Smoothing Analysis of an All-at-Once Multigrid Approach for Optimal Control Problems Using Symbolic Computation

The numerical treatment of systems of partial differential equations (PDEs) is of great interest as many problems from applications, including the optimality system of optimal control problems that is discussed here, belong to this class. These problems are not elliptic and therefore both the construction of an efficient numerical solver and its analysis are hard. In this work we will use all-at-once multigrid methods as solvers. For sake of simplicity, we will only analyze the smoothing properties of a well-known smoother. Local Fourier analysis (or local mode analysis) is a widely-used tool to analyze numerical methods for solving discretized systems of PDEs which has also been used in particular to analyze multigrid methods. The rates that can be computed with local Fourier analysis are typically the supremum of some rational function. In several publications this supremum was merely approximated numerically by interpolation. We show that it can be resolved exactly using cylindrical algebraic decomposition which is a well established method in symbolic computation.

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