In this thesis we present a Regularization Network approach to
reconstruct a continuous function
ƒ:[0,1]^{n}→**R** from its function values
ƒ(**x**_{j}) on discrete data points **x**_{j}, j=1,…,P. The ansatz is based on a new
constructive version of Kolmogorov's superposition theorem.

Typically, the numerical solution of mathematical problems underlies
the so--called *curse of dimensionality*. This term describes the
exponential dependency of the involved numerical costs on the
dimensionality n. To circumvent the curse at least to some extend,
typically higher regularity assumptions on the function ƒ are
made which however are unrealistic in most cases. Therefore, we employ
a representation of the function as superposition of one--dimensional
functions which does not require higher smoothness assumptions on
ƒ than continuity. To this end, a constructive version of
Kolmogorov's superposition theorem which is based on D. Sprecher is
adapted in such a manner that one single outer function Φ and a
universal inner function ψ suffice to represent the function
ƒ. Here, ψ is the extension of a function which was defined
by M. Köppen on a dense subset of the real line. The proofs of
existence, continuity, and monotonicity are presented in this
thesis. To compute the outer function Φ, we adapt a constructive
algorithm by Sprecher such that in each iteration step, depending on
ƒ, an element of a sequence of univariate functions {
Φ^{r}}_{r} is computed. It will be shown that
this sequence converges to a continuous limit
Φ:**R**→**R**. This constructively proves
Kolmogorov's superposition theorem with a single outer and inner
function.

Due to the fact that the numerical complexity to compute the outer
function Φ by the algorithm grows exponentially with the
dimensionality, we alternatively present a Regularization Network
approach which is based on this representation. Here, the outer
function is computed from discrete function samples
(**x**_{j},ƒ(**x**_{j})),
j=1,…,P. The model to reconstruct ƒ will be introduced in
two steps. First, the outer function Φ is represented in a
finite basis with unknown coefficients which are then determined by a
variational formulation, i.e. by the minimization of a regularized
empirical error functional. A detailed numerical analysis of this
model shows that the dimensionality of ƒ is transformed by
Kolmogorov's representation into oscillations of Φ. Thus, the
use of locally supported basis functions leads to an exponential
growth of the complexity since the spatial mesh resolution has to
resolve the strong oscillations. Furthermore, a numerical analysis of
the Fourier transform of Φ shows that the locations of the
relevant frequencies in Fourier space can be determined a priori and
are independent of ƒ. It also reveals a product structure of the
outer function and directly motivates the definition of the final
model. Therefore, Φ is replaced in the second step by a product
of functions for which each factor is expanded in a Fourier basis with
appropriate frequency numbers. Again, the coefficients in the
expansions are determined by the minimization of a regularized
empirical error functional.

For both models, the underlying approximation spaces are developed by means of reproducing kernel Hilbert spaces and the corresponding norms are the respective regularization terms in the empirical error functionals. Thus, both approaches can be interpreted as Regularization Networks. However, it is important to note that the error functional for the second model is not convex and that nonlinear minimizers have to be used for the computation of the model parameters. A detailed numerical analysis of the product model shows that it is capable of reconstructing functions which depend on up to ten variables.

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