{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T21:40:23Z","timestamp":1772574023321,"version":"3.50.1"},"reference-count":33,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2024,2,5]],"date-time":"2024-02-05T00:00:00Z","timestamp":1707091200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,2,5]],"date-time":"2024-02-05T00:00:00Z","timestamp":1707091200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100018934","name":"Universit\u00e4t Greifswald","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100018934","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2024,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We investigate the Helmholtz equation with suitable boundary conditions and uncertainties in the wavenumber. Thus the wavenumber is modeled as a random variable or a random field. We discretize the Helmholtz equation using finite differences in space, which leads to a linear system of algebraic equations including random variables. A stochastic Galerkin method yields a deterministic linear system of algebraic equations. This linear system is high-dimensional, sparse and complex symmetric but, in general, not hermitian. We therefore solve this system iteratively with GMRES and propose two preconditioners: a complex shifted Laplace preconditioner and a mean value preconditioner. Both preconditioners reduce the number of iteration steps as well as the computation time in our numerical experiments.\n<\/jats:p>","DOI":"10.1007\/s10915-024-02450-3","type":"journal-article","created":{"date-parts":[[2024,2,5]],"date-time":"2024-02-05T06:02:16Z","timestamp":1707112936000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["The Helmholtz Equation with Uncertainties in the Wavenumber"],"prefix":"10.1007","volume":"98","author":[{"given":"Roland","family":"Pulch","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3107-3053","authenticated-orcid":false,"given":"Olivier","family":"S\u00e8te","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,2,5]]},"reference":[{"issue":"1","key":"2450_CR1","doi-asserted-by":"publisher","first-page":"1196","DOI":"10.1016\/j.jcp.2007.05.013","volume":"226","author":"T Airaksinen","year":"2007","unstructured":"Airaksinen, T., Heikkola, E., Pennanen, A., Toivanen, J.: An algebraic multigrid based shifted-Laplacian preconditioner for the Helmholtz equation. J. Comput. Phys. 226(1), 1196\u20131210 (2007). https:\/\/doi.org\/10.1016\/j.jcp.2007.05.013","journal-title":"J. Comput. Phys."},{"key":"2450_CR2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-4942-3","volume-title":"Inverse Acoustic and Electromagnetic Scattering Theory","author":"D Colton","year":"2013","unstructured":"Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory, 3rd edn. Springer, New York (2013)","edition":"3"},{"issue":"4","key":"2450_CR3","doi-asserted-by":"publisher","first-page":"575","DOI":"10.1002\/nla.1881","volume":"20","author":"S Cools","year":"2013","unstructured":"Cools, S., Vanroose, W.: Local Fourier analysis of the complex shifted Laplacian preconditioner for Helmholtz problems. Numer. Linear Algebra Appl. 20(4), 575\u2013597 (2013). https:\/\/doi.org\/10.1002\/nla.1881","journal-title":"Numer. Linear Algebra Appl."},{"key":"2450_CR4","unstructured":"Davis, T.A.: UMFPACK user guide (version 5.7.7). Tech. rep. (2018)"},{"issue":"1","key":"2450_CR5","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1007\/s11831-007-9013-7","volume":"15","author":"YA Erlangga","year":"2008","unstructured":"Erlangga, Y.A.: Advances in iterative methods and preconditioners for the Helmholtz equation. Arch. Comput. Methods Eng. 15(1), 37\u201366 (2008). https:\/\/doi.org\/10.1007\/s11831-007-9013-7","journal-title":"Arch. Comput. Methods Eng."},{"issue":"3\u20134","key":"2450_CR6","doi-asserted-by":"publisher","first-page":"409","DOI":"10.1016\/j.apnum.2004.01.009","volume":"50","author":"YA Erlangga","year":"2004","unstructured":"Erlangga, Y.A., Vuik, C., Oosterlee, C.W.: On a class of preconditioners for solving the Helmholtz equation. Appl. Numer. Math. 50(3\u20134), 409\u2013425 (2004). https:\/\/doi.org\/10.1016\/j.apnum.2004.01.009","journal-title":"Appl. Numer. Math."},{"issue":"3","key":"2450_CR7","doi-asserted-by":"publisher","first-page":"567","DOI":"10.1007\/s00211-015-0700-2","volume":"131","author":"MJ Gander","year":"2015","unstructured":"Gander, M.J., Graham, I.G., Spence, E.A.: Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: what is the largest shift for which wavenumber-independent convergence is guaranteed? Numer. Math. 131(3), 567\u2013614 (2015). https:\/\/doi.org\/10.1007\/s00211-015-0700-2","journal-title":"Numer. Math."},{"issue":"1","key":"2450_CR8","doi-asserted-by":"publisher","first-page":"262","DOI":"10.1137\/16M108361X","volume":"39","author":"L Garc\u00eda Ramos","year":"2018","unstructured":"Garc\u00eda Ramos, L., Nabben, R.: On the spectrum of deflated matrices with applications to the deflated shifted Laplace preconditioner for the Helmholtz equation. SIAM J. Matrix Anal. Appl. 39(1), 262\u2013286 (2018). https:\/\/doi.org\/10.1137\/16M108361X","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"2450_CR9","doi-asserted-by":"publisher","first-page":"534","DOI":"10.1553\/etna_vol54s534","volume":"54","author":"L Garc\u00eda Ramos","year":"2021","unstructured":"Garc\u00eda Ramos, L., S\u00e8te, O., Nabben, R.: Preconditioning the Helmholtz equation with the shifted Laplacian and Faber polynomials. Electron. Trans. Numer. Anal. 54, 534\u2013557 (2021)","journal-title":"Electron. Trans. Numer. Anal."},{"key":"2450_CR10","doi-asserted-by":"publisher","first-page":"289","DOI":"10.1016\/0045-7825(95)00909-4","volume":"129","author":"RG Ghanem","year":"1996","unstructured":"Ghanem, R.G., Kruger, R.M.: Numerical solution of spectral stochastic finite element systems. Comput. Meth. Appl. Mech. Engrg. 129, 289\u2013303 (1996)","journal-title":"Comput. Meth. Appl. Mech. Engrg."},{"key":"2450_CR11","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3094-6","volume-title":"Stochastic Finite Elements: A Spectral Method Approach","author":"RG Ghanem","year":"1991","unstructured":"Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements: A Spectral Method Approach. Springer, New York (1991)"},{"issue":"5","key":"2450_CR12","doi-asserted-by":"publisher","first-page":"1942","DOI":"10.1137\/060661491","volume":"29","author":"MB van Gijzen","year":"2007","unstructured":"van Gijzen, M.B., Erlangga, Y.A., Vuik, C.: Spectral analysis of the discrete Helmholtz operator preconditioned with a shifted Laplacian. SIAM J. Sci. Comput. 29(5), 1942\u20131958 (2007). https:\/\/doi.org\/10.1137\/060661491","journal-title":"SIAM J. Sci. Comput."},{"issue":"283","key":"2450_CR13","doi-asserted-by":"publisher","first-page":"1515","DOI":"10.1090\/S0025-5718-2013-02654-3","volume":"82","author":"CJ Gittelson","year":"2013","unstructured":"Gittelson, C.J.: An adaptive stochastic Galerkin method for random elliptic operators. Math. Comput. 82(283), 1515\u20131541 (2013)","journal-title":"Math. Comput."},{"issue":"2","key":"2450_CR14","first-page":"505","volume":"3","author":"D Gottlieb","year":"2008","unstructured":"Gottlieb, D., Xiu, D.: Galerkin method for wave equations with uncertain coefficients. Comm. Comput. Phys. 3(2), 505\u2013518 (2008)","journal-title":"Comm. Comput. Phys."},{"key":"2450_CR15","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-22569-2","volume-title":"Essential Partial Differential Equations. Springer Undergraduate Mathematics Series","author":"DF Griffiths","year":"2015","unstructured":"Griffiths, D.F., Dold, J.W., Silvester, D.J.: Essential Partial Differential Equations. Springer Undergraduate Mathematics Series. Springer, Cham (2015)"},{"key":"2450_CR16","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-71584-9","volume-title":"Numerical Treatment of Partial Differential Equations","author":"C Grossmann","year":"2007","unstructured":"Grossmann, C., Roos, H.G., Stynes, M.: Numerical Treatment of Partial Differential Equations. Springer, Berlin (2007)"},{"key":"2450_CR17","doi-asserted-by":"crossref","DOI":"10.1007\/b98828","volume-title":"Finite Element Analysis of Acoustic Scattering","author":"F Ihlenburg","year":"1998","unstructured":"Ihlenburg, F.: Finite Element Analysis of Acoustic Scattering, vol. 132. Springer, New York (1998)"},{"key":"2450_CR18","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-28832-1","volume-title":"Modern solvers for Helmholtz problems","year":"2017","unstructured":"Lahaye, D., Tang, J., Vuik, K. (eds.): Modern solvers for Helmholtz problems. Birkh\u00e4user\/Springer, Cham (2017). https:\/\/doi.org\/10.1007\/978-3-319-28832-1"},{"issue":"1","key":"2450_CR19","doi-asserted-by":"publisher","first-page":"136","DOI":"10.4208\/nmtma.2015.w03si","volume":"8","author":"I Livshits","year":"2015","unstructured":"Livshits, I.: Use of shifted Laplacian operators for solving indefinite Helmholtz equations. Numer. Math. Theory Methods Appl. 8(1), 136\u2013148 (2015). https:\/\/doi.org\/10.4208\/nmtma.2015.w03si","journal-title":"Numer. Math. Theory Methods Appl."},{"key":"2450_CR20","volume-title":"Handbook of Linear Partial Differential Equations for Engineers and Scientists","author":"AD Polyanin","year":"2002","unstructured":"Polyanin, A.D.: Handbook of Linear Partial Differential Equations for Engineers and Scientists. Chapman & Hall\/CRC, Boca Raton (2002)"},{"key":"2450_CR21","doi-asserted-by":"publisher","DOI":"10.1186\/s13362-019-0067-6","author":"R Pulch","year":"2019","unstructured":"Pulch, R.: Stability-preserving model order reduction for linear stochastic Galerkin systems. J. Math. Ind. (2019). https:\/\/doi.org\/10.1186\/s13362-019-0067-6","journal-title":"J. Math. Ind."},{"issue":"2","key":"2450_CR22","doi-asserted-by":"publisher","first-page":"245","DOI":"10.1016\/j.matcom.2009.05.008","volume":"80","author":"R Pulch","year":"2009","unstructured":"Pulch, R., van Emmerich, C.: Polynomial chaos for simulating random volatilities. Math. Comput. Simul. 80(2), 245\u2013255 (2009). https:\/\/doi.org\/10.1016\/j.matcom.2009.05.008","journal-title":"Math. Comput. Simul."},{"key":"2450_CR23","doi-asserted-by":"publisher","unstructured":"Pulch, R., S\u00e8te, O.: The Helmholtz equation with uncertainties in the wavenumber. arXiv preprint: 2209.14740v1 (2022). https:\/\/doi.org\/10.48550\/arXiv.2209.14740","DOI":"10.48550\/arXiv.2209.14740"},{"issue":"2","key":"2450_CR24","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1007\/s10915-011-9511-5","volume":"51","author":"R Pulch","year":"2012","unstructured":"Pulch, R., Xiu, D.: Generalised polynomial chaos for a class of linear conservation laws. J. Sci. Comput. 51(2), 293\u2013312 (2012)","journal-title":"J. Sci. Comput."},{"issue":"22","key":"2450_CR25","doi-asserted-by":"publisher","first-page":"8384","DOI":"10.1016\/j.jcp.2010.07.022","volume":"229","author":"B Reps","year":"2010","unstructured":"Reps, B., Vanroose, W., Bin Zubair, H.: On the indefinite Helmholtz equation: complex stretched absorbing boundary layers, iterative analysis, and preconditioning. J. Comput. Phys. 229(22), 8384\u20138405 (2010). https:\/\/doi.org\/10.1016\/j.jcp.2010.07.022","journal-title":"J. Comput. Phys."},{"key":"2450_CR26","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718003","volume-title":"Iterative Methods for Sparse Linear Systems","author":"Y Saad","year":"2003","unstructured":"Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2003). https:\/\/doi.org\/10.1137\/1.9780898718003","edition":"2"},{"issue":"3","key":"2450_CR27","doi-asserted-by":"publisher","first-page":"856","DOI":"10.1137\/0907058","volume":"7","author":"Y Saad","year":"1986","unstructured":"Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 7(3), 856\u2013869 (1986). https:\/\/doi.org\/10.1137\/0907058","journal-title":"SIAM J. Sci. Statist. Comput."},{"key":"2450_CR28","doi-asserted-by":"publisher","first-page":"473","DOI":"10.1016\/j.jcp.2016.06.025","volume":"322","author":"AH Sheikh","year":"2016","unstructured":"Sheikh, A.H., Lahaye, D., Garcia Ramos, L., Nabben, R., Vuik, C.: Accelerating the shifted Laplace preconditioner for the Helmholtz equation by multilevel deflation. J. Comput. Phys. 322, 473\u2013490 (2016). https:\/\/doi.org\/10.1016\/j.jcp.2016.06.025","journal-title":"J. Comput. Phys."},{"key":"2450_CR29","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-21738-3","volume-title":"Introduction to Numerical Analysis","author":"J Stoer","year":"2002","unstructured":"Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis, 3rd edn. Springer, New York (2002)","edition":"3"},{"issue":"5","key":"2450_CR30","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1615\/Int.J.UncertaintyQuantification.2021034247","volume":"11","author":"G Wang","year":"2021","unstructured":"Wang, G., Xue, F., Liao, Q.: Localized stochastic Galerkin methods for Helmholtz problems close to resonance. Int. J. Uncertain. Quantif. 11(5), 77\u201399 (2021)","journal-title":"Int. J. Uncertain. Quantif."},{"key":"2450_CR31","doi-asserted-by":"publisher","DOI":"10.1515\/9781400835348","volume-title":"Numerical Methods for Stochastic Computations: A Spectral Method Approach","author":"D Xiu","year":"2010","unstructured":"Xiu, D.: Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press, Princeton (2010)"},{"key":"2450_CR32","doi-asserted-by":"publisher","first-page":"266","DOI":"10.1016\/j.jcp.2008.09.008","volume":"228","author":"D Xiu","year":"2009","unstructured":"Xiu, D., Shen, J.: Efficient stochastic Galerkin methods for random diffusion equations. J. Comput. Phys. 228, 266\u2013281 (2009)","journal-title":"J. Comput. Phys."},{"key":"2450_CR33","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-021-01498-9","author":"M Youssef","year":"2021","unstructured":"Youssef, M., Pulch, R.: Poly-Sinc solution of stochastic elliptic differential equations. J. Sci. Comput. (2021). https:\/\/doi.org\/10.1007\/s10915-021-01498-9","journal-title":"J. Sci. Comput."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-024-02450-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-024-02450-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-024-02450-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,24]],"date-time":"2024-02-24T11:11:47Z","timestamp":1708773107000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-024-02450-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,2,5]]},"references-count":33,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2024,3]]}},"alternative-id":["2450"],"URL":"https:\/\/doi.org\/10.1007\/s10915-024-02450-3","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,2,5]]},"assertion":[{"value":"3 January 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 October 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 January 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 February 2024","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no competing interests to declare that are relevant to the content of this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"60"}}