Abstract Sensitivity Analysis for Nonlinear Equations and Applications

A. Chernov ()
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A. Chernov: ETH, Seminar for Applied Mathematics

A chapter in Numerical Mathematics and Advanced Applications, 2008, pp 407-414 from Springer

Abstract: Abstract Let $$u \in Y$$ be a unique solution of a nonlinear equation J(α,u) = 0 for $$\alpha \in X$$ , where X and Y are suitable Banach spaces. We investigate sensitivity of u with respect to small perturbations of α = α0 + r. The Implicit Function Theorem postulates that the Fréchet differentiability of u = S(α) w.r.t. α is determined by the differentiability properties of J. Then S(α0 + r) is approximated by its truncated Taylor series at α0. We derive a sequence of problems, which characterizes the Fréchet derivative d k S(α0) via the Fréchet derivatives of lower order. This gives a recursive procedure for computing the Taylor series of S(α0 + r). We illustrate the above approach on an elliptic PDE, an elliptic integral equation and on an abstract strongly elliptic problem with small randomly perturbed parameter.

Date: 2008
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DOI: 10.1007/978-3-540-69777-0_48

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