Question

Define a linear problem with \(n=500\) using the script \(A=\operatorname{randn}(500,500) ; x t=\operatorname{randn}(500,1) ; b=A * x t\) Now we save xt away and solve \(A \mathbf{x}=\mathbf{b} .\) Set tol \(=1 . \mathrm{e}-6\) and maximum iteration limit of 2000\. Run three solvers for this problem: (a) \(\mathrm{CG}\) on the normal equations: \(A^{T} A \mathbf{x}=A^{T} \mathbf{b}\). (b) GMRES(500). (c) GMRES(100), i.e., restarted GMRES with \(m=100\). Record residual norm and solution error norm for each run. What are your conclusions?

Short answer

Question: Compare the efficiency and accuracy of the Conjugate Gradient method, GMRES(500), and GMRES(100) based on their residual norms and solution error norms for a 500x500 linear system generated by a given script.

Step-by-step solution

Define the linear problem

Use the provided script to create the matrix A, true solution vector xt, and right-hand side vector b for a 500x500 linear system: 1. Generate a random 500x500 matrix A using $$A=\operatorname{randn}(500,500)$$ 2. Generate a random 500x1 vector xt using $$x t=\operatorname{randn}(500,1)$$ 3. Calculate the right-hand side vector b using $$b=A * x t$$ After completing this process, save the true solution vector xt.

Solving the system using CG on normal equations

Solve the system using the Conjugate Gradient method on the normal equations: 1. Form the normal equations by calculating $$A^{T}A$$ and $$A^{T}b$$. 2. Solve the linear system of equations $$A^{T}A\mathbf{x}=A^{T}\mathbf{b}$$ using the Conjugate Gradient method with a tolerance of $$1\mathrm{e}-6$$ and an iteration limit of 2000. 3. Record the residual norm (difference between right-hand side and the product of the matrix A and the calculated x for CG) and the solution error norm (difference between the true solution vector xt and the calculated x for CG).

Solving the system using GMRES(500)

Solve the system using GMRES(500) iterative method: 1. Apply GMRES with a restart value of 500 to the linear system $$A\mathbf{x}=\mathbf{b}$$, using a tolerance of $$1\mathrm{e}-6$$ and an iteration limit of 2000. 2. Record the residual norm (difference between right-hand side and the product of the matrix A and the calculated x for GMRES(500)) and the solution error norm (difference between the true solution vector xt and the calculated x for GMRES(500)).

Solving the system using GMRES(100)

Solve the system using GMRES(100) iterative method: 1. Apply GMRES with a restart value of 100 to the linear system $$A\mathbf{x}=\mathbf{b}$$, using a tolerance of $$1\mathrm{e}-6$$ and an iteration limit of 2000. 2. Record the residual norm (difference between right-hand side and the product of the matrix A and the calculated x for GMRES(100)) and the solution error norm (difference between the true solution vector xt and the calculated x for GMRES(100)).

Drawing conclusions

Based on the recorded residual norms and solution error norms for all three methods, compare their efficiency and accuracy in solving the linear system. Some possible conclusions might include: - Which method converges faster for this problem? - Which method gives better accuracy in terms of the solution error norm? - What are the trade-offs in using restarted GMRES with a smaller restart value (GMRES(100) vs GMRES(500))?

Key concepts

Conjugate Gradient Method

The Conjugate Gradient (CG) method is a powerful iterative method for solving systems of linear equations, particularly suited for large, sparse, symmetric, and positive-definite matrices. The objective of the CG method is to find an approximate solution by minimizing the inner matrix norms, without directly inverting the matrix.
To apply the CG method, linear equations are often transformed to normal equations, as seen with the system \(A^{T} A \mathbf{x} = A^{T} \mathbf{b}\). However, this transformation might amplify errors, so care is needed.

• CG is effective because it takes advantage of the properties of the matrix to efficiently explore the solution space, reducing computational overhead when directly solving \(A\mathbf{x} = \mathbf{b}\) could be costly.
• One key aspect of CG is its reliance on orthogonality and conjugation principles, ensuring that each step of iteration improves the accuracy of the solution.

Despite its strengths, CG might struggle with ill-conditioned systems when applied to normal equations, often requiring preconditioning or additional strategies to ensure robustness and convergence.

GMRES Method

Generalized Minimal Residual (GMRES) method is an iterative technique for solving nonsymmetric linear systems \(A\mathbf{x} = \mathbf{b}\). Unlike CG, GMRES does not require symmetry or positive definiteness, making it versatile for a broader range of applications.
The GMRES algorithm works by constructing an orthogonal basis for the Krylov subspace generated by the initial residual, which assists in minimization of the residual norm during iterations.
One notable feature of GMRES is its 'restart' capability. As applied in the exercise with GMRES(500) and GMRES(100), this entails calculating a solution up to a maximum number of iterations (500 or 100) and then restarting, which helps in managing memory usage in large systems.

• The restart parameter influences convergence rates and computational efficiency. Smaller restart values might reduce memory usage and improve computation time but could lead to slower convergence.
• Compared to CG, GMRES can handle a wider range of problem types due to its construction, but might require careful adjustment of restart values and preconditioning for best results.

Residual Norms

When solving linear systems iteratively, the residual norm is an essential metric to evaluate how far the current solution is from satisfying the equation \(A\mathbf{x} = \mathbf{b}\). It measures the magnitude of the residual vector, \(\mathbf{r} = \mathbf{b} - A\mathbf{x}\).A smaller residual norm indicates that the solution is closer to the true solution, providing a check to monitor convergence during iterations. Residual norms tend to decrease as the iterative method progresses.
The measure of the residual norm helps in:

• Evaluating the convergence speed of different methods; for instance, the behavior of residual norms in GMRES vs CG indicates how quickly each method approaches the solution.
• Guiding the decision to stop the algorithm, given a pre-specified residual tolerance. This ensures that computational resources are used effectively and that solutions are close to accurate within acceptable error bounds.

Solution Error Norms

The solution error norm is a crucial metric that quantifies the error in the calculated solution \(\mathbf{x}\) compared to the true solution \(\mathbf{x}_t\). It is defined as the norm of the vector difference \(\mathbf{x}_t - \mathbf{x}\).
In practical computations where an exact solution is known, as in this exercise, the solution error norm provides instructive insight into the accuracy of an iterative solver.

• This metric helps identify the precision of the computed solution relative to what is theoretically expected, and thus assess the overall effectiveness of the employed method.
• By analyzing the solution error norms for methods like CG and GMRES (with various restarts), users can determine which solver most closely approximates the true solution, taking into account factors such as iteration limits and tolerances.

Using both residual and solution error norms allows a comprehensive assessment of the performance of iterative solvers, giving meaningful data to optimize solver parameter selection and usage strategies.