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curl -X POST \
  'http://laksa036.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://ascelibrary.org/doi/10.1061/%28ASCE%290733-9399%282007%29133%3A7%28773%29\')), getAhrefsUnparsedNoserviceFromURL(\'https://ascelibrary.org/doi/10.1061/%28ASCE%290733-9399%282007%29133%3A7%28773%29\'))) FORMAT JSONEachRow'

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{"found_url":"https:\/\/ascelibrary.org\/doi\/10.1061\/%28ASCE%290733-9399%282007%29133%3A7%28773%29","crawl_time":1768449205,"first_indexed_time":1757426743,"http_code":200,"src_unparsed":"org,ascelibrary!\/doi\/10.1061\/%28ASCE%290733-9399%282007%29133%3A7%28773%29 s443","src_root_hash":"11928281331502450436","history_drop_reason":null,"meta_title":"Karhunen–Loéve Expansion of Stochastic Processes with a Modified Exponential Covariance Kernel | Journal of Engineering Mechanics | Vol 133, No 7","meta_descriptions":["The spectral representation of stationary stochastic processes via the Karhunen-Loéve (KL) expansion is examined from a numerical efficiency perspective. Attention is focused on processes which hav..."],"attrs_boilerpipe_text":"Authors\n:\nPol D.\nSpanos\n,\nF.ASCE\n,\nMichael\nBeer\n,\nM.ASCE\n, and\nJohn\nRed-Horse\nAuthor Affiliations\nPublication\n:\nJournal of Engineering Mechanics\nVolume\n133\n,\nIssue\n7\nAbstract\nThe spectral representation of stationary stochastic processes via the Karhunen-Loéve (KL) expansion is examined from a numerical efficiency perspective. Attention is focused on processes which have commonly been characterized by covariance kernels decaying exponentially versus the position\/time delay variable. By introducing a slight modification in the mathematical description of this covariance kernel, the nondifferentiability at its peak is eliminated, whereas most of its advantageous properties are retained. It is shown that compared to the common exponential model, the requisite number of terms for representing the process in context with the modified kernel is significantly smaller. The effect is demonstrated by means of a specific numerical example. This is done by first determining the eigenfunctions\/eigenvalues associated with the KL expansion for the modified kernel model, and by afterwards estimating the approximation errors corresponding to the two kernels considered for specific numerical values. Clearly, the enhanced computational efficiency of the KL expansion associated with the modified kernel can significantly expedite its incorporation in stochastic finite elements and other areas of stochastic mechanics.\nGet full access to this article\nView all available purchase options and get full access to this article.\nAcknowledgments\nThe writers gratefully acknowledge the partial support of this work from the Alexander von Humboldt (AvH) Foundation in Germany, the German Research Foundation (DFG) within the framework of the Collaborative Research Center SFB 528, and from the Sandia National Laboratories, DOE. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract No.\nDOE\nDE-AC04-94AL85000\n.\nReferences\nDeb, M. K., Babuska, I. M., and Oden, J. T. (2001). “Solution of stochastic partial differential equations using Galerkin finite element techniques.”\nComput. Methods Appl. Mech. Eng.\n, 190(48), 6359–6372.\nDu, S., Ellingwood, B. R., and Cox, J. V. (2005). “Initialization strategies in simulation-based SFE eigenvalue analysis.”\nComput. Aided Civ. Infrastruct. Eng.\n, 20(5), 304–315.\nField, R. V., Jr., and Grigoriu, M. (2004). “On the accuracy of the polynomial chaos approximation.”\nProbab. Eng. Mech.\n, 19(1–2), 65–80.\nGhanem, R. G., and Spanos, P. D. (2003).\nStochastic finite elements: A spectral approach\n, Revised Ed., Dover, Mineola, N.Y.\nGutierrez, E., and Zaldivar, J. M. (2000). “The application of Karhunen-Loéve, or principle component analysis method, to study the nonlinear seismic response of structures.”\nEarthquake Eng. Struct. Dyn.\n, 29(9), 1261–1286.\nIwan, W. D., and Jensen, H. (1993). “On the dynamic-response of continuous systems including model uncertainty.”\nJ. Appl. Mech.\n, 60(2), 484–490.\nJensen, H., and Iwan, W. D. (1991). “Response variability in structural dynamics.”\nEarthquake Eng. Struct. Dyn.\n, 20(10), 949–959.\nKanwal, R. P. (1971).\nLinear integral equations\n, Academic, New York.\nLoéve, M. (1977).\nProbability theory\n, 4th Ed., Springer, Berlin.\nMikhlin, S. G. (1957).\nIntegral equations\n, Pergamon, Oxford.\nNashed, M. Z., and Wahba, G. (1974). “Convergence rates of approximate least squares solution of linear integral and operator equations of the first kind.”\nMath. Comput.\n, 28(125), 69–80.\nPhoon, K. K., Huang, S. P., and Queck, S. T. (2002). “Implementation of Karhunen-Loéve expansion for simulation using a wavelet–Galerkin scheme.”\nProbab. Eng. Mech.\n, 17(3), 293–303.\nSchuëller, G. I. (2001). “Computational stochastic mechanics—Recent advances.”\nComput. Struct.\n, 79(22–25), 2225–2234.\nSchuëller, G. I., Pradlwarter, H. J., and Schenk, C. A. (2003). “Nonstationary response of large linear FE models under stochastic loading.”\nComput. Struct.\n, 81(8–11), 937–947.\nSpanos, P. D., and Ghanem, R. (1989). “Stochastic finite element expansion for random media.”\nJ. Eng. Mech.\n, 115(5), 1035–1053.\nVanmarcke, E. H. (1983).\nRandom fields\n, MIT Press, Cambridge, Mass.\nVan Trees, H. L. (1968).\nDetection, estimation, and modulation theory, Part I\n, Wiley, New York.\nZhang, J., and Ellingwood, B. (1996). “SFEM for reliability of structures with material nonlinearities.”\nJ. Struct. Eng.\n, 122(6), 701–704.","attrs_markdown":"[![Supporting UN Sustainable Development Goals](https:\/\/ascelibrary.org\/sda\/102018\/SDG_728x90.jpg)](https:\/\/ascelibrary.org\/action\/clickThrough?id=102018&url=https%3A%2F%2Fascelibrary.org%2Fsdg&loc=%2Fdoi%2F10.1061%2F%2528ASCE%25290733-9399%25282007%2529133%253A7%2528773%2529&pubId=41431307&placeholderId=1192&productId=1019 \"Supporting UN Sustainable Development Goals\")\n\nAdvertisement\n\n[Skip to main content](https:\/\/ascelibrary.org\/doi\/10.1061\/%28ASCE%290733-9399%282007%29133%3A7%28773%29#afterNav-zy2)\n\n[![ASCE Library logo, redirects to the Home Page](https:\/\/ascelibrary.org\/pb-assets\/images\/Logos\/primary-logo-1687376563187.svg)](https:\/\/ascelibrary.org\/ \"ASCE Library logo, redirects to the Home Page\")\n\n[Advanced search](https:\/\/ascelibrary.org\/search\/advanced)[Citation search](https:\/\/ascelibrary.org\/search\/advanced#citation)\n\n- [Shopping cart with 0 item](https:\/\/ascelibrary.org\/action\/showCart?FlowID=1 \"Cart\")\n- [Login \/ Register](https:\/\/ascelibrary.org\/action\/showLogin?uri=%2Fdoi%2F10.1061%2F%2528ASCE%25290733-9399%25282007%2529133%253A7%2528773%2529)\n  \n  - 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Spanos, F.ASCE](https:\/\/ascelibrary.org\/doi\/10.1061\/%28ASCE%290733-9399%282007%29133%3A7%28773%29#con1), [Michael Beer, M.ASCE](https:\/\/ascelibrary.org\/doi\/10.1061\/%28ASCE%290733-9399%282007%29133%3A7%28773%29#con2), and [John Red-Horse](https:\/\/ascelibrary.org\/doi\/10.1061\/%28ASCE%290733-9399%282007%29133%3A7%28773%29#con3)[Author Affiliations](https:\/\/ascelibrary.org\/doi\/10.1061\/%28ASCE%290733-9399%282007%29133%3A7%28773%29#tab-contributors)\n\nPublication: Journal of Engineering Mechanics\n\nVolume 133, Issue 7\n\n<https:\/\/doi.org\/10.1061\/(ASCE)0733-9399(2007)133:7(773)>\n\n[Get Access](https:\/\/ascelibrary.org\/doi\/10.1061\/%28ASCE%290733-9399%282007%29133%3A7%28773%29#core-collateral-purchase-access \"Go to purchase options\")\n\nContents\n\nJournal of Engineering Mechanics\n\n[Volume 133, Issue 7 July 2007](https:\/\/ascelibrary.org\/toc\/jenmdt\/133\/7)\n\n[PREVIOUS ARTICLE Simulation of a 3D Numerical Viscous Wave TankPrevious](https:\/\/ascelibrary.org\/doi\/10.1061\/%28ASCE%290733-9399%282007%29133%3A7%28761%29)\n\n[NEXT ARTICLE Model Selection in Applied Science and Engineering: A Decision-Theoretic ApproachNext](https:\/\/ascelibrary.org\/doi\/10.1061\/%28ASCE%290733-9399%282007%29133%3A7%28780%29)\n\n- [Abstract](https:\/\/ascelibrary.org\/doi\/10.1061\/%28ASCE%290733-9399%282007%29133%3A7%28773%29#abstract)\n- [Acknowledgments](https:\/\/ascelibrary.org\/doi\/10.1061\/%28ASCE%290733-9399%282007%29133%3A7%28773%29#acknowledgments)\n- [References](https:\/\/ascelibrary.org\/doi\/10.1061\/%28ASCE%290733-9399%282007%29133%3A7%28773%29#bibliography)\n\n- [Information & Authors](https:\/\/ascelibrary.org\/doi\/10.1061\/%28ASCE%290733-9399%282007%29133%3A7%28773%29#core-collateral-info \"Information & Authors\")\n- [Metrics & Citations](https:\/\/ascelibrary.org\/doi\/10.1061\/%28ASCE%290733-9399%282007%29133%3A7%28773%29#core-collateral-metrics \"Metrics & Citations\")\n- [Get Access](https:\/\/ascelibrary.org\/doi\/10.1061\/%28ASCE%290733-9399%282007%29133%3A7%28773%29#core-collateral-purchase-access \"Get Access\")\n- [References](https:\/\/ascelibrary.org\/doi\/10.1061\/%28ASCE%290733-9399%282007%29133%3A7%28773%29#core-collateral-references \"References\")\n- [Figures](https:\/\/ascelibrary.org\/doi\/10.1061\/%28ASCE%290733-9399%282007%29133%3A7%28773%29#core-collateral-figures \"Figures\")\n- [Tables](https:\/\/ascelibrary.org\/doi\/10.1061\/%28ASCE%290733-9399%282007%29133%3A7%28773%29#core-collateral-tables \"Tables\")\n- [Media](https:\/\/ascelibrary.org\/doi\/10.1061\/%28ASCE%290733-9399%282007%29133%3A7%28773%29#core-collateral-media \"Media\")\n- [Share](https:\/\/ascelibrary.org\/doi\/10.1061\/%28ASCE%290733-9399%282007%29133%3A7%28773%29#core-collateral-share \"Share\")\n\n## Abstract\nThe spectral representation of stationary stochastic processes via the Karhunen-Loéve (KL) expansion is examined from a numerical efficiency perspective. Attention is focused on processes which have commonly been characterized by covariance kernels decaying exponentially versus the position\/time delay variable. By introducing a slight modification in the mathematical description of this covariance kernel, the nondifferentiability at its peak is eliminated, whereas most of its advantageous properties are retained. It is shown that compared to the common exponential model, the requisite number of terms for representing the process in context with the modified kernel is significantly smaller. The effect is demonstrated by means of a specific numerical example. This is done by first determining the eigenfunctions\/eigenvalues associated with the KL expansion for the modified kernel model, and by afterwards estimating the approximation errors corresponding to the two kernels considered for specific numerical values. Clearly, the enhanced computational efficiency of the KL expansion associated with the modified kernel can significantly expedite its incorporation in stochastic finite elements and other areas of stochastic mechanics.\n\n## Get full access to this article\nView all available purchase options and get full access to this article.\n\n[Get Access](https:\/\/ascelibrary.org\/doi\/10.1061\/%28ASCE%290733-9399%282007%29133%3A7%28773%29#core-collateral-purchase-access)\n\nAlready a Subscriber? Sign in as an [individual](https:\/\/ascelibrary.org\/action\/showLogin?redirectUri=\/doi\/10.1061\/\$ASCE\$0733-9399\$2007\$133:7\$773\$) or via your [institution](https:\/\/ascelibrary.org\/action\/ssostart?uri=\/doi\/10.1061\/\$ASCE\$0733-9399\$2007\$133:7\$773\$)\n\n## Acknowledgments\nThe writers gratefully acknowledge the partial support of this work from the Alexander von Humboldt (AvH) Foundation in Germany, the German Research Foundation (DFG) within the framework of the Collaborative Research Center SFB 528, and from the Sandia National Laboratories, DOE. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract No. DOEDE-AC04-94AL85000.\n\n## References\nDeb, M. K., Babuska, I. M., and Oden, J. T. (2001). “Solution of stochastic partial differential equations using Galerkin finite element techniques.” *Comput. Methods Appl. Mech. Eng.*, 190(48), 6359–6372.\n\n[Crossref](https:\/\/doi.org\/10.1016\/S0045-7825\$01\$00237-7)\n\n[Google Scholar](https:\/\/scholar.google.com\/scholar_lookup?title=Solution+of+stochastic+partial+differential+equations+using+Galerkin+finite+element+techniques&author=M.+K.+Deb&author=I.+M.+Babuska&author=J.+T.+Oden&publication_year=2001&journal=Comput.+Methods+Appl.+Mech.+Eng.&pages=6359-6372&doi=10.1016%2FS0045-7825%2801%2900237-7)\n\nDu, S., Ellingwood, B. R., and Cox, J. V. (2005). “Initialization strategies in simulation-based SFE eigenvalue analysis.” *Comput. Aided Civ. Infrastruct. Eng.*, 20(5), 304–315.\n\n[Crossref](https:\/\/doi.org\/10.1111\/j.1467-8667.2005.00398.x)\n\n[Google Scholar](https:\/\/scholar.google.com\/scholar_lookup?title=Initialization+strategies+in+simulation-based+SFE+eigenvalue+analysis&author=S.+Du&author=B.+R.+Ellingwood&author=J.+V.+Cox&publication_year=2005&journal=Comput.+Aided+Civ.+Infrastruct.+Eng.&pages=304-315&doi=10.1111%2Fj.1467-8667.2005.00398.x)\n\nField, R. V., Jr., and Grigoriu, M. (2004). “On the accuracy of the polynomial chaos approximation.” *Probab. Eng. Mech.*, 19(1–2), 65–80.\n\n[Crossref](https:\/\/doi.org\/10.1016\/j.probengmech.2003.11.017)\n\n[Google Scholar](https:\/\/scholar.google.com\/scholar_lookup?title=On+the+accuracy+of+the+polynomial+chaos+approximation&author=R.+V.+Field&author=M.+Grigoriu&publication_year=2004&journal=Probab.+Eng.+Mech.&pages=65-80&doi=10.1016%2Fj.probengmech.2003.11.017)\n\nGhanem, R. G., and Spanos, P. D. 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Dyn.*, 29(9), 1261–1286.\n\n[Crossref](https:\/\/doi.org\/10.1002\/1096-9845\$200009\$29:9%3C1261::AID-EQE964%3E3.0.CO;2-N)\n\n[Google Scholar](https:\/\/scholar.google.com\/scholar_lookup?title=The+application+of+Karhunen-Lo%C3%A9ve%2C+or+principle+component+analysis+method%2C+to+study+the+nonlinear+seismic+response+of+structures&author=E.+Gutierrez&author=J.+M.+Zaldivar&publication_year=2000&journal=Earthquake+Eng.+Struct.+Dyn.&pages=1261-1286&doi=10.1002%2F1096-9845%28200009%2929%3A9%253C1261%3A%3AAID-EQE964%253E3.0.CO%3B2-N)\n\nIwan, W. D., and Jensen, H. (1993). “On the dynamic-response of continuous systems including model uncertainty.” *J. Appl. 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Meta Title	Karhunen–Loéve Expansion of Stochastic Processes with a Modified Exponential Covariance Kernel \| Journal of Engineering Mechanics \| Vol 133, No 7
Meta Description	The spectral representation of stationary stochastic processes via the Karhunen-Loéve (KL) expansion is examined from a numerical efficiency perspective. Attention is focused on processes which hav...
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Boilerpipe Text	Authors : Pol D. Spanos , F.ASCE , Michael Beer , M.ASCE , and John Red-Horse Author Affiliations Publication : Journal of Engineering Mechanics Volume 133 , Issue 7 Abstract The spectral representation of stationary stochastic processes via the Karhunen-Loéve (KL) expansion is examined from a numerical efficiency perspective. Attention is focused on processes which have commonly been characterized by covariance kernels decaying exponentially versus the position/time delay variable. By introducing a slight modification in the mathematical description of this covariance kernel, the nondifferentiability at its peak is eliminated, whereas most of its advantageous properties are retained. It is shown that compared to the common exponential model, the requisite number of terms for representing the process in context with the modified kernel is significantly smaller. The effect is demonstrated by means of a specific numerical example. This is done by first determining the eigenfunctions/eigenvalues associated with the KL expansion for the modified kernel model, and by afterwards estimating the approximation errors corresponding to the two kernels considered for specific numerical values. Clearly, the enhanced computational efficiency of the KL expansion associated with the modified kernel can significantly expedite its incorporation in stochastic finite elements and other areas of stochastic mechanics. Get full access to this article View all available purchase options and get full access to this article. Acknowledgments The writers gratefully acknowledge the partial support of this work from the Alexander von Humboldt (AvH) Foundation in Germany, the German Research Foundation (DFG) within the framework of the Collaborative Research Center SFB 528, and from the Sandia National Laboratories, DOE. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract No. DOE DE-AC04-94AL85000 . References Deb, M. K., Babuska, I. M., and Oden, J. T. (2001). “Solution of stochastic partial differential equations using Galerkin finite element techniques.” Comput. Methods Appl. Mech. Eng. , 190(48), 6359–6372. Du, S., Ellingwood, B. R., and Cox, J. V. (2005). “Initialization strategies in simulation-based SFE eigenvalue analysis.” Comput. Aided Civ. Infrastruct. Eng. , 20(5), 304–315. Field, R. V., Jr., and Grigoriu, M. (2004). “On the accuracy of the polynomial chaos approximation.” Probab. Eng. Mech. , 19(1–2), 65–80. Ghanem, R. G., and Spanos, P. D. (2003). Stochastic finite elements: A spectral approach , Revised Ed., Dover, Mineola, N.Y. Gutierrez, E., and Zaldivar, J. M. (2000). “The application of Karhunen-Loéve, or principle component analysis method, to study the nonlinear seismic response of structures.” Earthquake Eng. Struct. Dyn. , 29(9), 1261–1286. Iwan, W. D., and Jensen, H. (1993). “On the dynamic-response of continuous systems including model uncertainty.” J. Appl. Mech. , 60(2), 484–490. Jensen, H., and Iwan, W. D. (1991). “Response variability in structural dynamics.” Earthquake Eng. Struct. Dyn. , 20(10), 949–959. Kanwal, R. P. (1971). Linear integral equations , Academic, New York. Loéve, M. (1977). Probability theory , 4th Ed., Springer, Berlin. Mikhlin, S. G. (1957). Integral equations , Pergamon, Oxford. Nashed, M. Z., and Wahba, G. (1974). “Convergence rates of approximate least squares solution of linear integral and operator equations of the first kind.” Math. Comput. , 28(125), 69–80. Phoon, K. K., Huang, S. P., and Queck, S. T. (2002). “Implementation of Karhunen-Loéve expansion for simulation using a wavelet–Galerkin scheme.” Probab. Eng. Mech. , 17(3), 293–303. Schuëller, G. I. (2001). “Computational stochastic mechanics—Recent advances.” Comput. Struct. , 79(22–25), 2225–2234. Schuëller, G. I., Pradlwarter, H. J., and Schenk, C. A. (2003). “Nonstationary response of large linear FE models under stochastic loading.” Comput. Struct. , 81(8–11), 937–947. Spanos, P. D., and Ghanem, R. (1989). “Stochastic finite element expansion for random media.” J. Eng. Mech. , 115(5), 1035–1053. Vanmarcke, E. H. (1983). Random fields , MIT Press, Cambridge, Mass. Van Trees, H. L. (1968). Detection, estimation, and modulation theory, Part I , Wiley, New York. Zhang, J., and Ellingwood, B. (1996). “SFEM for reliability of structures with material nonlinearities.” J. Struct. Eng. , 122(6), 701–704.
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Spanos, F.ASCE](https://ascelibrary.org/doi/10.1061/%28ASCE%290733-9399%282007%29133%3A7%28773%29#con1), [Michael Beer, M.ASCE](https://ascelibrary.org/doi/10.1061/%28ASCE%290733-9399%282007%29133%3A7%28773%29#con2), and [John Red-Horse](https://ascelibrary.org/doi/10.1061/%28ASCE%290733-9399%282007%29133%3A7%28773%29#con3)[Author Affiliations](https://ascelibrary.org/doi/10.1061/%28ASCE%290733-9399%282007%29133%3A7%28773%29#tab-contributors) Publication: Journal of Engineering Mechanics Volume 133, Issue 7 <https://doi.org/10.1061/(ASCE)0733-9399(2007)133:7(773)> [Get Access](https://ascelibrary.org/doi/10.1061/%28ASCE%290733-9399%282007%29133%3A7%28773%29#core-collateral-purchase-access "Go to purchase options") Contents Journal of Engineering Mechanics [Volume 133, Issue 7 July 2007](https://ascelibrary.org/toc/jenmdt/133/7) [PREVIOUS ARTICLE Simulation of a 3D Numerical Viscous Wave TankPrevious](https://ascelibrary.org/doi/10.1061/%28ASCE%290733-9399%282007%29133%3A7%28761%29) [NEXT ARTICLE Model Selection in Applied Science and Engineering: A Decision-Theoretic ApproachNext](https://ascelibrary.org/doi/10.1061/%28ASCE%290733-9399%282007%29133%3A7%28780%29) - [Abstract](https://ascelibrary.org/doi/10.1061/%28ASCE%290733-9399%282007%29133%3A7%28773%29#abstract) - [Acknowledgments](https://ascelibrary.org/doi/10.1061/%28ASCE%290733-9399%282007%29133%3A7%28773%29#acknowledgments) - [References](https://ascelibrary.org/doi/10.1061/%28ASCE%290733-9399%282007%29133%3A7%28773%29#bibliography) - [Information & Authors](https://ascelibrary.org/doi/10.1061/%28ASCE%290733-9399%282007%29133%3A7%28773%29#core-collateral-info "Information & Authors") - [Metrics & Citations](https://ascelibrary.org/doi/10.1061/%28ASCE%290733-9399%282007%29133%3A7%28773%29#core-collateral-metrics "Metrics & Citations") - [Get Access](https://ascelibrary.org/doi/10.1061/%28ASCE%290733-9399%282007%29133%3A7%28773%29#core-collateral-purchase-access "Get Access") - [References](https://ascelibrary.org/doi/10.1061/%28ASCE%290733-9399%282007%29133%3A7%28773%29#core-collateral-references "References") - [Figures](https://ascelibrary.org/doi/10.1061/%28ASCE%290733-9399%282007%29133%3A7%28773%29#core-collateral-figures "Figures") - [Tables](https://ascelibrary.org/doi/10.1061/%28ASCE%290733-9399%282007%29133%3A7%28773%29#core-collateral-tables "Tables") - [Media](https://ascelibrary.org/doi/10.1061/%28ASCE%290733-9399%282007%29133%3A7%28773%29#core-collateral-media "Media") - [Share](https://ascelibrary.org/doi/10.1061/%28ASCE%290733-9399%282007%29133%3A7%28773%29#core-collateral-share "Share") ## Abstract The spectral representation of stationary stochastic processes via the Karhunen-Loéve (KL) expansion is examined from a numerical efficiency perspective. Attention is focused on processes which have commonly been characterized by covariance kernels decaying exponentially versus the position/time delay variable. By introducing a slight modification in the mathematical description of this covariance kernel, the nondifferentiability at its peak is eliminated, whereas most of its advantageous properties are retained. It is shown that compared to the common exponential model, the requisite number of terms for representing the process in context with the modified kernel is significantly smaller. The effect is demonstrated by means of a specific numerical example. This is done by first determining the eigenfunctions/eigenvalues associated with the KL expansion for the modified kernel model, and by afterwards estimating the approximation errors corresponding to the two kernels considered for specific numerical values. Clearly, the enhanced computational efficiency of the KL expansion associated with the modified kernel can significantly expedite its incorporation in stochastic finite elements and other areas of stochastic mechanics. ## Get full access to this article View all available purchase options and get full access to this article. [Get Access](https://ascelibrary.org/doi/10.1061/%28ASCE%290733-9399%282007%29133%3A7%28773%29#core-collateral-purchase-access) Already a Subscriber? 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(1996). “SFEM for reliability of structures with material nonlinearities.” J. Struct. Eng., 122(6), 701–704. [Crossref](https://ascelibrary.org/servlet/linkout?suffix=e_1_3_3_19_1&dbid=4&doi=10.1061%2F%28ASCE%290733-9399%282007%29133%3A7%28773%29&key=10.1061%2F%28ASCE%290733-9445%281996%29122%3A6%28701%29&site=asce-site) [Google Scholar](https://scholar.google.com/scholar_lookup?title=SFEM+for+reliability+of+structures+with+material+nonlinearities&author=J.+Zhang&author=B.+Ellingwood&publication_year=1996&journal=J.+Struct.+Eng.&pages=701-704&doi=10.1061%2F%28ASCE%290733-9445%281996%29122%3A6%28701%29) ## Information & Authors ### Information #### Published In [![Go to Journal of Engineering Mechanics ](https://ascelibrary.org/cms/asset/e2592a1f-db8b-49f4-8e01-de8f5a0f2bc3/default_cover.jpg)](https://ascelibrary.org/toc/jenmdt/133/7 "Go to Journal of Engineering Mechanics ") Journal of Engineering Mechanics Volume 133 • Issue 7 • July 2007 Pages: 773 - 779 #### Copyright © 2007 ASCE. #### History Received: Nov 15, 2005 Accepted: May 23, 2006 Published online: Jul 1, 2007 Published in print: Jul 2007 #### Permissions Request permissions for this article. [Request Permissions](https://marketplace.copyright.com/rs-ui-web/mp/search/all/10.1061%2F%28ASCE%290733-9399%282007%29133%3A7%28773%29) #### ASCE Technical Topics: 1. [Errors (statistics)](https://ascelibrary.org/topic/ascetechnicaltopics/770) 2. [Mathematical models](https://ascelibrary.org/topic/ascetechnicaltopics/1372) 3. [Mathematics](https://ascelibrary.org/topic/ascetechnicaltopics/1373) 4. [Numerical methods](https://ascelibrary.org/topic/ascetechnicaltopics/1493) 5. [Numerical models](https://ascelibrary.org/topic/ascetechnicaltopics/1494) 6. [Stationary processes](https://ascelibrary.org/topic/ascetechnicaltopics/2145) 7. [Stochastic processes](https://ascelibrary.org/topic/ascetechnicaltopics/2166) #### Notes Note. Associate Editor: Joel P. Conte ### Authors #### Affiliations Pol D. Spanos, F.ASCE Lewis B. Ryon Professor, Dept. of Mechanical Engineering and Civil and Environmental Engineering, Rice Univ., MS 321, P.O. Box 1892, Houston, TX 77251. E-mail: [\[email protected\]](https://ascelibrary.org/cdn-cgi/l/email-protection) [View all articles by this author](https://ascelibrary.org/authored-by/Spanos/Pol+D) Michael Beer, M.ASCE Assistant Professor, Dept. of Civil Engineering, National Univ. of Singapore, BLK E1A, 1 Engineering Dr. 2, Singapore 117576. E-mail: [\[email protected\]](https://ascelibrary.org/cdn-cgi/l/email-protection) [View all articles by this author](https://ascelibrary.org/authored-by/Beer/Michael) John Red-Horse Member of the Technical Staff, Sandia National Laboratories, Albuquerque, NM 87185. E-mail: [\[email protected\]](https://ascelibrary.org/cdn-cgi/l/email-protection) [View all articles by this author](https://ascelibrary.org/authored-by/Red-Horse/John) ## Metrics & Citations ### Metrics ### Citations #### Download citation If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. 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