subject:(Multigrid Krylov)

Delft University of Technology

1. Diao, H. (author). Fourier Analysis of Iterative Methods for the Helmholtz Problem.

Degree: 2012, Delft University of Technology

URL: http://resolver.tudelft.nl/uuid:d82de64b-b446-4df6-b335-36a3e058c8f8

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This thesis attempts to explain the convergence behaviour of solving Helmholtz problem by investigating its spectral properties. Fourier analysis is employ to solve the eigenvalues… (more)

Subjects/Keywords: Helmholtz problem; Krylov subspace methods; multigrid method; multilevel Krylov multigrid method; shifted Laplacian preconditioner; deflation operator; Fourier analysis

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Diao, H. (. (2012). Fourier Analysis of Iterative Methods for the Helmholtz Problem. (Masters Thesis). Delft University of Technology. Retrieved from http://resolver.tudelft.nl/uuid:d82de64b-b446-4df6-b335-36a3e058c8f8

Chicago Manual of Style (16^th Edition):

Diao, H (author). “Fourier Analysis of Iterative Methods for the Helmholtz Problem.” 2012. Masters Thesis, Delft University of Technology. Accessed March 02, 2021. http://resolver.tudelft.nl/uuid:d82de64b-b446-4df6-b335-36a3e058c8f8.

MLA Handbook (7^th Edition):

Diao, H (author). “Fourier Analysis of Iterative Methods for the Helmholtz Problem.” 2012. Web. 02 Mar 2021.

Vancouver:

Diao H(. Fourier Analysis of Iterative Methods for the Helmholtz Problem. [Internet] [Masters thesis]. Delft University of Technology; 2012. [cited 2021 Mar 02]. Available from: http://resolver.tudelft.nl/uuid:d82de64b-b446-4df6-b335-36a3e058c8f8.

Council of Science Editors:

Diao H(. Fourier Analysis of Iterative Methods for the Helmholtz Problem. [Masters Thesis]. Delft University of Technology; 2012. Available from: http://resolver.tudelft.nl/uuid:d82de64b-b446-4df6-b335-36a3e058c8f8

INP Toulouse

2. Pinel, Xavier. A perturbed two-level preconditioner for the solution of three-dimensional heterogeneous Helmholtz problems with applications to geophysics : Un preconditionnement perturbé à deux niveaux pour la résolution de problèmes d'Helmholtz hétérogènes dans le cadre d'une application en géophysique.

Degree: Docteur es, Mathématiques, Informatiques et Télécommunication, 2010, INP Toulouse

URL: http://www.theses.fr/2010INPT0033

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Le sujet de cette thèse est le développement de méthodes itératives permettant la résolution degrands systèmes linéaires creux d'équations présentant plusieurs seconds membres simultanément. Ces… (more)

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Le sujet de cette thèse est le développement de méthodes itératives permettant la résolution degrands systèmes linéaires creux d'équations présentant plusieurs seconds membres simultanément. Ces méthodes seront en particulier utilisées dans le cadre d'une application géophysique : la migration sismique visant à simuler la propagation d'ondes sous la surface de la terre. Le problème prend la forme d'une équation d'Helmholtz dans le domaine fréquentiel en trois dimensions, discrétisée par des différences finies et donnant lieu à un système linéaire creux, complexe, non-symétrique, non-hermitien. De plus, lorsque de grands nombres d'onde sont considérés, cette matrice possède une taille élevée et est indéfinie. Du fait de ces propriétés, nous nous proposons d'étudier des méthodes de Krylov préconditionnées par des techniques hiérarchiques deux niveaux. Un tel pre-conditionnement s'est montré particulièrement efficace en deux dimensions et le but de cette thèse est de relever le défi de l'adapter au cas tridimensionel. Pour ce faire, des méthodes de Krylov sont utilisées à la fois comme lisseur et comme méthode de résolution du problème grossier. Ces derniers choix induisent l'emploi de méthodes de Krylov dites flexibles.

The topic of this PhD thesis is the development of iterative methods for the solution of large sparse linear systems of equations with possibly multiple right-hand sides given at once. These methods will be used for a specific application in geophysics - seismic migration - related to the simulation of wave propagation in the subsurface of the Earth. Here the three-dimensional Helmholtz equation written in the frequency domain is considered. The finite difference discretization of the Helmholtz equation with the Perfect Matched Layer formulation produces, when high frequencies are considered, a complex linear system which is large, non-symmetric, non-Hermitian, indefinite and sparse. Thus we propose to study preconditioned flexible Krylov subspace methods, especially minimum residual norm methods, to solve this class of problems. As a preconditioner we consider multi-level techniques and especially focus on a two-level method. This twolevel preconditioner has shown efficient for two-dimensional applications and the purpose of this thesis is to extend this to the challenging three-dimensional case. This leads us to propose and analyze a perturbed two-level preconditioner for a flexible Krylov subspace method, where Krylov methods are used both as smoother and as approximate coarse grid solver.

Advisors/Committee Members: Gratton, Serge (thesis director).

Subjects/Keywords: Equation d'Helmholtz; Méthodes de Krylov; Multigrille; Analyse de Fourier; Programmation parrallèle; Seconds membres multiples; Krylov methods; Multigrid; Helmholtz problems; Fourier analysis; Super computers; Geophysics; Multiple right-hand sides problems

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Pinel, X. (2010). A perturbed two-level preconditioner for the solution of three-dimensional heterogeneous Helmholtz problems with applications to geophysics : Un preconditionnement perturbé à deux niveaux pour la résolution de problèmes d'Helmholtz hétérogènes dans le cadre d'une application en géophysique. (Doctoral Dissertation). INP Toulouse. Retrieved from http://www.theses.fr/2010INPT0033

Chicago Manual of Style (16^th Edition):

Pinel, Xavier. “A perturbed two-level preconditioner for the solution of three-dimensional heterogeneous Helmholtz problems with applications to geophysics : Un preconditionnement perturbé à deux niveaux pour la résolution de problèmes d'Helmholtz hétérogènes dans le cadre d'une application en géophysique.” 2010. Doctoral Dissertation, INP Toulouse. Accessed March 02, 2021. http://www.theses.fr/2010INPT0033.

MLA Handbook (7^th Edition):

Pinel, Xavier. “A perturbed two-level preconditioner for the solution of three-dimensional heterogeneous Helmholtz problems with applications to geophysics : Un preconditionnement perturbé à deux niveaux pour la résolution de problèmes d'Helmholtz hétérogènes dans le cadre d'une application en géophysique.” 2010. Web. 02 Mar 2021.

Vancouver:

Pinel X. A perturbed two-level preconditioner for the solution of three-dimensional heterogeneous Helmholtz problems with applications to geophysics : Un preconditionnement perturbé à deux niveaux pour la résolution de problèmes d'Helmholtz hétérogènes dans le cadre d'une application en géophysique. [Internet] [Doctoral dissertation]. INP Toulouse; 2010. [cited 2021 Mar 02]. Available from: http://www.theses.fr/2010INPT0033.

Council of Science Editors:

Pinel X. A perturbed two-level preconditioner for the solution of three-dimensional heterogeneous Helmholtz problems with applications to geophysics : Un preconditionnement perturbé à deux niveaux pour la résolution de problèmes d'Helmholtz hétérogènes dans le cadre d'une application en géophysique. [Doctoral Dissertation]. INP Toulouse; 2010. Available from: http://www.theses.fr/2010INPT0033

3. Yang, Zhao, 1983-. A multigrid Krylov method for eigenvalue problems.

Degree: PhD, Baylor University. Dept. of Mathematics., 2015, Baylor University

URL: http://hdl.handle.net/2104/9514

► We are interested in computing eigenvalues and eigenvectors of matrices derived from differential equations. They are often large sparse matrices, including both symmetric and non… (more)

Subjects/Keywords: Krylov subspaces. Arnoldi. Multigrid. Eigenvalue problems.

…Krylov subspace. These methods use the Rayleigh-Ritz procedure to extract eigenvalue… …information from the Krylov subspace. Arnoldi and Lanczos methods have several difficulties. The… …Arnoldi method computes the orthogonal projection of A onto an m dimensional Krylov subspace… …x5B;18], where the Ritz eigenvectors are attached to a Krylov subspace. It is proved to… …multigrid methods. Multigrid methods can tackle the original operator and exploit discretizations…

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Yang, Zhao, 1. (2015). A multigrid Krylov method for eigenvalue problems. (Doctoral Dissertation). Baylor University. Retrieved from http://hdl.handle.net/2104/9514

Chicago Manual of Style (16^th Edition):

Yang, Zhao, 1983-. “A multigrid Krylov method for eigenvalue problems.” 2015. Doctoral Dissertation, Baylor University. Accessed March 02, 2021. http://hdl.handle.net/2104/9514.

MLA Handbook (7^th Edition):

Yang, Zhao, 1983-. “A multigrid Krylov method for eigenvalue problems.” 2015. Web. 02 Mar 2021.

Vancouver:

Yang, Zhao 1. A multigrid Krylov method for eigenvalue problems. [Internet] [Doctoral dissertation]. Baylor University; 2015. [cited 2021 Mar 02]. Available from: http://hdl.handle.net/2104/9514.

Council of Science Editors:

Yang, Zhao 1. A multigrid Krylov method for eigenvalue problems. [Doctoral Dissertation]. Baylor University; 2015. Available from: http://hdl.handle.net/2104/9514

Delft University of Technology

4. Bin Zubair, H. Efficient Multigrid Methods based on Improved Coarse Grid Correction Techniques.

Degree: 2009, Delft University of Technology

URL: http://resolver.tudelft.nl/uuid:e0b43b38-ad07-4076-baf1-aa02579c397f ; urn:NBN:nl:ui:24-uuid:e0b43b38-ad07-4076-baf1-aa02579c397f ; urn:NBN:nl:ui:24-uuid:e0b43b38-ad07-4076-baf1-aa02579c397f ; http://resolver.tudelft.nl/uuid:e0b43b38-ad07-4076-baf1-aa02579c397f

► Multigrid efficiency often suffers from inadequate coarse grid correction in different prototypic situations. We select a few problems, where coarse grid correction issues arise because… (more)

▼ Multigrid efficiency often suffers from inadequate coarse grid correction in different prototypic situations. We select a few problems, where coarse grid correction issues arise because of anisotropic coefficients, non-equidistant or non-uniform grid stretching, or inherent indefiniteness in the partial differential equation. Most of the work in this thesis can be classified as an attempt to increase multigrid efficiency by analysing and developing novel grid coarsening techniques that ensure sufficient coarse grid correction for the multigrid algorithm. Anisotropy in discrete systems can stem from various continuous and discrete features of the problem and has to have its negative effects countered before a successful multigrid solution can be brought about. We select multidimensional stationary diffusion equation as the first important problem to be treated in this context. The work for dimensions higher than three, is aimed at developing grid coarsening strategies for discretization on rectangular hyper-grids that differ greatly in their dimensions, and thus induce the so-called grid-aligned anisotropies in the system. Coarse grids formed through standard coarsening fail to provide sufficient coarse grid correction, and alternative block relaxation techniques are expensive in high dimensions. We also investigate and test coarsening strategies with the aim that their use would allow point based relaxation to stay effective in this non-equidistant multigrid scenario. Through local Fourier analysis we also analyze w-RB Jacobi, and implement a computer program through which we compute the optimal relaxation parameters. There are three important inferences in this regard. (1) Partial (and grid dependent) coarsening strategies allow the successful use of point relaxation methods for this problem. (2) Quadrupling along a few dimensions is a very attractive partial coarsening choice. (3) Optimal relaxation parameters have a significant enhancement effect on multigrid convergence in high dimensions. The efficient solution of time-dependent multidimensional equations (discretized with implicit time-integration schemes) is also a challenge. We first use the sparse grid technique to reduce the exponential complexity of the discrete problem, and then use the d-dimensional multigrid techniques to solve the sparse grid sub problems. In this situation, i.e., with a multitude of different non equidistant grids, evaluating and using optimal relaxation parameters on the fly is not an option any more. As a multigrid solver in high dimensions depends on optimal attributes to quite a large extent, we employ the method as a preconditioner, instead of a solver. This results in a very robust and efficient multigrid preconditioned Bi-CGSTAB solver. Another coarsening strategy that we develop in this thesis is aimed at two-dimensional grids that are non-uniformly stretched. We investigated different experimental coarsening strategies. A strategy based on improving individual mesh aspect ratios of grid cells proves successful both… Advisors/Committee Members: Oosterlee, C.W..

Subjects/Keywords: multigrid; d-multigrid; sparse grids; multidimensional partial differential equations; multigrid preconditioned Krylov methods; grid-stretching; indefinite Helmholtz; adaptive mesh-refinement

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Bin Zubair, H. (2009). Efficient Multigrid Methods based on Improved Coarse Grid Correction Techniques. (Doctoral Dissertation). Delft University of Technology. Retrieved from http://resolver.tudelft.nl/uuid:e0b43b38-ad07-4076-baf1-aa02579c397f ; urn:NBN:nl:ui:24-uuid:e0b43b38-ad07-4076-baf1-aa02579c397f ; urn:NBN:nl:ui:24-uuid:e0b43b38-ad07-4076-baf1-aa02579c397f ; http://resolver.tudelft.nl/uuid:e0b43b38-ad07-4076-baf1-aa02579c397f

Chicago Manual of Style (16^th Edition):

Bin Zubair, H. “Efficient Multigrid Methods based on Improved Coarse Grid Correction Techniques.” 2009. Doctoral Dissertation, Delft University of Technology. Accessed March 02, 2021. http://resolver.tudelft.nl/uuid:e0b43b38-ad07-4076-baf1-aa02579c397f ; urn:NBN:nl:ui:24-uuid:e0b43b38-ad07-4076-baf1-aa02579c397f ; urn:NBN:nl:ui:24-uuid:e0b43b38-ad07-4076-baf1-aa02579c397f ; http://resolver.tudelft.nl/uuid:e0b43b38-ad07-4076-baf1-aa02579c397f.

MLA Handbook (7^th Edition):

Bin Zubair, H. “Efficient Multigrid Methods based on Improved Coarse Grid Correction Techniques.” 2009. Web. 02 Mar 2021.

Vancouver:

Bin Zubair H. Efficient Multigrid Methods based on Improved Coarse Grid Correction Techniques. [Internet] [Doctoral dissertation]. Delft University of Technology; 2009. [cited 2021 Mar 02]. Available from: http://resolver.tudelft.nl/uuid:e0b43b38-ad07-4076-baf1-aa02579c397f ; urn:NBN:nl:ui:24-uuid:e0b43b38-ad07-4076-baf1-aa02579c397f ; urn:NBN:nl:ui:24-uuid:e0b43b38-ad07-4076-baf1-aa02579c397f ; http://resolver.tudelft.nl/uuid:e0b43b38-ad07-4076-baf1-aa02579c397f.

Council of Science Editors:

Bin Zubair H. Efficient Multigrid Methods based on Improved Coarse Grid Correction Techniques. [Doctoral Dissertation]. Delft University of Technology; 2009. Available from: http://resolver.tudelft.nl/uuid:e0b43b38-ad07-4076-baf1-aa02579c397f ; urn:NBN:nl:ui:24-uuid:e0b43b38-ad07-4076-baf1-aa02579c397f ; urn:NBN:nl:ui:24-uuid:e0b43b38-ad07-4076-baf1-aa02579c397f ; http://resolver.tudelft.nl/uuid:e0b43b38-ad07-4076-baf1-aa02579c397f

Delft University of Technology

5. Sheikh, A.H. Development Of The Helmholtz Solver Based On A Shifted Laplace Preconditioner And A Multigrid Deflation Technique.

Degree: 2014, Delft University of Technology

URL: http://resolver.tudelft.nl/uuid:1020f418-b488-4435-81ee-2b4f6a5024e1 ; urn:NBN:nl:ui:24-uuid:1020f418-b488-4435-81ee-2b4f6a5024e1 ; urn:NBN:nl:ui:24-uuid:1020f418-b488-4435-81ee-2b4f6a5024e1 ; http://resolver.tudelft.nl/uuid:1020f418-b488-4435-81ee-2b4f6a5024e1

► The Helmholtz equation is the simplest possible model for the wave propagation. Perhaps this is the reason, despite denying traditional iterative methods like Krylov sub-space… (more)

▼ The Helmholtz equation is the simplest possible model for the wave propagation. Perhaps this is the reason, despite denying traditional iterative methods like Krylov sub-space methods, Multigrids, etcetera, numerical solution of the Helmholtz equation has been an interesting and abundant problem to researchers since years. The work in this dissertation is also classified as an attempt to develop fast and robust iterative methods for the solution of the Helmholtz equation. This works is specified for applications in seismic imaging-Geophysics, where usually high frequency are used. Thus we will be targeting large wavenumber Helmholtz problems. The finite difference discretization of the Helmholtz equation with typically given Absorbing (Sommerfeld) boundary conditions gives rise to symmetric, non-Hermitian, indefinite linear systems. Resolution of large wavenumber requires larger number of grid points, thus large linear systems. Many (sparse) direct solvers and hybrid (direct and iterative) solvers have been proposed, but it is quite obvious for very large problems that (sparse) direct solvers have been too much depending upon memory, which makes them less acceptable. Quite a lot of work has been invested in researching iterative solution methods for the Helmholtz equation since many decades. The indefiniteness, which increases with respect to an increase in the wavenumber, poses extra problems for iterative solvers and robust solution of indefinite (large) linear system forms an important research activity. Many iterative techniques like domain decomposition methods, multigrid methods and preconditioners for Krylov subspace methods have been proposed but non of them has been quite robust. For multigrid methods, indefiniteness arises difficulties in having both good smoothing property and constructing appropriate coarse-grid approximations of the problem, which are responsible for further reduction of low frequency errors. Many attempts have been spent in algebraic variants of multigrid methods. Some of them works well with limitation of homogeneity. Most of them fails to show satisfactory convergence. The same holds for Krylov subspace methods. One of the difficulties for Krylov methods is to find a cheap and performing preconditioner for the indefinite Helmholtz equation. An overview of preconditioners, ranging from classical to matrix based, for indefinite Helmholtz linear system has been give in this thesis. A matrix-based complex shifted Laplace preconditioner (CSLP) has been seen as best in the available ones. However, with increasing wavenumbers CSLP shows a slow convergence behavior. We address this issue continuing using CSLP while taking care of its requirement of specific complex shifts. The projection-type preconditioners have been widely investigated by researchers in numerical analysis community. We propose the projection-type deflation preconditioner to tackle the near-singular nodes, which are the cause of the decay the convergence of, this otherwise well performing, CSLP. Like multigrid, this… Advisors/Committee Members: Vuik, C..

Subjects/Keywords: Helmholtz; multigrid methods; Krylov; iterative solvers; wave equations; deflation method; multilevel methods

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Sheikh, A. H. (2014). Development Of The Helmholtz Solver Based On A Shifted Laplace Preconditioner And A Multigrid Deflation Technique. (Doctoral Dissertation). Delft University of Technology. Retrieved from http://resolver.tudelft.nl/uuid:1020f418-b488-4435-81ee-2b4f6a5024e1 ; urn:NBN:nl:ui:24-uuid:1020f418-b488-4435-81ee-2b4f6a5024e1 ; urn:NBN:nl:ui:24-uuid:1020f418-b488-4435-81ee-2b4f6a5024e1 ; http://resolver.tudelft.nl/uuid:1020f418-b488-4435-81ee-2b4f6a5024e1

Chicago Manual of Style (16^th Edition):

Sheikh, A H. “Development Of The Helmholtz Solver Based On A Shifted Laplace Preconditioner And A Multigrid Deflation Technique.” 2014. Doctoral Dissertation, Delft University of Technology. Accessed March 02, 2021. http://resolver.tudelft.nl/uuid:1020f418-b488-4435-81ee-2b4f6a5024e1 ; urn:NBN:nl:ui:24-uuid:1020f418-b488-4435-81ee-2b4f6a5024e1 ; urn:NBN:nl:ui:24-uuid:1020f418-b488-4435-81ee-2b4f6a5024e1 ; http://resolver.tudelft.nl/uuid:1020f418-b488-4435-81ee-2b4f6a5024e1.

MLA Handbook (7^th Edition):

Sheikh, A H. “Development Of The Helmholtz Solver Based On A Shifted Laplace Preconditioner And A Multigrid Deflation Technique.” 2014. Web. 02 Mar 2021.

Vancouver:

Sheikh AH. Development Of The Helmholtz Solver Based On A Shifted Laplace Preconditioner And A Multigrid Deflation Technique. [Internet] [Doctoral dissertation]. Delft University of Technology; 2014. [cited 2021 Mar 02]. Available from: http://resolver.tudelft.nl/uuid:1020f418-b488-4435-81ee-2b4f6a5024e1 ; urn:NBN:nl:ui:24-uuid:1020f418-b488-4435-81ee-2b4f6a5024e1 ; urn:NBN:nl:ui:24-uuid:1020f418-b488-4435-81ee-2b4f6a5024e1 ; http://resolver.tudelft.nl/uuid:1020f418-b488-4435-81ee-2b4f6a5024e1.

Council of Science Editors:

Sheikh AH. Development Of The Helmholtz Solver Based On A Shifted Laplace Preconditioner And A Multigrid Deflation Technique. [Doctoral Dissertation]. Delft University of Technology; 2014. Available from: http://resolver.tudelft.nl/uuid:1020f418-b488-4435-81ee-2b4f6a5024e1 ; urn:NBN:nl:ui:24-uuid:1020f418-b488-4435-81ee-2b4f6a5024e1 ; urn:NBN:nl:ui:24-uuid:1020f418-b488-4435-81ee-2b4f6a5024e1 ; http://resolver.tudelft.nl/uuid:1020f418-b488-4435-81ee-2b4f6a5024e1

6. Duminil, Sébastien. Extrapolation vectorielle et applications aux équations aux dérivées partielles : Vector extrapolation and applications to partial differential equations.

Degree: Docteur es, Mathématiques appliquées, 2012, Littoral

URL: http://www.theses.fr/2012DUNK0336

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Nous nous intéressons, dans cette thèse, à l'étude des méthodes d'extrapolation polynômiales et à l'application de ces méthodes dans l'accélération de méthodes de points fixes… (more)

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Nous nous intéressons, dans cette thèse, à l'étude des méthodes d'extrapolation polynômiales et à l'application de ces méthodes dans l'accélération de méthodes de points fixes pour des problèmes donnés. L'avantage de ces méthodes d'extrapolation est qu'elles utilisent uniquement une suite de vecteurs qui n'est pas forcément convergente, ou qui converge très lentement pour créer une nouvelle suite pouvant admettreune convergence quadratique. Le développement de méthodes cycliques permet, deplus, de limiter le coût de calculs et de stockage. Nous appliquons ces méthodes à la résolution des équations de Navier-Stokes stationnaires et incompressibles, à la résolution de la formulation Kohn-Sham de l'équation de Schrödinger et à la résolution d'équations elliptiques utilisant des méthodes multigrilles. Dans tous les cas, l'efficacité des méthodes d'extrapolation a été montrée.Nous montrons que lorsqu'elles sont appliquées à la résolution de systèmes linéaires, les méthodes d'extrapolation sont comparables aux méthodes de sous espaces de Krylov. En particulier, nous montrons l'équivalence entre la méthode MMPE et CMRH. Nous nous intéressons enfin, à la parallélisation de la méthode CMRH sur des processeurs à mémoire distribuée et à la recherche de préconditionneurs efficaces pour cette même méthode.

In this thesis, we study polynomial extrapolation methods. We discuss the design and implementation of these methods for computing solutions of fixed point methods. Extrapolation methods transform the original sequance into another sequence that converges to the same limit faster than the original one without having explicit knowledge of the sequence generator. Restarted methods permit to keep the storage requirement and the average of computational cost low. We apply these methods for computing steady state solutions of incompressible flow problems modelled by the Navier-Stokes equations, for solving the Schrödinger equation using the Kohn-Sham formulation and for solving elliptic equations using multigrid methods. In all cases, vector extrapolation methods have a useful role to play. We show that, when applied to linearly generated vector sequences, extrapolation methods are related to Krylov subspace methods. For example, we show that the MMPE approach is mathematically equivalent to CMRH method. We present an implementation of the CMRH iterative method suitable for parallel architectures with distributed memory. Finally, we present a preconditioned CMRH method.

Advisors/Committee Members: Sadok, Hassane (thesis director).

Subjects/Keywords: Extrapolation vectorielle; RRE; MPE; MMPE; Systèmes linéaires; Méthodes de Krylov; CMRH; Implémentation parallèle; CMRH préconditionnée; Systèmes non linéaires; Équations de Navier-Stokes; Équations de Schrödinger; Méthodes multigrilles; Vector extrapolation; Reduced Rank Extrapolation; Minimal Polynomial Extrapolation; Modified Minimal Polynomial Extrapolation; Linear systems; Krylov method; CMRH; Parallel implementation; Preconditioned CMRH; Nonlinear systems; Navier-Stokes problem; Schrödinger equation; Multigrid method

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Duminil, S. (2012). Extrapolation vectorielle et applications aux équations aux dérivées partielles : Vector extrapolation and applications to partial differential equations. (Doctoral Dissertation). Littoral. Retrieved from http://www.theses.fr/2012DUNK0336

Chicago Manual of Style (16^th Edition):

Duminil, Sébastien. “Extrapolation vectorielle et applications aux équations aux dérivées partielles : Vector extrapolation and applications to partial differential equations.” 2012. Doctoral Dissertation, Littoral. Accessed March 02, 2021. http://www.theses.fr/2012DUNK0336.

MLA Handbook (7^th Edition):

Duminil, Sébastien. “Extrapolation vectorielle et applications aux équations aux dérivées partielles : Vector extrapolation and applications to partial differential equations.” 2012. Web. 02 Mar 2021.

Vancouver:

Duminil S. Extrapolation vectorielle et applications aux équations aux dérivées partielles : Vector extrapolation and applications to partial differential equations. [Internet] [Doctoral dissertation]. Littoral; 2012. [cited 2021 Mar 02]. Available from: http://www.theses.fr/2012DUNK0336.

Council of Science Editors:

Duminil S. Extrapolation vectorielle et applications aux équations aux dérivées partielles : Vector extrapolation and applications to partial differential equations. [Doctoral Dissertation]. Littoral; 2012. Available from: http://www.theses.fr/2012DUNK0336

7. Liu, Jun. NEW COMPUTATIONAL METHODS FOR OPTIMAL CONTROL OF PARTIAL DIFFERENTIAL EQUATIONS.

Degree: PhD, Mathematics, 2015, Southern Illinois University

URL: https://opensiuc.lib.siu.edu/dissertations/1076

► Partial differential equations are the chief means of providing mathematical models in science, engineering and other fields. Optimal control of partial differential equations (PDEs)… (more)

▼ Partial differential equations are the chief means of providing mathematical models in science, engineering and other fields. Optimal control of partial differential equations (PDEs) has tremendous applications in engineering and science, such as shape optimization, image processing, fluid dynamics, and chemical processes. In this thesis, we develop and analyze several efficient numerical methods for the optimal control problems governed by elliptic PDE, parabolic PDE, and wave PDE, respectively. The thesis consists of six chapters. In Chapter 1, we briefly introduce a few motivating applications and summarize some theoretical and computational foundations of our following developed approaches. In Chapter 2, we establish a new multigrid algorithm to accelerate the semi-smooth Newton method that is applied to the first-order necessary optimality system arising from semi-linear control-constrained elliptic optimal control problems. Under suitable assumptions, the discretized Jacobian matrix is proved to have a uniformly bounded inverse with respect to mesh size. Different from current available approaches, a new strategy that leads to a robust multigrid solver is employed to define the coarse grid operator. Numerical simulations are provided to illustrate the efficiency of the proposed method, which shows to be computationally more efficient than the popular full approximation storage (FAS) multigrid method. In particular, our proposed approach achieves a mesh-independent convergence and its performance is highly robust with respect to the regularization parameter. In Chaper 3, we present a new second-order leapfrog finite difference scheme in time for solving the first-order necessary optimality system of the linear parabolic optimal control problems. The new leapfrog scheme is shown to be unconditionally stable and it provides a second-order accuracy, while the classical leapfrog scheme usually is well-known to be unstable. A mathematical proof for the convergence of the proposed scheme is provided under a suitable norm. Moreover, the proposed leapfrog scheme gives a favorable structure that leads to an effective implementation of a fast solver under the multigrid framework. Numerical examples show that the proposed scheme significantly outperforms the widely used second-order backward time differentiation approach, and the resultant fast solver demonstrates a mesh-independent convergence as well as a linear time complexity. In Chapter 4, we develop a new semi-smooth Newton multigrid algorithm for solving the discretized first-order necessary optimality system that characterizes the optimal solution of semi-linear parabolic PDE optimal control problems with control constraints. A new leapfrog discretization scheme in time associated with the standard five-point stencil in space is established to achieve a second-order accuracy. The convergence (or unconditional stability) of the proposed scheme is proved when time-periodic solutions are considered. Moreover, the derived well-structured discretized Jacobian… Advisors/Committee Members: Xiao, Mingqing.

Subjects/Keywords: finite difference scheme; multigrid method; optimal control; partial differential equations; preconditioned Krylov subspace method; semi-smooth Newton method

…multigrid, Algebraic multigrid. • Non-stationary methods: Krylov subspace methods, etc.; We will… …102.14 1.4.2 Krylov subspace method In the case of hyperbolic PDEs, multigrid methods turn… …16 1.4.1 Multigrid method… …16 1.4.2 Krylov subspace method… …23 1.5.2 Full approximation scheme (FAS) multigrid method…

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APA (6^th Edition):

Liu, J. (2015). NEW COMPUTATIONAL METHODS FOR OPTIMAL CONTROL OF PARTIAL DIFFERENTIAL EQUATIONS. (Doctoral Dissertation). Southern Illinois University. Retrieved from https://opensiuc.lib.siu.edu/dissertations/1076

Chicago Manual of Style (16^th Edition):

Liu, Jun. “NEW COMPUTATIONAL METHODS FOR OPTIMAL CONTROL OF PARTIAL DIFFERENTIAL EQUATIONS.” 2015. Doctoral Dissertation, Southern Illinois University. Accessed March 02, 2021. https://opensiuc.lib.siu.edu/dissertations/1076.

MLA Handbook (7^th Edition):

Liu, Jun. “NEW COMPUTATIONAL METHODS FOR OPTIMAL CONTROL OF PARTIAL DIFFERENTIAL EQUATIONS.” 2015. Web. 02 Mar 2021.

Vancouver:

Liu J. NEW COMPUTATIONAL METHODS FOR OPTIMAL CONTROL OF PARTIAL DIFFERENTIAL EQUATIONS. [Internet] [Doctoral dissertation]. Southern Illinois University; 2015. [cited 2021 Mar 02]. Available from: https://opensiuc.lib.siu.edu/dissertations/1076.

Council of Science Editors:

Liu J. NEW COMPUTATIONAL METHODS FOR OPTIMAL CONTROL OF PARTIAL DIFFERENTIAL EQUATIONS. [Doctoral Dissertation]. Southern Illinois University; 2015. Available from: https://opensiuc.lib.siu.edu/dissertations/1076

8. Köster, Michael. A Hierarchical Flow Solver for Optimisation with PDE Constraints.

Degree: 2011, Technische Universität Dortmund

URL: http://hdl.handle.net/2003/29239

► Active flow control plays a central role in many industrial applications such as e.g. control of crystal growth processes, where the flow in the melt… (more)

▼ Active flow control plays a central role in many industrial applications such as e.g. control of crystal growth processes, where the flow in the melt has a significant impact on the quality of the crystal. Optimal control of the flow by electro-magnetic fields and/or boundary temperatures leads to optimisation problems with PDE constraints, which are frequently governed by the time-dependent Navier-Stokes equations. The mathematical formulation is a minimisation problem with PDE constraints. By exploiting the special structure of the first order necessary optimality conditions, the so called Karush-Kuhn-Tucker (KKT)-system, this thesis develops a special hierarchical solution approach for such equations, based on the distributed control of the Stokes – and Navier – Stokes. The numerical costs for solving the optimisation problem are only about 20-50 times higher than a pure forward simulation, independent of the refinement level. Utilising modern multigrid techniques in space, it is possible to solve a forward simulation with optimal complexity, i.e., an appropriate solver for a forward simulation needs O(N) operations, N denoting the total number of unknowns for a given computational mesh in space and time. Such solvers typically apply appropriate multigrid techniques for the linear subproblems in space. As a consequence, the developed solution approach for the optimal control problem has complexity O(N) as well. This is achieved by a combination of a space-time Newton approach for the nonlinearity and a monolithic space-time multigrid approach for 'global' linear subproblems. A second inner monolithic multigrid method is applied for subproblems in space, utilising local Pressure-Schur complement techniques to treat the saddle-point structure. The numerical complexity of this algorithm distinguishes this approach from adjoint-based steepest descent methods used to solve optimisation problems in many practical applications, which in general do not satisfy this complexity requirement. Advisors/Committee Members: Turek, Stefan.

Subjects/Keywords: Block-Glätter; Block smoother; CFD; Crank-Nicolson; Crystal growth; Czochralski; Distributed Control; Edge-oriented stabilisation; Elliptic; Elliptisch; EOJ stabilisation; EOJ Stabilisierung; FEAT; FEATFLOW; Finite Elemente; Finite Elements; First discretise then optimise; First discretize then optimize; First optimise then discretise; First optimize then discretize; Flow-Around-Cylinder; Full Newton-SAND; Heat equation; Hierarchical; Hierarchical solution concept; Hierarchisch; Hierarchisches Lösungskonzept; Inexact Newton; Inexaktes Newton-Verfahren; Instationär; Inverse Probleme; Inverse Problems; Kantenbasierte Stabilisierung; KKT system; Kristallwachstum; Krylov; Large-Scale; linear complexity; lineare Komplexität; Mehrgitter; Mehrgitter-Krylov; Monolithic; Monolithisch; Multigrid; Multigrid-Krylov; Multilevel; Navier-Stokes; Nichtparametrische Finite Elemente; Nonparametric finite elements; Nonstationary; OPTFLOW; Optimierung; Optimisation; Optimization; PDE Constraints; Raum-Zeit; saddle point; SAND; Sattelpunkt; Schur complement preconditioning; Schurkomplement-Vorkonditionierer; Space-time; SQP; Stokes; Theta schema; Theta scheme; Time-dependent; Transient; Unstructured Grids; Unstrukturierte Gitter; Vanka; Verteilte Kontrolle; Wärmeleitung; Wärmeleitungsgleichung; 510

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Köster, M. (2011). A Hierarchical Flow Solver for Optimisation with PDE Constraints. (Thesis). Technische Universität Dortmund. Retrieved from http://hdl.handle.net/2003/29239

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

Köster, Michael. “A Hierarchical Flow Solver for Optimisation with PDE Constraints.” 2011. Thesis, Technische Universität Dortmund. Accessed March 02, 2021. http://hdl.handle.net/2003/29239.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

Köster, Michael. “A Hierarchical Flow Solver for Optimisation with PDE Constraints.” 2011. Web. 02 Mar 2021.

Vancouver:

Köster M. A Hierarchical Flow Solver for Optimisation with PDE Constraints. [Internet] [Thesis]. Technische Universität Dortmund; 2011. [cited 2021 Mar 02]. Available from: http://hdl.handle.net/2003/29239.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Köster M. A Hierarchical Flow Solver for Optimisation with PDE Constraints. [Thesis]. Technische Universität Dortmund; 2011. Available from: http://hdl.handle.net/2003/29239

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

9. Köster, Michael. A Hierarchical Flow Solver for Optimisation with PDE Constraints.

Degree: 2011, Technische Universität Dortmund

URL: http://dx.doi.org/10.17877/DE290R-6950

► Active flow control plays a central role in many industrial applications such as e.g. control of crystal growth processes, where the flow in the melt… (more)

▼ Active flow control plays a central role in many industrial applications such as e.g. control of crystal growth processes, where the flow in the melt has a significant impact on the quality of the crystal. Optimal control of the flow by electro-magnetic fields and/or boundary temperatures leads to optimisation problems with PDE constraints, which are frequently governed by the time-dependent Navier-Stokes equations. The mathematical formulation is a minimisation problem with PDE constraints. By exploiting the special structure of the first order necessary optimality conditions, the so called Karush-Kuhn-Tucker (KKT)-system, this thesis develops a special hierarchical solution approach for such equations, based on the distributed control of the Stokes – and Navier – Stokes. The numerical costs for solving the optimisation problem are only about 20-50 times higher than a pure forward simulation, independent of the refinement level. Utilising modern multigrid techniques in space, it is possible to solve a forward simulation with optimal complexity, i.e., an appropriate solver for a forward simulation needs O(N) operations, N denoting the total number of unknowns for a given computational mesh in space and time. Such solvers typically apply appropriate multigrid techniques for the linear subproblems in space. As a consequence, the developed solution approach for the optimal control problem has complexity O(N) as well. This is achieved by a combination of a space-time Newton approach for the nonlinearity and a monolithic space-time multigrid approach for 'global' linear subproblems. A second inner monolithic multigrid method is applied for subproblems in space, utilising local Pressure-Schur complement techniques to treat the saddle-point structure. The numerical complexity of this algorithm distinguishes this approach from adjoint-based steepest descent methods used to solve optimisation problems in many practical applications, which in general do not satisfy this complexity requirement. Advisors/Committee Members: Turek, Stefan (advisor), Meyer, Christian (referee).

Subjects/Keywords: Block-Glätter; Czochralski; Elliptisch; EOJ Stabilisierung; Finite Elemente; Hierarchisch; Hierarchisches Lösungskonzept; Inexaktes Newton-Verfahren; Instationär; Inverse Probleme; Kantenbasierte Stabilisierung; Kristallwachstum; Krylov; lineare Komplexität; Mehrgitter; Mehrgitter-Krylov; Monolithisch; Navier-Stokes; Nichtparametrische Finite Elemente; Optimierung; Raum-Zeit; Sattelpunkt; Schurkomplement-Vorkonditionierer; Stokes; Unstrukturierte Gitter; Vanka; Verteilte Kontrolle; Wärmeleitung; Wärmeleitungsgleichung; Block smoother; CFD; Crank-Nicolson; Crystal growth; Distributed Control; Edge-oriented stabilisation; Elliptic; EOJ stabilisation; FEAT; FEATFLOW; Finite Elements; First discretise then optimise; First discretize then optimize; First optimise then discretise; First optimize then discretize; Flow-Around-Cylinder; Full Newton-SAND; Heat equation; Hierarchical; Hierarchical solution concept; Inexact Newton; Inverse Problems; KKT system; Large-Scale; linear complexity; Monolithic; Multigrid; Multigrid-Krylov; Multilevel; Nonparametric finite elements; Nonstationary; OPTFLOW; Optimisation; Optimization; PDE Constraints; saddle point; SAND; Schur complement preconditioning; Space-time; SQP; Theta schema; Theta scheme; Time-dependent; Transient; Unstructured Grids; 510

Record Details Similar Records Cite « Share

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Köster, M. (2011). A Hierarchical Flow Solver for Optimisation with PDE Constraints. (Doctoral Dissertation). Technische Universität Dortmund. Retrieved from http://dx.doi.org/10.17877/DE290R-6950

Chicago Manual of Style (16^th Edition):

Köster, Michael. “A Hierarchical Flow Solver for Optimisation with PDE Constraints.” 2011. Doctoral Dissertation, Technische Universität Dortmund. Accessed March 02, 2021. http://dx.doi.org/10.17877/DE290R-6950.

MLA Handbook (7^th Edition):

Köster, Michael. “A Hierarchical Flow Solver for Optimisation with PDE Constraints.” 2011. Web. 02 Mar 2021.

Vancouver:

Köster M. A Hierarchical Flow Solver for Optimisation with PDE Constraints. [Internet] [Doctoral dissertation]. Technische Universität Dortmund; 2011. [cited 2021 Mar 02]. Available from: http://dx.doi.org/10.17877/DE290R-6950.

Council of Science Editors:

Köster M. A Hierarchical Flow Solver for Optimisation with PDE Constraints. [Doctoral Dissertation]. Technische Universität Dortmund; 2011. Available from: http://dx.doi.org/10.17877/DE290R-6950

10. Wobker, Hilmar. Efficient multilevel solvers and high performance computing techniques for the finite element simulation of large-scale elasticity problems.

Degree: 2010, Technische Universität Dortmund

URL: http://hdl.handle.net/2003/26998

► Bei der Simulation realistischer strukturmechanischer Probleme können Gleichungssysteme mit mehreren hundert Millionen Unbekannten entstehen. Für die effiziente Lösung solcher Systeme sind parallele Multilevel-Methoden unerlässlich, die… (more)

▼ Bei der Simulation realistischer strukturmechanischer Probleme können Gleichungssysteme mit mehreren hundert Millionen Unbekannten entstehen. Für die effiziente Lösung solcher Systeme sind parallele Multilevel-Methoden unerlässlich, die in der Lage sind, die Leistung moderner Hardware-Technologien auszuschöpfen. Die Finite-Elemente- und Löser-Toolbox FEAST, die auf die Behandlung skalarer Gleichungen ausgelegt ist, verfolgt genau dieses Ziel. FEAST kombiniert Hardware-orientierte Implementierungstechniken mit einer Multilevel-Gebietszerlegungsmethode namens ScaRC. In der vorliegenden Arbeit wird ein Konzept entwickelt, multivariate Elastizitätsprobleme basierend auf der FEAST-Bibliothek zu lösen. Die generelle Herangehensweise besteht darin, die Lösung multivariater Probleme auf die Lösung einer Reihe von skalaren Problemen zurückzuführen. Dieser Ansatz ermöglicht eine strikte Trennung von skalaren "low level" Kernfunktionalitäten (in Form der FEAST-Bibliothek) und multivariatem "high level" Anwendungscode (in Form des Elastizitätsproblems), was aus Sicht der Softwareentwicklungstechnik sehr vorteilhaft ist: Alle Bemühungen zur Verbesserung der Hardware-Effizienz, sowie Anpassungen an zukünftige technologische Entwicklungen können auf skalare Operationen beschränkt werden, während die multivariate Anwendung automatisch von diesen Erweiterungen profitiert. Im ersten Teil der Arbeit werden substantielle Verbesserungen der skalaren ScaRC-Löser entwickelt, die dann als essentielle Bausteine zur Lösung multivariater Elastizitätsprobleme eingesetzt werden. Ausführliche numerische Untersuchungen zeigen, wie sich die Effizienz der skalaren FEAST-Bibliothek auf den multivariaten Lösungsprozess überträgt. Die Löserstrategie wird dann auf nichtlineare Probleme der Elastizität mit finiter Deformation angewandt. Durch Einsatz einer Liniensuche-Methode wird die Robustheit des Newton-Raphson-Verfahrens signifikant erhöht. Es werden verschiedene Strategien miteinander verglichen, wie genau die linearen Probleme innerhalb der nichtlinearen Iteration zu lösen sind. Zur Behandlung der wichtigen Klasse von (fast) inkompressiblen Materialien wird eine gemischte Verschiebung/Druck-Formulierung gewählt, die mit Hilfe von stabilisierten bilinearen finiten Elementen (Q1/Q1) diskretisiert wird. Eine erweiterte Version der klassischen "Druck-Poisson"-Stabilisierung wird präsentiert, die auch für hochgradig irreguläre Gitter geeignet ist. Es werden Vor- und Nachteile der Q1/Q1-Diskretisierung erörtert, insbesondere in Bezug auf zeitabhängige Rechnungen. Zwei Löser-Klassen zur Behandlung der entstehenden Sattelpunkt-Probleme werden beschrieben und miteinander verglichen: einerseits verschiedene Arten von (beschleunigten) entkoppelten Lösern (Uzawa, Druck-Schurkomplement-Methoden, Block-Vorkonditionierer), andererseits gekoppelte Mehrgitter-Verfahren mit Vanka-Glättern. Effiziente Schurkomplement-Vorkonditionierer, die für die erste Löser-Klasse notwendig sind, werden im Rahmen statischer und zeitabhängiger Probleme besprochen. Die zentrale… Advisors/Committee Members: Turek, S..

Subjects/Keywords: Adaptive coarse grid correction; Adaptive Grobgitterkorrektur; Damped Newton-Raphson; Domain decomposition; Elasticity; Elastizität; Elastodynamic; Elastodynamisch; Elastostatic; Elastostatisch; Equal-order finite elements; FEAST; FEAST; Festkörpermechanik; Finite deformation; Finite Deformation; Finite-Elemente-Methode; Finite element method; Gebietszerlegung; Gedämpftes Newton-Raphson; Gemischte Formulierung; Globales Newton-Raphson; Global Newton-Raphson; Große Deformation; Großskalig; Hardware-oriented; Hardware-orientiert; High performance computing; Incompressible material; Inexact Newton-Raphson; Inexaktes Newton-Raphson; Inkompressibles Material; Irreguläres Gitter; Irregular grids; Iterativer Löser; Iterative solver; Large deformation; Large-scale; LBB stabilisation; LBB Stabilisierung; Line-search; Liniensuche; Mehrgitter; Mehrgitter-Krylov; Minimale Überlappung; Minimal overlap; Mixed formulation; Multigrid; Multigrid-Krylov; Multilevel; Multilevel; Newton-Raphson; Nicht-konformes Mehrgitter; Nonconforming multigrid; Parallel computing; Parallele Effizienz; Parallel efficiency; Paralleles Rechnen; Saddle point problem; Sattelpunkt-Problem; ScaRC; ScaRC; Schubversteifung; Schur complement preconditioning; Schurkomplement-Vorkonditionierer; Shear locking; Solid mechanics; Structural mechanics; Strukturmechanik; Transient; Vanka; Vanka; Volume locking; Volumenversteifung; Zeitabhängig; 510

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Wobker, H. (2010). Efficient multilevel solvers and high performance computing techniques for the finite element simulation of large-scale elasticity problems. (Thesis). Technische Universität Dortmund. Retrieved from http://hdl.handle.net/2003/26998

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

Wobker, Hilmar. “Efficient multilevel solvers and high performance computing techniques for the finite element simulation of large-scale elasticity problems.” 2010. Thesis, Technische Universität Dortmund. Accessed March 02, 2021. http://hdl.handle.net/2003/26998.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

Wobker, Hilmar. “Efficient multilevel solvers and high performance computing techniques for the finite element simulation of large-scale elasticity problems.” 2010. Web. 02 Mar 2021.

Vancouver:

Wobker H. Efficient multilevel solvers and high performance computing techniques for the finite element simulation of large-scale elasticity problems. [Internet] [Thesis]. Technische Universität Dortmund; 2010. [cited 2021 Mar 02]. Available from: http://hdl.handle.net/2003/26998.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Wobker H. Efficient multilevel solvers and high performance computing techniques for the finite element simulation of large-scale elasticity problems. [Thesis]. Technische Universität Dortmund; 2010. Available from: http://hdl.handle.net/2003/26998

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

11. Rudi, Johann. Global convection in Earth's mantle : advanced numerical methods and extreme-scale simulations.

Degree: PhD, Computational Science, Engineering, and Mathematics, 2019, University of Texas – Austin

URL: http://dx.doi.org/10.26153/tsw/1258

► The thermal convection of rock in Earth's mantle and associated plate tectonics are modeled by nonlinear incompressible Stokes and energy equations. This dissertation focuses on… (more)

▼ The thermal convection of rock in Earth's mantle and associated plate tectonics are modeled by nonlinear incompressible Stokes and energy equations. This dissertation focuses on the development of advanced, scalable linear and nonlinear solvers for numerical simulations of realistic instantaneous mantle flow, where we must overcome several computational challenges. The most notable challenges are the severe nonlinearity, heterogeneity, and anisotropy due to the mantle's rheology as well as a wide range of spatial scales and highly localized features. Resolving the crucial small scale features efficiently necessitates adaptive methods, while computational results greatly benefit from a high accuracy per degree of freedom and local mass conservation. Consequently, the discretization of Earth's mantle is carried out by high-order finite elements on aggressively adaptively refined hexahedral meshes with a continuous, nodal velocity approximation and a discontinuous, modal pressure approximation. These velocity – pressure pairings yield optimal asymptotic convergence rates of the finite element approximation to the infinite-dimensional solution with decreasing mesh element size, are inf-sup stable on general, non-conforming hexahedral meshes with "hanging nodes,'' and have the advantage of preserving mass locally at the element level due to the discontinuous pressure. However, because of the difficulties cited above and the desired accuracy, the large implicit systems to be solved are extremely poorly conditioned and sophisticated linear and nonlinear solvers including powerful preconditioning techniques are required. The nonlinear Stokes system is solved using a grid continuation, inexact Newton – Krylov method. We measure the residual of the momentum equation in the H⁻¹-norm for backtracking line search to avoid overly conservative update steps that are significantly reduced from one. The Newton linearization is augmented by a perturbation of a highly nonlinear term in mantle's rheology, resulting in dramatically improved nonlinear convergence. We present a new Schur complement-based Stokes preconditioner, weighted BFBT, that exhibits robust fast convergence for Stokes problems with smooth but highly varying (up to 10 orders of magnitude) viscosities, optimal algorithmic scalability with respect to mesh refinement, and only a mild dependence on the polynomial order of high-order finite element discretizations. In addition, we derive theoretical eigenvalue bounds to prove spectral equivalence of our inverse Schur complement approximation. Finally, we present a parallel hybrid spectral – geometric – algebraic multigrid (HMG) to approximate the inverses of the Stokes system's viscous block and variable-coefficient pressure Poisson operators within weighted BFBT. Building on the parallel scalability of HMG, our Stokes solver demonstrates excellent parallel scalability to 1.6 million CPU cores without sacrificing algorithmic optimality. Advisors/Committee Members: Ghattas, Omar N. (advisor), Stadler, Georg, Ph. D. (advisor), Gurnis, Michael (committee member), Ren, Kui (committee member), Biros, George (committee member), Hesse, Marc (committee member).

Subjects/Keywords: Partial differential equations; Stokes; Numerical methods; Variable viscosity; Schur complement; Preconditioning; Multigrid; Scalable algorithms; Parallel computing; Iterative methods; Nonlinear problems; Inexact Newton-Krylov methods; Mantle convection

…42 5 Inexact Newton–Krylov Methods 45 5.1 Inexact Newton–Krylov methods for nonlinear… …77 7 Multigrid Preconditioning with HMG 80 7.1 An abstract multigrid method… …81 7.2 Hybrid spectral–geometric–algebraic multigrid (HMG)… …77 7.1 Hybrid spectral–geometric–algebraic multigrid (HMG) hierarchy and V… …104 8.2 Algorithmic scalability of inexact Newton–Krylov method…

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Rudi, J. (2019). Global convection in Earth's mantle : advanced numerical methods and extreme-scale simulations. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://dx.doi.org/10.26153/tsw/1258

Chicago Manual of Style (16^th Edition):

Rudi, Johann. “Global convection in Earth's mantle : advanced numerical methods and extreme-scale simulations.” 2019. Doctoral Dissertation, University of Texas – Austin. Accessed March 02, 2021. http://dx.doi.org/10.26153/tsw/1258.

MLA Handbook (7^th Edition):

Rudi, Johann. “Global convection in Earth's mantle : advanced numerical methods and extreme-scale simulations.” 2019. Web. 02 Mar 2021.

Vancouver:

Rudi J. Global convection in Earth's mantle : advanced numerical methods and extreme-scale simulations. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2019. [cited 2021 Mar 02]. Available from: http://dx.doi.org/10.26153/tsw/1258.

Council of Science Editors:

Rudi J. Global convection in Earth's mantle : advanced numerical methods and extreme-scale simulations. [Doctoral Dissertation]. University of Texas – Austin; 2019. Available from: http://dx.doi.org/10.26153/tsw/1258

12. Wobker, Hilmar. Efficient multilevel solvers and high performance computing techniques for the finite element simulation of large-scale elasticity problems.

Degree: 2010, Technische Universität Dortmund

URL: http://dx.doi.org/10.17877/DE290R-497

► In the simulation of realistic solid mechanical problems, linear equation systems with hundreds of million unknowns can arise. For the efficient solution of such systems,… (more)

▼ In the simulation of realistic solid mechanical problems, linear equation systems with hundreds of million unknowns can arise. For the efficient solution of such systems, parallel multilevel methods are mandatory that are able to exploit the capabilities of modern hardware technologies. The finite element and solution toolbox FEAST, which is designed to solve scalar equations, pursues exactly this goal. It combines hardware-oriented implementation techniques with a multilevel domain decomposition method called ScaRC that achieves high numerical and parallel efficiency. In this thesis a concept is developed to solve multivariate elasticity problems based on the FEAST library. The general strategy is to reduce the solution of multivariate problems to the solution of a series of scalar problems. This approach facilitates a strict separation of 'low level' scalar kernel functionalities (in the form of the FEAST library) and 'high level' multivariate application code (in the form of the elasticity problem), which is very attractive from a software-engineering point of view: All efforts to improve hardware-efficiency and adaptations to future technology trends can be restricted to scalar operations, and the multivariate application automatically benefits from these enhancements. In the first part of the thesis, substantial improvements of the scalar ScaRC solvers are developed, which are then used as essential building blocks for the efficient solution of multivariate elasticity problems. Extensive numerical studies demonstrate how the efficiency of the scalar FEAST library transfers to the multivariate solution process. The solver strategy is then applied to treat nonlinear problems of finite deformation elasticity. A line-search method is used to significantly increase the robustness of the Newton-Raphson method, and different strategies are compared how to choose the accuracy of the linear system solves within the nonlinear iteration. In order to treat the important class of (nearly) incompressible material, a mixed displacement/pressure formulation is used which is discretised with stabilised bilinear finite elements (Q1/Q1). An enhanced version of the classical 'pressure Poisson' stabilisation is presented which is suitable for highly irregular meshes. Advantages and disadvantages of the Q1/Q1 discretisation are discussed, especially in the context of transient computations. Two solver classes for the resulting saddle point systems are described and compared: on the one hand various kinds of (accelerated) segregated solvers (Uzawa, pressure Schur complement methods, block preconditioners), and on the other hand coupled multigrid solvers with Vanka-smoothers. Efficient Schur complement preconditioners, which are required for the former class, are discussed for the static and the transient case. The main strategy to reduce the solution of multivariate systems to the solution of scalar systems is only applicable in the case of segregated methods. It is shown that for the class of elasticity problems considered in this… Advisors/Committee Members: Turek, S. (advisor), Suttmeier, F.-T. (referee).

Subjects/Keywords: Iterativer Löser; Multilevel; Mehrgitter; Gebietszerlegung; Mehrgitter-Krylov; Nicht-konformes Mehrgitter; ScaRC; Adaptive Grobgitterkorrektur; Minimale Überlappung; Sattelpunkt-Problem; Schurkomplement-Vorkonditionierer; Vanka; Gedämpftes Newton-Raphson; Globales Newton-Raphson; Inexaktes Newton-Raphson; Liniensuche; FEAST; Hardware-orientiert; Großskalig; Paralleles Rechnen; Parallele Effizienz; Finite-Elemente-Methode; Gemischte Formulierung; LBB Stabilisierung; Irreguläres Gitter; Festkörpermechanik; Strukturmechanik; Elastizität; Elastostatisch; Elastodynamisch; Zeitabhängig; Inkompressibles Material; Finite Deformation; Große Deformation; Volumenversteifung; Schubversteifung; Iterative solver; Multilevel; Multigrid; Domain decomposition; Multigrid-Krylov; Nonconforming multigrid; ScaRC; Adaptive coarse grid correction; Minimal overlap; Saddle point problem; Schur complement preconditioning; Vanka; Newton-Raphson; Damped Newton-Raphson; Global Newton-Raphson; Inexact Newton-Raphson; Line-search; FEAST; High performance computing; Hardware-oriented; Large-scale; Parallel computing; Parallel efficiency; Finite element method; Mixed formulation; LBB stabilisation; Equal-order finite elements; Irregular grids; Solid mechanics; Structural mechanics; Elasticity; Elastostatic; Elastodynamic; Transient; Incompressible material; Finite deformation; Large deformation; Volume locking; Shear locking; 510

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APA (6^th Edition):

Wobker, H. (2010). Efficient multilevel solvers and high performance computing techniques for the finite element simulation of large-scale elasticity problems. (Doctoral Dissertation). Technische Universität Dortmund. Retrieved from http://dx.doi.org/10.17877/DE290R-497

Chicago Manual of Style (16^th Edition):

Wobker, Hilmar. “Efficient multilevel solvers and high performance computing techniques for the finite element simulation of large-scale elasticity problems.” 2010. Doctoral Dissertation, Technische Universität Dortmund. Accessed March 02, 2021. http://dx.doi.org/10.17877/DE290R-497.

MLA Handbook (7^th Edition):

Wobker, Hilmar. “Efficient multilevel solvers and high performance computing techniques for the finite element simulation of large-scale elasticity problems.” 2010. Web. 02 Mar 2021.

Vancouver:

Wobker H. Efficient multilevel solvers and high performance computing techniques for the finite element simulation of large-scale elasticity problems. [Internet] [Doctoral dissertation]. Technische Universität Dortmund; 2010. [cited 2021 Mar 02]. Available from: http://dx.doi.org/10.17877/DE290R-497.

Council of Science Editors:

Wobker H. Efficient multilevel solvers and high performance computing techniques for the finite element simulation of large-scale elasticity problems. [Doctoral Dissertation]. Technische Universität Dortmund; 2010. Available from: http://dx.doi.org/10.17877/DE290R-497

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