ML
M. Looije
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Isogeometric Analysis (IgA) is an extension of the more well known Finite Element Method (FEM). It allows for more accurate descriptions of boundary value problems on irregular domains. However, many of the traditional iterative solution strategies that are known to work well in FEM do not show the same behavior in IgA, especially for increasing order of basis functions p.
A method shown to have fast convergence in this situation is a p-multigrid method with a smoother based on a Block ILUT factorization. Most of the blocks of this factorization are efficiently calculated. The same holds for the smoothing steps.
It is therefore our objective to make changes to the Block ILUT smoother. Inspiration is taken from methods where ILU factorizations are constructed using a fixed-point iteration. We combine these with the existing Block ILUT smoother.
This ultimately leads to two new proposed methods we will call Block Fixed-point ILU and Block ParILUT.
The existing methods as well as the newly suggested methods are tested and compared on computational costs of the factorization, the number of nonzero entries in this factorization and the number of multigrid iterations needed
to reach convergence, if these factorizations are to be used as smoother. The benchmark used for these tests is a convection diffusion reaction (CDR) equation on a multipatch geometry with 4, 16 or 64 patches.
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A method shown to have fast convergence in this situation is a p-multigrid method with a smoother based on a Block ILUT factorization. Most of the blocks of this factorization are efficiently calculated. The same holds for the smoothing steps.
It is therefore our objective to make changes to the Block ILUT smoother. Inspiration is taken from methods where ILU factorizations are constructed using a fixed-point iteration. We combine these with the existing Block ILUT smoother.
This ultimately leads to two new proposed methods we will call Block Fixed-point ILU and Block ParILUT.
The existing methods as well as the newly suggested methods are tested and compared on computational costs of the factorization, the number of nonzero entries in this factorization and the number of multigrid iterations needed
to reach convergence, if these factorizations are to be used as smoother. The benchmark used for these tests is a convection diffusion reaction (CDR) equation on a multipatch geometry with 4, 16 or 64 patches.
...
Isogeometric Analysis (IgA) is an extension of the more well known Finite Element Method (FEM). It allows for more accurate descriptions of boundary value problems on irregular domains. However, many of the traditional iterative solution strategies that are known to work well in FEM do not show the same behavior in IgA, especially for increasing order of basis functions p.
A method shown to have fast convergence in this situation is a p-multigrid method with a smoother based on a Block ILUT factorization. Most of the blocks of this factorization are efficiently calculated. The same holds for the smoothing steps.
It is therefore our objective to make changes to the Block ILUT smoother. Inspiration is taken from methods where ILU factorizations are constructed using a fixed-point iteration. We combine these with the existing Block ILUT smoother.
This ultimately leads to two new proposed methods we will call Block Fixed-point ILU and Block ParILUT.
The existing methods as well as the newly suggested methods are tested and compared on computational costs of the factorization, the number of nonzero entries in this factorization and the number of multigrid iterations needed
to reach convergence, if these factorizations are to be used as smoother. The benchmark used for these tests is a convection diffusion reaction (CDR) equation on a multipatch geometry with 4, 16 or 64 patches.
A method shown to have fast convergence in this situation is a p-multigrid method with a smoother based on a Block ILUT factorization. Most of the blocks of this factorization are efficiently calculated. The same holds for the smoothing steps.
It is therefore our objective to make changes to the Block ILUT smoother. Inspiration is taken from methods where ILU factorizations are constructed using a fixed-point iteration. We combine these with the existing Block ILUT smoother.
This ultimately leads to two new proposed methods we will call Block Fixed-point ILU and Block ParILUT.
The existing methods as well as the newly suggested methods are tested and compared on computational costs of the factorization, the number of nonzero entries in this factorization and the number of multigrid iterations needed
to reach convergence, if these factorizations are to be used as smoother. The benchmark used for these tests is a convection diffusion reaction (CDR) equation on a multipatch geometry with 4, 16 or 64 patches.
Voor grote random matrices bestaat er een verband tussen de eigenwaarden van deze random matrices en halve cirkels. Om meer precies te zijn bestaan er voor random matrices die aan een aantal voorwaarden voldoen een vorm van convergentie tussen de verdeling van eigenwaarden van de matrices en een halve cirkel verdeling. Een stelling die deze convergentie beschrijft is Wigners halve cirkel stelling. Het doel van dit project is het begrijpen van deze stelling en zijn bewijs om het vervolgens op een voor andere studenten toegankelijke manier te reproduceren.
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Voor grote random matrices bestaat er een verband tussen de eigenwaarden van deze random matrices en halve cirkels. Om meer precies te zijn bestaan er voor random matrices die aan een aantal voorwaarden voldoen een vorm van convergentie tussen de verdeling van eigenwaarden van de matrices en een halve cirkel verdeling. Een stelling die deze convergentie beschrijft is Wigners halve cirkel stelling. Het doel van dit project is het begrijpen van deze stelling en zijn bewijs om het vervolgens op een voor andere studenten toegankelijke manier te reproduceren.