CS675_Project: A survey of Krylov Subspace methods for nonsymmetric linear systems.

Project in CS675 - Computational Linear Algebra

Features:

1. The folder 'Krylov_subspace_method' contains implementation of GMRES, BiCG, and IDR(s) methods.

2. The folder 'PoissonPDE' builds the finite difference scheme for {1D,2D,3D} for Poisson PDE equations with 2 heat sources temperature. The Laplacian returns the matrix A corresponds to the lhs of finite difference scheme and matrix b of source terms:

• Laplacian takes 2 inputs: m: number of discretized points corresponding to each dimensions and dim: 1D, 2D, 3D:
  • [A,b] = Laplacian(10, '1D') returns A is a 10-by-10 matrix and b is a 10-by-1 vector.
  • [A,b] = Laplacian(10, '2D') returns A is a 100-by-100 matrix and b is a 100-by-1 vector.
  • [A,b] = Laplacian(10, '3D') returns A is a 1000-by-1000 matrix and b is a 1000-by-1 vector.

3. The folder 'Image_Denoising' builds the PDE-constrained equation to denoise the image for grid size [64,128].

• The FormMatrix is to form the Total Variation Scheme Matrix of PDE-constrained equation.
• The Denoise is to run the Krylov subspace method to denoise the image for a given grid size.
• The DenoiseFunc is to run the algorithm of image denoising.

References:

@book{saad_book,
author = {Saad, Y.},
title = {Iterative Methods for Sparse Linear Systems},
year = {2003},
isbn = {0898715342},
publisher = {Society for Industrial and Applied Mathematics},
address = {USA},
edition = {2nd}
}


@article{Saad1986GMRESAG,
  title={GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems},
  author={Y. Saad and M. Schultz},
  journal={Siam Journal on Scientific and Statistical Computing},
  year={1986},
  volume={7},
  pages={856-869}
}

@article{idr_method,
author = {Sonneveld, Peter and Gijzen, M.B.},
year = {2008},
month = {01},
pages = {1035-1062},
title = {IDR( s ): A Family of Simple and Fast Algorithms for Solving Large Nonsymmetric Systems of Linear Equations},
volume = {31},
journal = {SIAM J. Scientific Computing},
doi = {10.1137/070685804}
}