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Bernold Fiedler

2011-11-25
13:30:00 - 14:20:00

308 , Mathematics Research Center Building (ori. New Math. Bldg.)

When can a function of several variables be written as a finite composition of functions of fewer variables? This is the question of Hilbert’s problem XIII. The answer by young V.I. Arnold in the form given by Kolmogorov says: always, by functions of two variables, if we assume continuity. The answer by Vitushkin says: almost never, if we require differentiability. Modeling, for example in chemical engineering, frequently provides input-output relations as compositions of input-output relations of smaller production units, or ”reactors”. The individual reactor models, or functions, may be known (”white box”) or unknown (”black box”). The hybrid model is the composition of such black- and whitebox functions. Let black boxes have at most d inputs – typically much less than the total number of inputs to the composition network. Assuming sufficient, and at times prohibitive, differentiability we indicate how to uniquely identify all unknown ”black-box” functions in the network from only d-dimensional data on their composition. This addresses the ”curse of dimension” in data analysis, and provides extrapolability. Results are joint work with Stefan Liebscher, Andreas Schuppert, and others. See also http://dynamics.mi.fu-berlin.de/