Projects in Sparse Grids and High Dimensional Approximation

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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.
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

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 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

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 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