Project Description
The far-field expansion is the key part of the treecode algorithm. In the current used treecode algorithm, Taylor expansion with recurrence relation is the one that has been used. However, the recurrence relation for computing the Taylor coefficients efficiently is not available for a general kernel function, but polynomial interpolation is always doable. The PI propose to develop polynomial interpolation based far-field approximation. Here is the objective and the plan: (1) the convergence rate of polynomial approximation; comparison with Taylor approximation; (2) stabilities. High order polynomial interpolation always has stabilities issues, but it can be improved with Horner's rule, and different forms (Lagrange, Newton, baycentric) of polynomial interpolation shows different performance. (3) efficiency. As part of the treecode algorithm, whose main goal is to save CPU time and memory usage, the We will also investigate the efficiency of the polynomial far-field.
Tasks and Responsibilites
Tasks and responsibilities are: (1) Write Matlab code to test the conversgence of the polynimal far-field expansion with one particle one cluster interaction, with different interpolation points (uniform points, Chebyshev points), starting from a 1 dimension problem, then extend it to a 3D cluster. Compare the conergence rate with the Taylor approximation. (2) Stable the algorithm with Horner's rule, implement different forms of the polynomial interpolation, and compare the performance. (3) Optimize the Matlab code to record the CPU time of the polynomial interpolation and Compare with the Taylor aprpoximation.
Desired Qualifications
For advanced undergraduates.