form. That is, substantial modifications of deterministic finite element codes are required to accommodate stochastic analysis. In view of this, a technique that uses the techniques of stochastic finite elements in a non-intrusive manner is required. This research provides one such approach. Specifically, the problem is divided into three parts: (1) model structural dynamic excitations using traditional approaches, and model physical system randomness using techniques of stochastic finite elements, namely, the Karhunen-Loeve expansion and polynomial chaos; (2) generate stochastic structural realizations and realizations of the random excitation using a Monte Carlo approach, and analyze structural responses with parallel computation in a suitable, large-scale finite element code; and (3) analyze structural dynamic responses using the techniques of stochastic finite elements, namely, the Karhunen-Loeve expansion and polynomial chaos. This paper supplies the details of the analytical approach. A numerical example is presented.

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20

Huang, S. P., S. T. Quek, and K. K. Phoon. "Convergence study of the truncated Karhunen–Loeve expansion for simulation of stochastic processes." International Journal for Numerical Methods in Engineering 52, no. 9 (November 30, 2001): 1029–43. http://dx.doi.org/10.1002/nme.255.

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21

MA, XIA, and GEORGE EM KARNIADAKIS. "A low-dimensional model for simulating three-dimensional cylinder flow." Journal of Fluid Mechanics 458 (May 10, 2002): 181–90. http://dx.doi.org/10.1017/s0022112002007991.

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We investigate the stability and dynamics of three-dimensional limit-cycle states in flow past a circular cylinder using low-dimensional modelling. High-resolution direct numerical simulations are employed to obtain flow snapshots from which the most energetic modes are extracted using proper orthogonal decomposition. We show that the limit cycle is reproduced very accurately with only twenty three-dimensional modes. The addition of two-dimensional modes to the Karhunen–Loeve expansion basis improves the ability of the model to capture the three-dimensional bifurcation, including the discontinuity in the Strouhal number discovered experimentally.

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22

La, Viet Duc. "Combination of Partial Stochastic Linearization and Karhunen-Loeve Expansion to Design Coriolis Dynamic Vibration Absorber." Mathematical Problems in Engineering 2017 (2017): 1–11. http://dx.doi.org/10.1155/2017/1615859.

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Coriolis dynamic vibration absorber is a device working in nonlinear zone. In stochastic design of this device, the Monte Carlo simulation requires large computation time. A simplified model of the system is built to retain the most important nonlinear term, the Coriolis damping of the dynamic vibration absorber. Applying the full equivalent linearization technique to the simplified model is inaccurate to describe the nonlinear behavior. This paper proposes a combination of partial stochastic linearization and Karhunen-Loeve expansion to solve the problem. The numerical demonstration of a ropeway gondola induced by wind load is presented. A design example based on the partial linearization supports the advantage of the proposed approach.

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23

Yan, Wei, Zhi Hua Chen, Tai Jiao Du, and Jian Guo Wang. "Atmospheric Phase Screen Simulation Using Zernike Polynomial." Applied Mechanics and Materials 182-183 (June 2012): 1002–6. http://dx.doi.org/10.4028/www.scientific.net/amm.182-183.1002.

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An algorithm is described which simulates atmospheric phase screen (PS) distorted by von Karman atmospheric turbulence by using Zernike expansion of randomly weighted Karhunen-Loeve function. The statistics of the PS generated by using power spectrum method which is most commonly used for generating PS poorly match up with the theoretical structure function, especially at low spatial frequency, while the algorithm described in the this paper can compensate for this shortcoming. And the accuracy of the PS is verified by comparing with the theoretical results. Furthermore, comparing with the existed methods which also use the Zernike expansion to simulate PS, algorithm presented in this paper is more accurate because the effects of the finite outer scale L0.is included

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24

Naseri, R., and A. Malek. "Numerical Optimal Control for Problems with Random Forced SPDE Constraints." ISRN Applied Mathematics 2014 (February 20, 2014): 1–11. http://dx.doi.org/10.1155/2014/974305.

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A numerical algorithm for solving optimization problems with stochastic diffusion equation as a constraint is proposed. First, separation of random and deterministic variables is done via Karhunen-Loeve expansion. Then, the problem is discretized, in spatial part, using the finite element method and the polynomial chaos expansion in the stochastic part of the problem. This process leads to the optimal control problem with a large scale system in its constraint. To overcome these difficulties the adjoint technique for derivative computation to implementation of the optimal control issue in preconditioned Newton’s conjugate gradient method is used. By some numerical simulation, it is shown that this hybrid approach is efficient and simple to implement.

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25

Savoji, M. H., and R. E. Burge. "On different methods based on the Karhunen-Loeve expansion and used in image analysis." Computer Vision, Graphics, and Image Processing 29, no. 2 (February 1985): 259–69. http://dx.doi.org/10.1016/0734-189x(85)90121-5.

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26

Yip, K. W., and T. S. Ng. "Karhunen-Loeve expansion of the WSSUS channel output and its application to efficient simulation." IEEE Journal on Selected Areas in Communications 15, no. 4 (May 1997): 640–46. http://dx.doi.org/10.1109/49.585774.

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27

Liao, Jun, Da Fu Xu, and Bing Yan Jiang. "A Method for Double Random Vibration Analysis." Applied Mechanics and Materials 577 (July 2014): 119–24. http://dx.doi.org/10.4028/www.scientific.net/amm.577.119.

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A numerical procedure to compute the mean and covariance matrix of the random response of stochastic structures modeled by FE models is presented. With the help of Gegenbauer polynomial approximation, the calculation of dynamic response of random parameter system is transformed into an equivalent certainty expansion order system's response calculation. Non-stationary, non-white, non-zero means, Gaussian distributed excitation is represented by the well-known Karhunen-Loeve (K-L) expansion. The Precise Integration Method is employed to obtain the K-L decomposition of the non-stationary filtered white noise random excitation. A very accurate result is obtained by a small amount of K-L vectors with the vector characteristic of energy concentration, especially for the small band-width excitation. Correctness of the method is verified by the simulations.

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28

Khonina, S. N., S. G. Volotovskiy, and M. S. Kirilenko. "A method of generating a random optical field using the Karhunen-Loeve expansion to simulate atmospheric turbulence." Computer Optics 44, no. 1 (February 2020): 53–59. http://dx.doi.org/10.18287/2412-6179-co-680.

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It is proposed to use the random field generation in the numerical simulation of the propagation of radiation through a random medium using method based on the Karhunen–Loeve expansion with various types of correlation operators to describe turbulence simulators. The properties of the calculated simulators of a random medium with a Gaussian correlation function were investigated in modeling the propagation of Laguerre-Gaussian vortex beams. The simulation results showed that an increase in the order of the optical vortex leads, as in the experiment, to lower stability of the phase singularity of the beams to random optical fluctuations. The similarity of the simulation results and the optical experiments indicates the promise of the proposed approach for the synthesis of random environment simulators.

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29

Hussein, A., and M. M. Selim. "A general analytical solution for the stochastic Milne problem using Karhunen–Loeve (K–L) expansion." Journal of Quantitative Spectroscopy and Radiative Transfer 125 (August 2013): 84–92. http://dx.doi.org/10.1016/j.jqsrt.2013.03.018.

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30

Uenohara, M., and T. Kanade. "Optimal approximation of uniformly rotated images: relationship between Karhunen-Loeve expansion and discrete cosine transform." IEEE Transactions on Image Processing 7, no. 1 (1998): 116–19. http://dx.doi.org/10.1109/83.650856.

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31

Chang, Haibin, and Dongxiao Zhang. "History matching of statistically anisotropic fields using the Karhunen-Loeve expansion-based global parameterization technique." Computational Geosciences 18, no. 2 (March 1, 2014): 265–82. http://dx.doi.org/10.1007/s10596-014-9409-z.

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32

Ma, Shaojuan, and Qianling Zhang. "Hopf Bifurcation of Compound Stochastic van der Pol System." Mathematical Problems in Engineering 2016 (2016): 1–7. http://dx.doi.org/10.1155/2016/4246916.

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Hopf bifurcation analysis for compound stochastic van der Pol system with a bound random parameter and Gaussian white noise is investigated in this paper. By the Karhunen-Loeve (K-L) expansion and the orthogonal polynomial approximation, the equivalent deterministic van der Pol system can be deduced. Based on the bifurcation theory of nonlinear deterministic system, the critical value of bifurcation parameter is obtained and the influence of random strengthδand noise intensityσon stochastic Hopf bifurcation in compound stochastic system is discussed. At last we found that increasedδcan relocate the critical value of bifurcation parameter forward while increasedσmakes it backward and the influence ofδis more sensitive thanσ. The results are verified by numerical simulations.

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33

Ghanem, Roger, and Bernard Hayek. "Probabilistic Modeling of Flow Over Rough Terrain." Journal of Fluids Engineering 124, no. 1 (November 12, 2001): 42–50. http://dx.doi.org/10.1115/1.1445138.

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This paper presents a method for the propagation of uncertainty, modeled in a probabilistic framework, through a model-based simulation of rainflow on a rough terrain. The adopted model involves a system of conservation equations with associated nonlinear state equations. The topography, surface runoff coefficient, and precipitation data are all modeled as spatially varying random processes. The Karhunen-Loeve expansion is used to represent these processes in terms of a denumerable set of random variables. The predicted state variables in the model are identified with their coordinates with respect to the basis formed by the Polynomial Chaos random variables. A system of linear algebraic deterministic equations are derived for estimating these coordinates.

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34

Watabe, Shinji, Hiroshi Hayashi, Yoshiji Yamada, Kyoichi Miyaji, Seitaro Yabe, Iwao Sotobata, Akira Iwata, and Nobuo Suzumura. "Application of the Karhunen-Loeve expansion to evaluate regional cardiac excitation in body surface potential maps." Journal of Electrocardiology 23, no. 1 (January 1990): 33–40. http://dx.doi.org/10.1016/0022-0736(90)90148-u.

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35

Khorshidi, Hossein, Nasser Talebbeydokhti, and Gholamreza Rakhshandehroo. "High-Order Perturbation Approach for Wave Transformation by Applying Advection-Diffusion Equation via Karhunen–Loeve Expansion." ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering 3, no. 1 (March 2017): 04016011. http://dx.doi.org/10.1061/ajrua6.0000891.

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36

Dong, D., P. Fang, Y. Bock, F. Webb, L. Prawirodirdjo, S. Kedar, and P. Jamason. "Spatiotemporal filtering using principal component analysis and Karhunen-Loeve expansion approaches for regional GPS network analysis." Journal of Geophysical Research: Solid Earth 111, B3 (March 2006): n/a. http://dx.doi.org/10.1029/2005jb003806.

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37

Stadnyk, Maria, Mykhailo Fryz, and Leonid Scherbak. "The feature extraction and estimation of a steady-state visual evoked potential by the Karhunen-Loeve expansion." Eastern-European Journal of Enterprise Technologies 1, no. 4 (85) (February 13, 2017): 56–62. http://dx.doi.org/10.15587/1729-4061.2017.91861.

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38

Yazdanpanah, M., L. Allard, L.-G. Durand, and R. Guardo. "Evaluation of Karhunen-Loeve expansion for feature selection in computer-assisted classification of bioprosthetic heart-valve status." Medical & Biological Engineering & Computing 37, no. 4 (July 1999): 504–10. http://dx.doi.org/10.1007/bf02513337.

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39

Yang, Jinzhong, Dongxiao Zhang, and Zhiming Lu. "Stochastic analysis of saturated–unsaturated flow in heterogeneous media by combining Karhunen-Loeve expansion and perturbation method." Journal of Hydrology 294, no. 1-3 (July 2004): 18–38. http://dx.doi.org/10.1016/j.jhydrol.2003.10.023.

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40

KREUZER, W., H. WAUBKE, G. RIECKH, and P. BALAZS. "A 3D MODEL TO SIMULATE VIBRATIONS IN A LAYERED MEDIUM WITH STOCHASTIC MATERIAL PARAMETERS." Journal of Computational Acoustics 19, no. 02 (June 2011): 139–54. http://dx.doi.org/10.1142/s0218396x11004419.

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A major problem for the simulation of the propagation of vibrations in ground layers is the fact that it is almost impossible to determine the material parameters needed for a numerical model exactly. In this work, we present a 3D model for layered soil, where in each layer the shear modulus is modeled as a stochastic process. Using the Karhunen Loeve expansion, the polynomial chaos expansion, and the Fourier transform, the stochastic system can be transformed into a linear system of equations in the wavenumber frequency domain. Unfortunately, the size of this system becomes very large and — contrary to a deterministic system — the stochastic system can no longer be decoupled for every wavenumber in the spatial Fourier domain. To solve this system efficiently, we propose an iteration procedure where the system is split into a deterministic and a stochastic part. As an external load on top of the ground, we use a vibrating box load moving along the x-axis. We discuss implementational details and present simulation results.

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41

Xiao, M. Y., and H. P. Hong. "Unconditional and conditional simulation of nonstationary and non-Gaussian vector and field with prescribed marginal and correlation by using iteratively matched correlation." Disaster Prevention and Resilience 1, no. 1 (2022): 5. http://dx.doi.org/10.20517/dpr.2022.01.

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In many probabilistic analysis problems, the homogeneous/nonhomogeneous non-Gaussian field is represented as a mapped Gaussian field based on the Nataf translation system. We propose a new sample-based iterative procedure to estimate the underlying Gaussian correlation for homogeneous/nonhomogeneous non-Gaussian vector or field. The numerical procedure takes advantage that the range of feasible correlation coefficients for non-Gaussian random variables is bounded if the translation system is adopted. The estimated underlying Gaussian correlation is then employed for unconditional as well as conditional simulation of the non-Gaussian vector or field according to the theory of the translation process. We then present the steps for augmenting the simulated non-Gaussian field through the Karhunen-Loeve expansion for a refined discretized grid of the field. In addition, the steps to extend the procedure described in the previous section to the multi-dimensional field are highlighted. The application of the proposed algorithms is presented through numerical examples.

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42

Feng, Fan, Fanglin Huang, Weibin Wen, Zhe Liu, and Xiang Liu. "Evaluating the Dynamic Response of the Bridge-Vehicle System considering Random Road Roughness Based on the Moment Method." Advances in Civil Engineering 2021 (November 1, 2021): 1–12. http://dx.doi.org/10.1155/2021/9923592.

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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.

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43

BALAZS, PETER, WOLFGANG KREUZER, and HOLGER WAUBKE. "A STOCHASTIC 2D-MODEL FOR CALCULATING VIBRATIONS IN RANDOM LAYERS." Journal of Computational Acoustics 15, no. 03 (September 2007): 271–83. http://dx.doi.org/10.1142/s0218396x07003354.

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Vibrations induced by machinery and traffic have become of increasing concern in the last years, for example, when constructing buildings near railway lines. In this paper we will present a model designed to predict the vibration level in the ground. Since in practice it is nearly impossible to determine exact material parameters for soil layers, we use a model with a stochastic shear modulus G. Under moderate assumptions G can be split with the Karhunen–Loeve expansion into a mean value G0 and a stochastic part G stoch . Using a combination method of finite elements, Fourier transformation and Polynomial Chaos, it is possible to transform the partial differential equation describing the system into a matrix-vector formulation Kx = b which can be split into a deterministic and a stochastic part (K0 + Ks) x = b analog to the shear modulus. To keep the dimensions of the matrices involved with this system small, we use a Neumann-like iteration to solve it. Finally, results for a small example are presented.

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44

Wu, Yuching, and Jianzhuang Xiao. "The Multiscale Spectral Stochastic Finite Element Method for Chloride Diffusion in Recycled Aggregate Concrete." International Journal of Computational Methods 15, no. 01 (September 27, 2017): 1750078. http://dx.doi.org/10.1142/s0219876217500785.

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In this study, the multiscale stochastic finite element method (MsSFEM) was developed based on a novel digital image kernel to make analysis for chloride diffusion in recycled aggregate concrete (RAC). It is significant to study the chloride diffusivity in RAC, because when RAC was applied in coastal areas, chloride-induced rebar corrosion became a common problem for concrete infrastructures. The MsSFEM was an efficient tool to examine the effect of microscopic randomness of RAC on the chloride diffusivity. Based on the proposed digital image kernel, the Karhunen–Loeve expansion and the polynomial chaos were used in the stochastic homogenization process. To investigate advantages and disadvantages of both generation and application of the proposed digital image kernel, it was compared with many other kernels. The comparisons were made between the method to develop the digital image kernel, which is called the pixel-matrix method, and other methods, and between the application of the kernel and various other kernels. It was shown that the proposed digital image kernel is superior to other kernels in many aspects.

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45

Barber, Jared, Roxana Tanase, and Ivan Yotov. "Kalman filter parameter estimation for a nonlinear diffusion model of epithelial cell migration using stochastic collocation and the Karhunen–Loeve expansion." Mathematical Biosciences 276 (June 2016): 133–44. http://dx.doi.org/10.1016/j.mbs.2016.03.018.

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Wu, Yuching, and Jianzhuang Xiao. "Implementation of the Multiscale Stochastic Finite Element Method on Elliptic PDE Problems." International Journal of Computational Methods 14, no. 01 (January 11, 2017): 1750003. http://dx.doi.org/10.1142/s0219876217500037.

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In this study, a multi-scale finite element method was proposed to solve two linear scale-coupling stochastic elliptic PDE problems, a tightly stretched wire and flow through porous media. At microscopic level, the main idea was to form coarse-scale equations with a prescribed analytic form that may differ from the underlying fine-scale equations. The relevant stochastic homogenization theory was proposed to model the effective global material coefficient matrix. At the macroscopic level, the Karhunen–Loeve decomposition was coupled with a Polynomial Chaos expansion in conjunction with a Galerkin projection to achieve an efficient implementation of the randomness into the solution procedure. Various stochastic methods were used to plug the microscopic cell to the global system. Strategy and relevant algorithms were developed to boost computational efficiency and to break the curse of dimension. The results of numerical examples were shown consistent with ones from literature. It indicates that the proposed numerical method can act as a paradigm for general stochastic partial differential equations involving multi-scale stochastic data. After some modification, the proposed numerical method could be extended to diverse scientific disciplines such as geophysics, material science, biological systems, chemical physics, oceanography, and astrophysics, etc.

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Schuster, Andreas, Richard Degenhardt, Christian Willberg, and Tobias Wille. "Influence of Spatially Distributed Out-of-Plane CFRP Fiber Waviness on the Estimation of Knock-Down Factors Based on Stochastic Numerical Analysis." Journal of Composites Science 6, no. 12 (November 22, 2022): 353. http://dx.doi.org/10.3390/jcs6120353.

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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.

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Agrawal, O. P. "Application of Wavelets in Modeling Stochastic Dynamic Systems." Journal of Vibration and Acoustics 120, no. 3 (July 1, 1998): 763–69. http://dx.doi.org/10.1115/1.2893895.

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This paper presents a wavelet based model for stochastic dynamic systems. In this model, the state variables and their variations are approximated using truncated linear sums of orthogonal polynomials, and a modified Hamilton’s law of varying action is used to reduce the integral equations representing dynamics of the system to a set of algebraic equations. For deterministic systems, the coefficients of the polynomials are constant, but for stochastic systems, the coefficients are random variables. The external forcing functions are treated as stationary Gaussian processes with specified mean and correlation functions. Using Karhunen-Loeve (K-L) expansion, the random input processes are represented in terms of linear sums of finite number of orthonormal eigenfunctions with uncorrelated random coefficients. A wavelet based technique is used to solve the integral eigenvalue problem. Application of wavelets and K-L expansion reduces the infinite dimensional input force vector to one with finite dimensions. Orthogonal properties of the polynomials and the wavelets are utilized to make the algebraic equations sparse and computationally efficient. A method to compute the mean and the variance functions for the state processes is developed. A single degree of freedom spring-mass-damper system subjected to a random forcing function is considered to show the feasibility and effectiveness of the formulation. Studies show that the results of this formulation agree well with those obtained using other schemes.

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AbdelFattah, Hesham, Amnah Al-Johani, and Mohamed El-Beltagy. "Analysis of the Stochastic Quarter-Five Spot Problem Using Polynomial Chaos." Molecules 25, no. 15 (July 24, 2020): 3370. http://dx.doi.org/10.3390/molecules25153370.

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Analysis of fluids in porous media is of great importance in many applications. There are many mathematical models that can be used in the analysis. More realistic models should account for the stochastic variations of the model parameters due to the nature of the porous material and/or the properties of the fluid. In this paper, the standard porous media problem with random permeability is considered. Both the deterministic and stochastic problems are analyzed using the finite volume technique. The solution statistics of the stochastic problem are computed using both Polynomial Chaos Expansion (PCE) and the Karhunen-Loeve (KL) decomposition with an exponential correlation function. The results of both techniques are compared with the Monte Carlo sampling to verify the efficiency. Results have shown that PCE with first order polynomials provides higher accuracy for lower (less than 20%) permeability variance. For higher permeability variance, using higher-order PCE considerably improves the accuracy of the solution. The PCE is also combined with KL decomposition and faster convergence is achieved. The KL-PCE combination should carefully choose the number of KL decomposition terms based on the correlation length of the random permeability. The suggested techniques are successfully applied to the quarter-five spot problem.

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Hamrouni, Adam, Daniel Dias, and Xiangfeng Guo. "Behavior of Shallow Circular Tunnels—Impact of the Soil Spatial Variability." Geosciences 12, no. 2 (February 21, 2022): 97. http://dx.doi.org/10.3390/geosciences12020097.

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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.

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