Efficient preconditioning strategies for accelerating GMRES in block-structured nonlinear systems for image
Rizwan Khalid1, Shahbaz Ahmad1, Mohamed Medani2
1Abdus Salam School of Mathematical Sciences, Government College University, Lahore, Pakistan.
Abstract:
We propose an efficient preconditioning strategy to accelerate the convergence of Krylov subspace methods, specifically for solving complex nonlinear systems with a block five-by-five structure, commonly found in cell-centered finite difference discretizations for image deblurring using mean curvature techniques. Our method introduces two innovative preconditioned matrices, analyzed spectrally to show a favorable eigenvalue distribution that accelerates convergence in the Generalized Minimal Residual (GMRES) method. This technique significantly improves image quality, as measured by peak signal-to-noise ratio (PSNR), and demonstrates faster convergence compared to traditional GMRES, requiring minimal CPU time and few iterations for exceptional deblurring performance. The preconditioned matrices' eigenvalues cluster around 1, indicating a beneficial spectral distribution. The source code is available at https://github.com/shahbaz1982/Precondition-Matrix.
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