Computational Methods for Parametrized Equations

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The purpose of this investigation is to continue work on the development of effective computational methods for nonlinear, parameter dependent equations. The equations involve a state variable, which is usually infinite dimensional, a finite dimensional parameter vector. Under appropriate conditions, the solutions of the state space and the parameter space. Problems of this type arise in numerous, practically important equilibrium studies in physics and engineering. For example, the equation may describe the equilibrium positions of a mechanical structure, in which case the state variable characterizes, say, the deformations while the parameter vector incorporates information about the load points, load directions, material properties, geometrical data, etc. In this, as in most applications of this type, interest centers rarely on the computation of a few solutions for fixed choices of the parameters, but instead one has to address, what is often called, the sensitivity problem. This concerns the question of the change of the solutions under variation of the parameters. In particular, one needs to determine those values of the parameters where the character of the solution changes, for instance, where stability is lost. Thus, in essence, the sensitivity problem requires a computational study of the shape and properties of the solution manifold. The assumption of the existence of the solution manifold presumes that the problem has been suitably unfolded. This turns out to be a very natural assumption in most applications and to have considerable advantages for the numerical solution of parametrized equation. Up to now, the numerical analysis literature has paid little attention to this differential-geometric aspect of multi-parameter nonlinear equations. But a consistent utilization of differential-geometric concepts and results can provide a powerful tool in the development of effective computational methods for a unified analysis of the solution properties of the parametrized equation.