4.2 Successive Over-Relaxation (SOR)

Successive Over-Relaxation (SOR) is a powerful method for solving large linear systems. It builds on the Gauss-Seidel method, adding a relaxation parameter to speed up convergence. This technique is especially useful for systems arising from discretized partial differential equations.

SOR's effectiveness lies in its ability to balance between previous solutions and new updates. By carefully choosing the relaxation parameter, SOR can dramatically reduce the number of iterations needed compared to other methods, making it a go-to choice for many large-scale problems.

Successive Over-Relaxation (SOR)

Concept and Application

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• SOR accelerates iterative methods for solving large linear equation systems
• Variant of Gauss-Seidel method introduces relaxation parameter ω to improve convergence speed
• Combines previous iteration's solution with Gauss-Seidel solution using weighted average
• Relaxation parameter ω optimizes convergence rate (typically 0 < ω < 2)
• Effective for large, sparse systems from discretized partial differential equations (finite difference methods)
• Applies to symmetric and non-symmetric systems with different optimal relaxation parameters
• Extends Gauss-Seidel by introducing overrelaxation factor to extrapolate the change made by Gauss-Seidel
• Can significantly reduce number of iterations required for convergence compared to standard methods

Mathematical Formulation

• SOR iteration formula derived from Gauss-Seidel method with relaxation parameter ω
• Splits coefficient matrix A into diagonal (D), strictly lower (L), and strictly upper (U) triangular parts
• Expresses iteration as $x^{(k+1)} = (1-ω)x^{(k)} + ω(D+ωL)^{-1}(b - Ux^{(k)})$
• Requires solving linear system $(D+ωL)x^{(k+1)} = ωb + (ωU + (ω-1)D)x^{(k)}$ at each iteration
• Generalizes to $x^{(k+1)} = x^{(k)} + ωD^{-1}(b - Ax^{(k)})$ for point-wise updates
• Introduces parameter ω to balance between previous solution and Gauss-Seidel update

Practical Considerations

• Efficient implementation uses forward substitution to solve system at each iteration
• Convergence check based on specified tolerance or maximum iteration count
• Special treatment needed for boundary conditions in discretized PDE applications
• Choice of ω crucial for performance (too small: slow convergence, too large: divergence)
• Can be adapted for parallel computing environments with appropriate modifications
• Often combined with other techniques (preconditioning, multigrid methods) for enhanced performance

SOR Implementation for Linear Systems

Algorithm Structure

• Initialize solution vector x^(0) (often set to zero or an initial guess)
• Set relaxation parameter ω (0 < ω < 2)
• Iterate until convergence or maximum iterations reached:
  • Compute new solution components using SOR formula
  • Update solution vector
  • Check convergence criteria
• Implement efficient forward substitution to solve (D+ωL)x^(k+1) = ωb + (ωU + (ω-1)D)x^(k)
• Use sparse matrix storage formats (CSR, CSC) for large systems to reduce memory usage
• Incorporate convergence checks (relative residual, absolute change in solution)

Code Implementation

• Pseudocode for basic SOR iteration:

for k = 1, 2, ..., until convergence
    for i = 1, 2, ..., n
        sigma = 0
        for j = 1, 2, ..., i-1
            sigma += a[i][j] * x_new[j]
        for j = i+1, ..., n
            sigma += a[i][j] * x_old[j]
        x_new[i] = (1 - omega) * x_old[i] + omega * (b[i] - sigma) / a[i][i]
    check convergence

• Efficient implementation avoids explicit matrix splitting
• Utilize BLAS (Basic Linear Algebra Subprograms) for optimized vector operations
• Implement parallel versions using OpenMP or MPI for shared or distributed memory systems

Optimization Techniques

• Use red-black ordering for certain problem types to enable parallelization
• Implement adaptive ω selection to improve convergence throughout iteration process
• Combine SOR with multigrid methods for highly efficient solvers (Full Multigrid V-cycle)
• Apply SOR as a smoother in multigrid methods for elliptic PDEs
• Utilize block SOR for systems with natural block structure (multiple unknowns per grid point)
• Implement SOR preconditioner for Krylov subspace methods (GMRES, BiCGSTAB)

Optimal Relaxation Parameter for SOR

Analytical Determination

• Optimal ω_opt depends on spectral properties of iteration matrix
• For symmetric positive definite matrices, estimate using Jacobi iteration matrix spectral radius
• Formula: $ω_{opt} = \frac{2}{1 + \sqrt{1 - ρ(B_J)^2}}$ where ρ(B_J) is Jacobi iteration matrix spectral radius
• Spectral radius ρ(B_J) can be approximated for specific problem classes (Poisson equation)
• For 2D Poisson equation with h step size: $ω_{opt} ≈ \frac{2}{1 + \sin(πh)}$
• Theoretical optimal ω minimizes spectral radius of SOR iteration matrix

Numerical Estimation

• Experimental determination involves running SOR with different ω values
• Compare convergence rates for various ω to find optimal value
• Use techniques like bisection or golden section search to narrow ω range
• Adaptive methods estimate and update optimal ω during iteration process
• Dynamic ω selection based on monitoring residual reduction or solution change
• Chebyshev acceleration technique for estimating optimal ω in certain cases

Special Cases and Considerations

• Non-symmetric systems require different approaches for ω_opt determination
• Young's theory provides insights for consistently ordered matrices
• Block SOR methods may have different optimal ω than point SOR
• Consider problem-specific properties (matrix structure, boundary conditions) for ω selection
• Balance between optimal convergence rate and numerical stability in ω choice
• Sensitivity analysis of convergence rate to ω variations near optimal value

SOR Convergence vs Jacobi and Gauss-Seidel

Convergence Properties

• SOR converges for 0 < ω < 2 if coefficient matrix A is symmetric positive definite
• Convergence rate typically faster than Jacobi and Gauss-Seidel with optimal ω
• Spectral radius of SOR iteration matrix: $ρ(L_ω) = (ω - 1)$ when ω is optimal
• SOR can converge for systems where Jacobi and Gauss-Seidel fail or converge slowly
• Asymptotic convergence factor for SOR proportional to 1 - O(h) (h: mesh size)
• Gauss-Seidel asymptotic convergence factor proportional to 1 - O(h^2)
• SOR converges in O(N^(1/2)) iterations for 2D problems, compared to O(N) for Gauss-Seidel

Comparative Analysis

• SOR generally outperforms Jacobi and Gauss-Seidel in terms of iteration count
• Computational cost per iteration slightly higher for SOR due to ω calculations
• Memory requirements similar for all three methods
• Jacobi method easier to parallelize compared to Gauss-Seidel and SOR
• SOR more sensitive to choice of relaxation parameter than Jacobi or Gauss-Seidel
• Gauss-Seidel equivalent to SOR with ω = 1
• SOR can be viewed as extrapolation between Gauss-Seidel iterate and previous iterate

Problem-Specific Considerations

• SOR effectiveness can be problem-dependent
• For some systems, other methods (conjugate gradient) may be more efficient
• Choice between SOR, Jacobi, and Gauss-Seidel depends on problem structure
• Implementation considerations (parallelization, memory access patterns) affect method selection
• SOR particularly effective for elliptic PDEs and certain structured linear systems
• Hybrid methods combining SOR with other techniques (Krylov subspace methods) often used in practice
• Preconditioning can significantly impact relative performance of iterative methods