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. 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