Evaluating derivatives: principles and techniques of algorithmic differentiation
Evaluating derivatives: principles and techniques of algorithmic differentiation
Foundations of Computational Mathematics
Methods and Applications of Interval Analysis (SIAM Studies in Applied and Numerical Mathematics) (Siam Studies in Applied Mathematics, 2.)
An efficient estimate based on FFT in topological verification method
Journal of Computational and Applied Mathematics - Special issue: Scientific computing, computer arithmetic, and validated numerics (SCAN 2004)
Validated Continuation for Equilibria of PDEs
SIAM Journal on Numerical Analysis
Validated continuation over large parameter ranges for equilibria of PDEs
Mathematics and Computers in Simulation
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We propose an efficient and generic algorithm for rigorous integration forward in time of partial differential equations written in the Fourier basis. By rigorous integration we mean a procedure which operates on sets and return sets which are guaranteed to contain the exact solution. The presented algorithm generates, in an efficient way, normalized derivatives which are used by the Lohner algorithm to produce a rigorous bound. The algorithm has been successfully tested on several partial differential equations (PDEs) including the Burgers equation, the Kuramoto-Sivashinsky equation, and the Swift-Hohenberg equation. The problem of rigorous integration in time of partial differential equations is a problem of large computational complexity and efficient algorithms are required to deal with PDEs on higher dimensional domains, like the Navier-Stokes equation. Technicalities regarding the various optimization techniques implemented in the software used in this paper will be reported elsewhere.