| aminer68@gmail.com: Aug 07 05:26PM -0700 Hello, Read again, i correct about SOR and Conjugate gradient mathematical methods.. I am a white arab, and i think i am smart since i have invented many scalable algorithms.. About SOR and Conjugate gradient mathematical methods.. I have just looked at SOR(Successive Overrelaxation Method), and i think it is much less powerful than Conjugate gradient method, read the following to notice it: COMPARATIVE PERFORMANCE OF THE CONJUGATE GRADIENT AND SOR METHODS FOR COMPUTATIONAL THERMAL HYDRAULICS https://inis.iaea.org/collection/NCLCollectionStore/_Public/19/055/19055644.pdf?r=1&r=1 This is why i have implemented in both C++ and Delphi my Parallel Conjugate Gradient Linear System Solver Library that scales very well, read my following thoughts about it to understand more: About the convergence properties of the conjugate gradient method The conjugate gradient method can theoretically be viewed as a direct method, as it produces the exact solution after a finite number of iterations, which is not larger than the size of the matrix, in the absence of round-off error. However, the conjugate gradient method is unstable with respect to even small perturbations, e.g., most directions are not in practice conjugate, and the exact solution is never obtained. Fortunately, the conjugate gradient method can be used as an iterative method as it provides monotonically improving approximations to the exact solution, which may reach the required tolerance after a relatively small (compared to the problem size) number of iterations. The improvement is typically linear and its speed is determined by the condition number κ(A) of the system matrix A: the larger is κ(A), the slower the improvement. Read more here: http://pages.stat.wisc.edu/~wahba/stat860public/pdf1/cj.pdf So i think my Conjugate Gradient Linear System Solver Library that scales very well is still very useful, read about it in my writing below: Read the following interesting news: The finite element method finds its place in games Read more here: https://translate.google.com/translate?hl=en&sl=auto&tl=en&u=https%3A%2F%2Fhpc.developpez.com%2Factu%2F288260%2FLa-methode-des-elements-finis-trouve-sa-place-dans-les-jeux-AMD-propose-la-bibliotheque-FEMFX-pour-une-simulation-en-temps-reel-des-deformations%2F But you have to be aware that finite element method uses Conjugate Gradient Method for Solution of Finite Element Problems, read here to notice it: Conjugate Gradient Method for Solution of Large Finite Element Problems on CPU and GPU https://pdfs.semanticscholar.org/1f4c/f080ee622aa02623b35eda947fbc169b199d.pdf This is why i have also designed and implemented my Parallel Conjugate Gradient Linear System Solver library that scales very well, here it is: My Parallel C++ Conjugate Gradient Linear System Solver Library that scales very well version 1.76 is here.. Author: Amine Moulay Ramdane Description: This library contains a Parallel implementation of Conjugate Gradient Dense Linear System Solver library that is NUMA-aware and cache-aware that scales very well, and it contains also a Parallel implementation of Conjugate Gradient Sparse Linear System Solver library that is cache-aware that scales very well. Sparse linear system solvers are ubiquitous in high performance computing (HPC) and often are the most computational intensive parts in scientific computing codes. A few of the many applications relying on sparse linear solvers include fusion energy simulation, space weather simulation, climate modeling, and environmental modeling, and finite element method, and large-scale reservoir simulations to enhance oil recovery by the oil and gas industry. Conjugate Gradient is known to converge to the exact solution in n steps for a matrix of size n, and was historically first seen as a direct method because of this. However, after a while people figured out that it works really well if you just stop the iteration much earlier - often you will get a very good approximation after much fewer than n steps. In fact, we can analyze how fast Conjugate gradient converges. The end result is that Conjugate gradient is used as an iterative method for large linear systems today. Please download the zip file and read the readme file inside the zip to know how to use it. You can download it from: https://sites.google.com/site/scalable68/scalable-parallel-c-conjugate-gradient-linear-system-solver-library Language: GNU C++ and Visual C++ and C++Builder Operating Systems: Windows, Linux, Unix and Mac OS X on (x86) -- As you have noticed i have just written above about my Parallel C++ Conjugate Gradient Linear System Solver Library that scales very well, but here is my Parallel Delphi and Freepascal Conjugate Gradient Linear System Solvers Libraries that scale very well: Parallel implementation of Conjugate Gradient Dense Linear System solver library that is NUMA-aware and cache-aware that scales very well https://sites.google.com/site/scalable68/scalable-parallel-implementation-of-conjugate-gradient-dense-linear-system-solver-library-that-is-numa-aware-and-cache-aware PARALLEL IMPLEMENTATION OF CONJUGATE GRADIENT SPARSE LINEAR SYSTEM SOLVER LIBRARY THAT SCALES VERY WELL https://sites.google.com/site/scalable68/scalable-parallel-implementation-of-conjugate-gradient-sparse-linear-system-solver Thank you, Amine Moulay Ramdane. |
| aminer68@gmail.com: Aug 07 04:56PM -0700 Hello, I am a white arab, and i think i am smart since i have invented many scalable algorithms.. About SOR and Conjugate gradient mathematical methods.. I have just looked at SOR(Successive Overrelaxation Method), and i think it is much less powerful than Conjugate gradient method, read the following to notice it: COMPARATIVE PERFORMANCE OF THE CONJUGATE GRADIENT AND SOR METHODS FOR COMPUTATIONAL THERMAL HYDRAULICS https://inis.iaea.org/collection/NCLCollectionStore/_Public/19/055/19055644.pdf?r=1&r=1 This is why i have implemented in both C++ and Delphi my Conjugate Gradient Linear System Solver Library that scales very well, read my following thoughts about it to understand more: About the convergence properties of the conjugate gradient method The conjugate gradient method can theoretically be viewed as a direct method, as it produces the exact solution after a finite number of iterations, which is not larger than the size of the matrix, in the absence of round-off error. However, the conjugate gradient method is unstable with respect to even small perturbations, e.g., most directions are not in practice conjugate, and the exact solution is never obtained. Fortunately, the conjugate gradient method can be used as an iterative method as it provides monotonically improving approximations to the exact solution, which may reach the required tolerance after a relatively small (compared to the problem size) number of iterations. The improvement is typically linear and its speed is determined by the condition number κ(A) of the system matrix A: the larger is κ(A), the slower the improvement. Read more here: http://pages.stat.wisc.edu/~wahba/stat860public/pdf1/cj.pdf So i think my Conjugate Gradient Linear System Solver Library that scales very well is still very useful, read about it in my writing below: Read the following interesting news: The finite element method finds its place in games Read more here: https://translate.google.com/translate?hl=en&sl=auto&tl=en&u=https%3A%2F%2Fhpc.developpez.com%2Factu%2F288260%2FLa-methode-des-elements-finis-trouve-sa-place-dans-les-jeux-AMD-propose-la-bibliotheque-FEMFX-pour-une-simulation-en-temps-reel-des-deformations%2F But you have to be aware that finite element method uses Conjugate Gradient Method for Solution of Finite Element Problems, read here to notice it: Conjugate Gradient Method for Solution of Large Finite Element Problems on CPU and GPU https://pdfs.semanticscholar.org/1f4c/f080ee622aa02623b35eda947fbc169b199d.pdf This is why i have also designed and implemented my Parallel Conjugate Gradient Linear System Solver library that scales very well, here it is: My Parallel C++ Conjugate Gradient Linear System Solver Library that scales very well version 1.76 is here.. Author: Amine Moulay Ramdane Description: This library contains a Parallel implementation of Conjugate Gradient Dense Linear System Solver library that is NUMA-aware and cache-aware that scales very well, and it contains also a Parallel implementation of Conjugate Gradient Sparse Linear System Solver library that is cache-aware that scales very well. Sparse linear system solvers are ubiquitous in high performance computing (HPC) and often are the most computational intensive parts in scientific computing codes. A few of the many applications relying on sparse linear solvers include fusion energy simulation, space weather simulation, climate modeling, and environmental modeling, and finite element method, and large-scale reservoir simulations to enhance oil recovery by the oil and gas industry. Conjugate Gradient is known to converge to the exact solution in n steps for a matrix of size n, and was historically first seen as a direct method because of this. However, after a while people figured out that it works really well if you just stop the iteration much earlier - often you will get a very good approximation after much fewer than n steps. In fact, we can analyze how fast Conjugate gradient converges. The end result is that Conjugate gradient is used as an iterative method for large linear systems today. Please download the zip file and read the readme file inside the zip to know how to use it. You can download it from: https://sites.google.com/site/scalable68/scalable-parallel-c-conjugate-gradient-linear-system-solver-library Language: GNU C++ and Visual C++ and C++Builder Operating Systems: Windows, Linux, Unix and Mac OS X on (x86) -- As you have noticed i have just wrote above my Parallel C++ Conjugate Gradient Linear System Solver Library that scales very well, but here is my Parallel Delphi and Freepascal Conjugate Gradient Linear System Solvers Libraries that scale very well: Parallel implementation of Conjugate Gradient Dense Linear System solver library that is NUMA-aware and cache-aware that scales very well https://sites.google.com/site/scalable68/scalable-parallel-implementation-of-conjugate-gradient-dense-linear-system-solver-library-that-is-numa-aware-and-cache-aware PARALLEL IMPLEMENTATION OF CONJUGATE GRADIENT SPARSE LINEAR SYSTEM SOLVER LIBRARY THAT SCALES VERY WELL https://sites.google.com/site/scalable68/scalable-parallel-implementation-of-conjugate-gradient-sparse-linear-system-solver Thank you, Amine Moulay Ramdane. |
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