Relevant bibliographies by topics / Karhunen-Loève theorem

Academic literature on the topic 'Karhunen-Loève theorem'

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Published: 4 June 2021
Last updated: 6 February 2022

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Journal articles on the topic "Karhunen-Loève theorem"

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Bay, Xavier, and Jean-Charles Croix. "Karhunen–Loève decomposition of Gaussian measures on Banach spaces." Probability and Mathematical Statistics 39, no. 2 (December 19, 2019): 279–97. http://dx.doi.org/10.19195/0208-4147.39.2.3.

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The study of Gaussian measures on Banach spaces is of active interest both in pure and applied mathematics. In particular, the spectral theorem for self-adjoint compact operators on Hilbert spaces provides a canonical decomposition of Gaussian measures on Hilbert spaces, the socalled Karhunen–Ločve expansion. In this paper, we extend this result to Gaussian measures on Banach spaces in a very similar and constructive manner. In some sense, this can also be seen as a generalization of the spectral theorem for covariance operators associated with Gaussian measures on Banach spaces. In the special case of the standardWiener measure, this decomposition matches with Lévy–Ciesielski construction of Brownian motion.
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Leite, F. E. A., Raúl Montagne, G. Corso, and L. S. Lucena. "Karhunen–Loève spectral analysis in multiresolution decomposition." Computational Geosciences 13, no. 2 (June 27, 2008): 165–70. http://dx.doi.org/10.1007/s10596-008-9091-0.

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Khoromskij, B. N., A. Litvinenko, and H. G. Matthies. "Application of hierarchical matrices for computing the Karhunen–Loève expansion." Computing 84, no. 1-2 (October 31, 2008): 49–67. http://dx.doi.org/10.1007/s00607-008-0018-3.

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Glösmann, Philipp. "Reduction of discrete element models by Karhunen–Loève transform: a hybrid model approach." Computational Mechanics 45, no. 4 (December 19, 2009): 375–85. http://dx.doi.org/10.1007/s00466-009-0456-6.

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Nakamura, Yukihiro, Hidekazu Kaneko, Tohru Kiryu, Shinya S. Suzuki, and Yoshiaki Saitoh. "Influence of motor unit firing patterns on evaluation of muscle activities by Karhunen-Loève expansion." Systems and Computers in Japan 34, no. 12 (September 11, 2003): 45–55. http://dx.doi.org/10.1002/scj.10326.

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Rauter, Natalie. "A computational modeling approach based on random fields for short fiber-reinforced composites with experimental verification by nanoindentation and tensile tests." Computational Mechanics 67, no. 2 (January 18, 2021): 699–722. http://dx.doi.org/10.1007/s00466-020-01958-3.

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AbstractIn this study a modeling approach for short fiber-reinforced composites is presented which allows one to consider information from the microstructure of the compound while modeling on the component level. The proposed technique is based on the determination of correlation functions by the moving window method. Using these correlation functions random fields are generated by the Karhunen–Loève expansion. Linear elastic numerical simulations are conducted on the mesoscale and component level based on the probabilistic characteristics of the microstructure derived from a two-dimensional micrograph. The experimental validation by nanoindentation on the mesoscale shows good conformity with the numerical simulations. For the numerical modeling on the component level the comparison of experimentally obtained Young’s modulus by tensile tests with numerical simulations indicate that the presented approach requires three-dimensional information of the probabilistic characteristics of the microstructure. Using this information not only the overall material properties are approximated sufficiently, but also the local distribution of the material properties shows the same trend as the results of conducted tensile tests.
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Blondeel, Philippe, Pieterjan Robbe, Cédric Van hoorickx, Stijn François, Geert Lombaert, and Stefan Vandewalle. "p-Refined Multilevel Quasi-Monte Carlo for Galerkin Finite Element Methods with Applications in Civil Engineering." Algorithms 13, no. 5 (April 28, 2020): 110. http://dx.doi.org/10.3390/a13050110.

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Civil engineering applications are often characterized by a large uncertainty on the material parameters. Discretization of the underlying equations is typically done by means of the Galerkin Finite Element method. The uncertain material parameter can be expressed as a random field represented by, for example, a Karhunen–Loève expansion. Computation of the stochastic responses, i.e., the expected value and variance of a chosen quantity of interest, remains very costly, even when state-of-the-art Multilevel Monte Carlo (MLMC) is used. A significant cost reduction can be achieved by using a recently developed multilevel method: p-refined Multilevel Quasi-Monte Carlo (p-MLQMC). This method is based on the idea of variance reduction by employing a hierarchical discretization of the problem based on a p-refinement scheme. It is combined with a rank-1 Quasi-Monte Carlo (QMC) lattice rule, which yields faster convergence compared to the use of random Monte Carlo points. In this work, we developed algorithms for the p-MLQMC method for two dimensional problems. The p-MLQMC method is first benchmarked on an academic beam problem. Finally, we use our algorithm for the assessment of the stability of slopes, a problem that arises in geotechnical engineering, and typically suffers from large parameter uncertainty. For both considered problems, we observe a very significant reduction in the amount of computational work with respect to MLMC.
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Jankoski, Radoslav, Ulrich Römer, and Sebastian Schöps. "Modeling of spatial uncertainties in the magnetic reluctivity." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 36, no. 4 (July 3, 2017): 1151–67. http://dx.doi.org/10.1108/compel-10-2016-0438.

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Purpose The purpose of this paper is to present a computationally efficient approach for the stochastic modeling of an inhomogeneous reluctivity of magnetic materials. These materials can be part of electrical machines such as a single-phase transformer (a benchmark example that is considered in this paper). The approach is based on the Karhunen–Loève expansion (KLE). The stochastic model is further used to study the statistics of the self-inductance of the primary coil as a quantity of interest (QoI). Design/methodology/approach The computation of the KLE requires solving a generalized eigenvalue problem with dense matrices. The eigenvalues and the eigenfunction are computed by using the Lanczos method that needs only matrix vector multiplications. The complexity of performing matrix vector multiplications with dense matrices is reduced by using hierarchical matrices. Findings The suggested approach is used to study the impact of the spatial variability in the magnetic reluctivity on the QoI. The statistics of this parameter are influenced by the correlation lengths of the random reluctivity. Both, the mean value and the standard deviation increase as the correlation length of the random reluctivity increases. Originality/value The KLE, computed by using hierarchical matrices, is used for uncertainty quantification of low frequency electrical machines as a computationally efficient approach in terms of memory requirement, as well as computation time.
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Dissertations / Theses on the topic "Karhunen-Loève theorem"

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Giambartolomei, Giordano. "The Karhunen-Loève theorem." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2016. http://amslaurea.unibo.it/10169/.

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La trasformata di Karhunen-Loève monodimensionale è la decomposizione di un processo stocastico del secondo ordine a parametrizzazione continua in coefficienti aleatori scorrelati. Nella presente dissertazione, la trasformata è ottenuta per via analitica, proiettando il processo, considerato in un intervallo di tempo limitato [a,b], su una base deterministica ottenuta dalle autofunzioni dell'operatore di Hilbert-Schmidt di covarianza corrispondenti ad autovalori positivi. Fondamentalmente l'idea del metodo è, dal primo, trovare gli autovalori positivi dell'operatore integrale di Hilbert-Schmidt, che ha in Kernel la funzione di covarianza del processo. Ad ogni tempo dell'intervallo, il processo è proiettato sulla base ortonormale dello span delle autofunzioni dell'operatore di Hilbert-Schmidt che corrispondono ad autovalori positivi. Tale procedura genera coefficienti aleatori che si rivelano variabili aleatorie centrate e scorrelate. L'espansione in serie che risulta dalla trasformata è una combinazione lineare numerabile di coefficienti aleatori di proiezione ed autofunzioni convergente in media quadratica al processo, uniformemente sull'intervallo temporale. Se inoltre il processo è Gaussiano, la convergenza è quasi sicuramente sullo spazio di probabilità (O,F,P). Esistono molte altre espansioni in serie di questo tipo, tuttavia la trasformata di Karhunen-Loève ha la peculiarità di essere ottimale rispetto all'errore totale in media quadratica che consegue al troncamento della serie. Questa caratteristica ha conferito a tale metodo ed alle sue generalizzazioni un notevole successo tra le discipline applicate.
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Ling, Hong. "Implementation of Stochastic Neural Networks for Approximating Random Processes." Master's thesis, Lincoln University. Environment, Society and Design Division, 2007. http://theses.lincoln.ac.nz/public/adt-NZLIU20080108.124352/.

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Artificial Neural Networks (ANNs) can be viewed as a mathematical model to simulate natural and biological systems on the basis of mimicking the information processing methods in the human brain. The capability of current ANNs only focuses on approximating arbitrary deterministic input-output mappings. However, these ANNs do not adequately represent the variability which is observed in the systems’ natural settings as well as capture the complexity of the whole system behaviour. This thesis addresses the development of a new class of neural networks called Stochastic Neural Networks (SNNs) in order to simulate internal stochastic properties of systems. Developing a suitable mathematical model for SNNs is based on canonical representation of stochastic processes or systems by means of Karhunen-Loève Theorem. Some successful real examples, such as analysis of full displacement field of wood in compression, confirm the validity of the proposed neural networks. Furthermore, analysis of internal workings of SNNs provides an in-depth view on the operation of SNNs that help to gain a better understanding of the simulation of stochastic processes by SNNs.
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Conference papers on the topic "Karhunen-Loève theorem"

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Stanescu, Daniela, Lucian Ionel Gaina, Bianca Gusita, and Ioana Ghergulescu. "Message Processing-based Steganographic Algorithm using Karhunen-Loève Transform." In 2019 23rd International Conference on System Theory, Control and Computing (ICSTCC). IEEE, 2019. http://dx.doi.org/10.1109/icstcc.2019.8886015.

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