Difference between revisions of "Projects in Sparse Grids and High Dimensional Approximation"

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| Coupling a general purpose PDE solver with a Combination Technique Framework
 
| Coupling a general purpose PDE solver with a Combination Technique Framework
|The solution of high-dimensional problems using grid based methods is a numerically demanding task with one reason being the curse of dimensionality. The sparse grid combination technique has been successfully applied for solving elliptic PDE's as the Poisson equation and hyperbolic problems as the advection equation. Nevertheless its coupling to general purpose PDE solvers will allow to test many different PDE's and to identify possible application areas. <br/> The work will include a comprehensive literature study and comparison of existing PDE solver frameworks, their coupling to the existing combination technique framework written in python, and a study of the numerical errors introduced by the combination technique for each PDE
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|The solution of high-dimensional problems using grid-based methods is a numerically demanding task. One reason is the curse of dimensionality. The sparse grid combination technique has been successfully applied to the solution of elliptic PDEs (such as the Poisson equation) and hyperbolic PDEs (such as the advection equation). Coupling it to general purpose PDE solvers will allow to test many different PDEs and to identify possible application areas. <br/> The work will include a comprehensive literature study and a comparison of existing PDE solver frameworks, their coupling to the existing combination technique framework written in Python, and a study of the numerical errors introduced by the combination technique for each PDE problem.
problem.
 
 
| [[Alfredo_Parra_Hinojosa,_M.Sc.|Alfredo Parra]]
 
| [[Alfredo_Parra_Hinojosa,_M.Sc.|Alfredo Parra]]
 
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| Applying the Optimized Sparse Grid Combination Technique on the Schroedinger Equation
 
| Applying the Optimized Sparse Grid Combination Technique on the Schroedinger Equation
|The sparse grid combination technique has proved to be a viable method for solving high-dimensional PDE problems like the Schroedinger equation. Therefore, a large eigenvalue problem is solved on meshes of varying resolution. All of these approximation are then combined to a single approximation. The computational effort of computing all partial approximations is substantially smaller than obtaining a single finely resolved approximation. A new method for obtaining
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|The sparse grid combination technique has proved to be a viable method for solving high-dimensional PDE problems such as the Schroedinger equation. With the combination technique one solves a large eigenvalue problem on various grids of varying resolution. The solutions on these various grids are then combined to approximate a high-resolution grid. The computational effort of computing the solution on the different grids is substantially smaller than obtaining a single, finely-resolved approximation. A new method for solving eigenvalue problems with the combination technique has been developed. Its performance for the Schroedinger equation will have to be compared with existing regular and combination technique approaches.<br/> The work includes the implementation of a solver of the Schroedinger equation on varying non-equidistant meshes. After its validation it will be used with the various traditional and new combination techniques for eigenvalue problems, where the existing results must be confirmed and the quality of the new method for solving eigenvalue problems will be evaluated.
solving eigenvalue problems with the combination technique has been developed. Its performance for the Schroedinger equation will have to be compared with existing regular and combination technique approaches.<br/> The work includes the implementation of a solver of the Schroedinger equation on varying non-equidistant meshes. After its validation it will be used with the various traditional and new combination techniques for eigenvalue problems, where the existing results shall, on the one hand, be confirmed and the quality of the new method for solving eigenvalue problems shall, on the other hand, be evaluated.
 
 
| [[Alfredo_Parra_Hinojosa,_M.Sc.|Alfredo Parra]]
 
| [[Alfredo_Parra_Hinojosa,_M.Sc.|Alfredo Parra]]
 
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Revision as of 15:26, 28 January 2016

Topic Description Contact
Coupling a general purpose PDE solver with a Combination Technique Framework The solution of high-dimensional problems using grid-based methods is a numerically demanding task. One reason is the curse of dimensionality. The sparse grid combination technique has been successfully applied to the solution of elliptic PDEs (such as the Poisson equation) and hyperbolic PDEs (such as the advection equation). Coupling it to general purpose PDE solvers will allow to test many different PDEs and to identify possible application areas.
The work will include a comprehensive literature study and a comparison of existing PDE solver frameworks, their coupling to the existing combination technique framework written in Python, and a study of the numerical errors introduced by the combination technique for each PDE problem.
Alfredo Parra
Applying the Optimized Sparse Grid Combination Technique on the Schroedinger Equation The sparse grid combination technique has proved to be a viable method for solving high-dimensional PDE problems such as the Schroedinger equation. With the combination technique one solves a large eigenvalue problem on various grids of varying resolution. The solutions on these various grids are then combined to approximate a high-resolution grid. The computational effort of computing the solution on the different grids is substantially smaller than obtaining a single, finely-resolved approximation. A new method for solving eigenvalue problems with the combination technique has been developed. Its performance for the Schroedinger equation will have to be compared with existing regular and combination technique approaches.
The work includes the implementation of a solver of the Schroedinger equation on varying non-equidistant meshes. After its validation it will be used with the various traditional and new combination techniques for eigenvalue problems, where the existing results must be confirmed and the quality of the new method for solving eigenvalue problems will be evaluated.
Alfredo Parra