Rigorous quantitative analysis of multigrid, I: constant coefficients two-level cycle with L2-norm
SIAM Journal on Numerical Analysis - Special issue: the articles in this issue are dedicated to Seymour V. Parter
Testing stability by quantifier elimination
Journal of Symbolic Computation - Special issue: applications of quantifier elimination
Solving systems of strict polynomial inequalities
Journal of Symbolic Computation
SIAM Journal on Control and Optimization
Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition
Proceedings of the 2nd GI Conference on Automata Theory and Formal Languages
A generic projection operator for partial cylindrical algebraic decomposition
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
QEPCAD B: a program for computing with semi-algebraic sets using CADs
ACM SIGSAM Bulletin
Journal of Computational Physics
Local Fourier Analysis of Multigrid for the Curl-Curl Equation
SIAM Journal on Scientific Computing
Fourier Analysis for Multigrid Methods on Triangular Grids
SIAM Journal on Scientific Computing
Symbolic methods for the element preconditioning technique
SNSC'01 Proceedings of the 2nd international conference on Symbolic and numerical scientific computation
Accuracy Measures and Fourier Analysis for the Full Multigrid Algorithm
SIAM Journal on Scientific Computing
A Robust Multigrid Method for Elliptic Optimal Control Problems
SIAM Journal on Numerical Analysis
Computing and Visualization in Science
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Cylindrical algebraic decomposition (CAD) is a standard tool in symbolic computation. In this paper we use it to compute a bound for the convergence rate for a numerical method that usually is merely resolved by numerical interpolation. Applying CAD allows us to determine an exact bound, but the given formula is too large to be simply plugged in. Hence a combination of reformulating, guess and prove and splitting into subproblems is necessary. In this paper we work out the details of a symbolic local Fourier analysis for a particular multigrid solver applied to a particular optimization problem constrained to a partial differential equation (PDE-constrained optimization problem), even though the proposed approach is applicable to different kinds of problems and different kinds of solvers. The approach is based on local Fourier analysis (or local mode analysis), a widely-used straight-forward method to analyze the convergence of numerical methods for solving discretized systems of partial differential equations (PDEs). Such an analysis requires to determine the supremum of some rational function, for which we apply CAD.