Topics in matrix analysis
Numerical calculation of invariant tori
SIAM Journal on Scientific and Statistical Computing
Block M-matrices and computation of invariant tori
SIAM Journal on Scientific and Statistical Computing
SIAM Journal on Scientific and Statistical Computing
Variants of BICGSTAB for matrices with complex spectrum
SIAM Journal on Scientific Computing
Solution of the systems associated with invariant tori approximation. II: multigrid methods
SIAM Journal on Scientific Computing
Applied Numerical Mathematics - Special issue on numerical methods for ordinary differential equations
Any Nonincreasing Convergence Curve is Possible for GMRES
SIAM Journal on Matrix Analysis and Applications
Computation of Invariant Tori by the Fourier Methods
SIAM Journal on Scientific Computing
Spectral methods in computing invariant tori
Applied Numerical Mathematics - Auckl numerical ordinary differential equations (ANODE 98 workshop)
The Best Circulant Preconditioners for Hermitian Toeplitz Systems
SIAM Journal on Numerical Analysis
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
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We propose an efficient preconditioning technique for the numerical solution of first-order partial differential equations (PDEs). This study has been motivated by the computation of an invariant torus of a system of ordinary differential equations. We find the torus by discretizing a nonlinear first-order PDE with a full two-dimensional Fourier spectral method and by applying Newton's method. This leads to large nonsymmetric linear algebraic systems. The sparsity pattern of these systems makes the use of direct solvers prohibitively expensive. Commonly used iterative methods, e.g., GMRes, BiCGStab and CGNR (Conjugate Gradient applied to the normal equations), are quite slow to converge. Our preconditioner is derived from the solution of a PDE with constant coefficients; it has a fast implementation based on the Fast Fourier Transform (FFT). It effectively increases the clustering of the spectrum, and speeds up convergence significantly. We demonstrate the performance of the preconditioner in a number of linear PDEs and the nonlinear PDE arising from the Van der Pol oscillator