On the Sensitivity of Solutions of Parametrized Equations
PITTSBURGH UNIV PA INST FOR COMPUTATIONAL MATHEMATICS AND APPLICATIONS
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The sensitivity of a solution of a parameterized equation Fz, lamda 0 with respect to the parameter vector lambda is usually defined as the change of the state z in dependence of lambda. In other words, for any solution expressible in the form zlambda, lambda with some smooth function z zlambda the sensitivity is the derivative Dzlambda. Typically the solutions form a manifold M in the product of the state-space and the parameter space and this sensitivity is available only at those points of M where the parameters can be used to define a local coordinate system. This paper introduces a general sensitivity concept which applies at all solutions on M and which includes the earlier definition. Some general geometric interpretations of the new measure are presented and it is shown that the sensitivity analysis can be easily integrated into the solution process. The theory also suggest the introduction of a readily computable second-order sensitivity measure reflecting the curvature-behavior of M. Two numerical examples illustrate the discussion.
- Theoretical Mathematics