Iterative Methods: Part 2

Last time, we looked at 2D Poisson’s equation and discussed how to arrive at a matrix equation using the finite difference method. The matrix, which represents the discrete Laplace operator, is sparse, so we can use an iterative method to solve the equation efficiently.

Today, we will look at Jacobi, Gauss-Seidel, Successive Over-Relaxation (SOR), and Symmetric SOR (SSOR), and a couple of related techniques—red-black ordering and Chebyshev acceleration. In addition, we will analyze the convergence of each method for the Poisson’s equation, both analytically and numerically.

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Iterative Methods: Part 1

Once again, we consider solving the equation A\vec{x} = \vec{b}, where A \in \mathbb{R}^{n \times n} is nonsingular. When we use a factorization method for dense matrices, such as LU, Cholesky, SVD, and QR, we make O(n^{3}) operations and store O(n^{2}) matrix entries. As a result, it’s hard to accurately model a real-life problem by adding more variables.

Luckily, when the matrix is sparse (e.g. because we used finite difference method or finite element method), we can use an iterative method to approximate the solution efficiently. Typically, an iterative method incurs O(n^{2}) operations, if not fewer, and requires O(n) storage.

Over the next few posts, we will look at 5 iterative methods:

  • Jacobi
  • Gauss-Seidel
  • Successive Over-Relaxation (SOR)
  • Symmetric SOR (SSOR)
  • Conjugate Gradient (CG).

In addition, we will solve 2D Poisson’s equation using these methods. We will compare the approximate solutions to the exact to illustrate the accuracy of each method.

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Hearing Perturbation Theory

In numerical linear algebra, we create ways for a computer to solve a linear system of equations A\vec{x} = \vec{b}. In doing so, we analyze how efficiently and accurately we can find the solution \vec{x}.

Perturbation theory concerns how much error we incur in the solution \vec{x} when we perturb (spoil) the data A and \vec{b}. A classic statement tells us that the amount of error depends on the condition number of the matrix A.

I will define and prove the statement, and help you understand it by “hearing” it.

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