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Reducing the condition number for microlocal discretization problems
Armel de La Bourdonnaye1, Marc Tolentino
1Caiman Project, INRIA--CERMICS, BP 93 06902 Sophia-Antipolis CEDEX, France. armel.de-la-bourdonnaye@equipement.gouv.fr
Summary
Researchers improved wave problem simulations by using wavelet transforms to reduce system size and enhance conditioning. This novel spectral domain approach significantly improves computational efficiency for scattering problems.
Area of Science:
- Computational physics
- Numerical analysis
- Wave propagation
Background:
- Microlocal discretizations commonly use oscillating basis functions for harmonic wave problems.
- These discretizations often result in ill-conditioned linear systems due to over-discretization and evanescent waves.
Purpose of the Study:
- To analyze and improve the conditioning of linear systems from microlocal discretizations.
- To develop a more efficient method for simulating wave scattering problems.
Main Methods:
- Investigated projecting the system onto the orthogonal of evanescent modes (limited success).
- Proposed transforming the linear system into a wavelet basis in the spectral domain.
- Applied thresholding to the transformed system to reduce matrix coefficients.
Main Results:
- Wavelet transformation effectively discriminates between large and small matrix coefficients.
- Thresholding the transformed system yields a smaller and better-conditioned system.
- The developed method requires only 1-2 degrees of freedom per wavelength for simulations.
Conclusions:
- Wavelet-based spectral domain transformation offers an original and effective solution for ill-conditioned microlocal discretization systems.
- This approach significantly enhances computational efficiency for simulating wave scattering problems.