{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,8]],"date-time":"2026-07-08T17:06:52Z","timestamp":1783530412376,"version":"3.55.0"},"reference-count":48,"publisher":"Society of Exploration Geophysicists","issue":"2","content-domain":{"domain":["library.seg.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2021,3,1]]},"abstract":"<jats:title>ABSTRACT<\/jats:title>\n                  <jats:p>We have developed a finite-difference iterative solver of the Helmholtz equation for seismic modeling and inversion in the frequency domain. The iterative solver involves the shifted Laplacian operator and two-level preconditioners. It is based on the application of the preconditioners to the Krylov subspace stabilized biconjugate gradient method. A critical factor for the iterative solver is the introduction of a new preconditioner into the Krylov subspace iteration method to solve the linear equation system resulting from the discretization of the Helmholtz equation. This new preconditioner is based on a reformulation of an integral equation-based convergent Born series for the Lippmann-Schwinger equation to an equivalent differential equation. We have determined that our iterative solver combined with the novel preconditioner when incorporated with the finite-difference method accelerates the convergence of the Krylov subspace iteration method for frequency-domain seismic wave modeling. A comparison of a direct solver, a one-level Krylov subspace iterative solver, and our two-level iterative solver verified the accuracy and accelerated convergence of the new scheme. Extensive tests in full-waveform inversion demonstrate the solver\u2019s applicability to such problems.<\/jats:p>","DOI":"10.1190\/geo2020-0411.1","type":"journal-article","created":{"date-parts":[[2020,12,24]],"date-time":"2020-12-24T12:11:12Z","timestamp":1608811872000},"page":"T107-T116","update-policy":"https:\/\/doi.org\/10.1190\/crossmark-policy","source":"Crossref","is-referenced-by-count":21,"title":["A finite-difference iterative solver of the Helmholtz equation for frequency-domain seismic wave modeling and full-waveform inversion"],"prefix":"10.1190","volume":"86","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9719-6297","authenticated-orcid":false,"given":"Xingguo","family":"Huang","sequence":"first","affiliation":[{"name":"Jilin University 1 , College of Instrumentation and Electrical Engineering, and Key Lab of Geo-Exploration Instrumentation (Ministry of Education), Changchun 130026, China. E-mail: xingguo.huang19@gmail.com (corresponding author)."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Stewart","family":"Greenhalgh","sequence":"additional","affiliation":[{"name":"Swiss Federal Institute of Technology (ETH) Zurich 2 , Institute of Geophysics, Department of Earth Sciences, 8092 Zurich, Switzerland. E-mail: gstewart@retired.ethz.ch ."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"186","published-online":{"date-parts":[[2021,2,23]]},"reference":[{"key":"2026041612394669400_r1","volume-title":"Quantitative seismology: Theory and methods","author":"Aki","year":"1980"},{"key":"2026041612394669400_r2","doi-asserted-by":"crossref","first-page":"501","DOI":"10.1016\/S0045-7825(99)00242-X","article-title":"Multifrontal parallel distributed symmetric and unsymmetric solvers","volume":"184","author":"Amestoy","year":"2000","journal-title":"Computer Methods in Applied Mechanics and Engineering","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"2026041612394669400_r3","doi-asserted-by":"crossref","first-page":"330","DOI":"10.1016\/j.jcp.2017.05.026","article-title":"An iterative solver for the 3D Helmholtz equation","volume":"345","author":"Belonosov","year":"2017","journal-title":"Journal of Computational 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