Krylov-secant methods for solving large-scale systems of coupled nonlinear parabolic equations

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dc.contributor.advisorWheeler, Mary F.en_US
dc.creatorKlie, Hector Manuelen_US
dc.date.accessioned2009-06-04T08:30:17Zen_US
dc.date.available2009-06-04T08:30:17Zen_US
dc.date.issued1997en_US
dc.description.abstractThis dissertation centers on two major aspects dictating the computational time of applications based on the solution of systems of coupled nonlinear parabolic equations: nonlinear and linear iterations. The former aspect leads to the conception of a novel way of reusing the Krylov information generated by GMRES for solving linear systems arising within a Newton method. The approach stems from theory recently developed on a nonlinear version of the Eirola-Nevanlinna, algorithm (originally for solving non-symmetric linear systems) which is capable of converging twice as fast as Broyden's method. A secant update strategy of the Hessenberg matrix resulting from the Arnoldi process in GMRES amounts to reflecting a secant update of the current Jacobian with the rank-one term projected onto the generated Krylov subspace (Krylov-Broyden update). This allows the design of a new nonlinear Krylov-Eirola-Nevanlinna (KEN) algorithm and a higher-order version of Newton's method (HOKN) as well. The underlying development is also auspicious to replace the use of GMRES by cheaper Richardson iterations for the sake of fulfilling the inexact Newton condition. Hence, three algorithms derived from Newton's method, Broyden's method and the nonlinear Eirola-Nevanlinna algorithm are proposed as a part of a new family of hybrid Krylov-secant methods. All five algorithms are shown to be computationally more economical than their Newton and quasi-Newton counterparts. The aspect of linear iterations complements the present research with an analysis on nested or inner-outer iterations to efficiently precondition Krylov subspace iterative solvers for linear systems arising from systems of coupled nonlinear equations. These preconditioners are called two-stage preconditioners and are developed on the basis of a simple but effective decoupling strategy. Their analysis is restricted to the particular class of problems arising in multi-phase flow phenomena modeled by systems of coupled nonlinear parabolic equations. The resulting approach outperforms fairly robust and standard preconditioners that "blindly" precondition the entire coupled linear system. Theoretical discussion and computational experiments show the suitability that both linear and nonlinear aspects undertaken in this research have for large scale implementations.en_US
dc.format.extent237 p.en_US
dc.format.mimetypeapplication/pdfen_US
dc.identifier.callnoTHESIS MATH.SCI. 1997 KLIEen_US
dc.identifier.citationKlie, Hector Manuel. "Krylov-secant methods for solving large-scale systems of coupled nonlinear parabolic equations." (1997) Diss., Rice University. <a href="https://hdl.handle.net/1911/19174">https://hdl.handle.net/1911/19174</a>.en_US
dc.identifier.urihttps://hdl.handle.net/1911/19174en_US
dc.language.isoengen_US
dc.rightsCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.en_US
dc.subjectMathematicsen_US
dc.subjectComputer scienceen_US
dc.titleKrylov-secant methods for solving large-scale systems of coupled nonlinear parabolic equationsen_US
dc.typeThesisen_US
dc.type.materialTexten_US
thesis.degree.departmentMathematical Sciencesen_US
thesis.degree.disciplineEngineeringen_US
thesis.degree.grantorRice Universityen_US
thesis.degree.levelDoctoralen_US
thesis.degree.nameDoctor of Philosophyen_US
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