Computation of high order derivatives in optimal shape design
In shape optimization problems, each computation of the cost function by the finite element method leads to an expensive analysis. The use of the second order derivative can help to reduce the number of analyses. \textit{N. Fujii} [SIAM J. Control Optimization 24, 346-360 (1986; Zbl 0587.49023)] was the first to study this problem, and \textit{J. Simon} [ISNM 91, 361-378 (1989; Zbl 0691.49023)] gave the second order derivative for the Navier-Stokes problem. We study higher order derivatives. But one can ask the following questions: Are they expensive to calculate? Are they complicated to use? Are they imprecise? Are they useless? At first sight, the answer seems to be positive, but classical results of \textit{V. Strassen} [Algebraic complexity theory, in J. van Leeuwen (ed.), Handbook of Theoretical Computer Science, Vol. A: Algorithms and complexity. Elsevier, Amsterdam (1990)] and \textit{J. Morgenstern} [SIGACT News 16, No. 4, 60-62 (1984; Zbl 0558.68033)] tell us that the higher order derivatives are not expensive to calculate, and can be computed automatically. We give an answer to the third question by proving that the higher order derivatives of a function can be computed with the same precision as the function itself. We also prove that the derivatives so computed are equal to the derivatives of the discrete problem. We call the discrete problem the finite dimensional problem processed by the computer. This result allows the use of automatic differentiation, which works only on discrete problems. Furthermore, the computations of Taylor's expansions which are obtained, could be a partial answer to the last question.
scientific article; zbMATH DE number 569097