Optimizing multigrid smoothers using GLT theory
Abstract: Multigrid algorithms are algorithms used to find numerical solutions to differential equations using a hierarchy of grids of different coarseness. This exploits the fact that short-wavelength components of the solutions converges at a faster rate than the long-wavelength components when using some basic iterative methods, such as the Jacobi method or the Gauss-Seidel method. These iterative methods are also called smoothers since they have the effect of smoothing out the errors. In this thesis, two classes of smoothers based on time integration methods are studied using the one dimensional linear advection equation as model problem. The first class is a class based on explicit Runge-Kutta methods, and the other class is derived from considering implicit Runge-Kutta methods. For both classes it is possible to derive a matrix M, dependent on the coefficient function in the linear advection problem and the parameters of the smoother, which describes the evolution of the error. To optimize the smoothers parameters are chosen so that the eigenvalues of M are optimized. If the coefficient function is constant it is possible to derive a closed form expression for the eigenvalues of the resulting matrix M. However, in the variable coefficient case it is not possible, and for large matrices it is impractical to compute the eigenvalues using iterative methods. Therefore, the theory of Generalized Locally Toeplitz (GLT) sequences is used to instead approximate the distribution of the eigenvalues. This results in an approximate optimization problem. The results show that this is an effective method for obtaining parameters for the smoothers in the variable coefficient case.
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