Computation of scattering from clusters of spheres using the fast multipole method

Area of Science:

• Computational physics
• Acoustics
• Electromagnetics

Background:

• Traditional T-matrix methods for multiple scattering are computationally intensive, scaling as O(N^3) with particle number (N) and O(k^6) with wave number (k).
• Previous work accelerated solutions using iterative techniques with the fast multipole method (FMM), but further optimization is needed for large-scale problems.

Purpose of the Study:

• To develop a significantly more efficient computational method for solving the multiple scattering problem using the T-matrix approach.
• To reduce the computational complexity of matrix-vector multiplication from O(N^3) to O(N log N) and the wave number dependence from O(k^6) to O(k^3).

Main Methods:

• Integration of preconditioned Krylov subspace iterative techniques.
• Utilization of fast multipole method (FMM) for accelerated matrix-vector products.
• Development of a novel FMM-based preconditioner and fast translation techniques.

Main Results:

• Achieved an overall algorithm with matrix-vector multiplication cost scaling as O(N log N) and O(k^3).
• Successfully applied the method to solve test problems for N ranging from approximately 10^2 to 10^4.
• Demonstrated efficient solutions for multiple scattering problems across various wave numbers.

Conclusions:

• The developed method offers a substantial improvement in computational efficiency for T-matrix based multiple scattering solutions.
• This approach enables the analysis of larger and more complex scattering problems than previously feasible.
• Further investigation into convergence, error analysis, and parameter selection is discussed.