| Three topics are investigated in this thesis. (1) In the first part of the thesis, the inverse problem of extracting a quantum mechanical potential from laboratory data is studied from the perspective of determining the amount and type of data capable of giving a unique answer. Bound state spectral information and expectation values of time-independent operators are used as data. The Schroedinger equation is treated as finite dimensional and for these types of data there are algebraic equations relating the unknowns in the system to the experimental data. As these equations are polynomials in the unknown parameters of the system, it is possible to determine the multiplicity of the solution set. With a fixed number of unknowns the effect of increasing the number of equations on the multiplicity of solutions is assessed. We show that if one has more equations than the number of unknowns, generically a unique solution exists. Several examples illustrating these results are provided. (2) In the second part of the thesis we introduce a family of approximation methods: High dimensional model representations. A systematic mapping procedure between the inputs and outputs is prescribed to reveal the hierarchy of correlations amongst the input variables. It is argued that for most well defined physical systems, only relatively low order correlations of the input variables are expected to have an impact upon the output. The high dimensional model representations (HDMR) utilize this property to present an exact hierarchical representation of the physical system. Application of the HDMR tools can dramatically reduce the computational effort needed in representing the input-output relationships of a physical system. Selected applications of the HDMR concept are presented along with a discussion of its general utility. HDMRs can be classified as non-regressive, non-parametric learning networks. (3) The last part of the thesis is on a fast algorithm for the solution of a polynomial system of equations. Its relation to Buchberger's algorithm of finding the variety defined by a system of polynomials over the field of real numbers is discussed. |
