  A basic issue in optimization, inverse theory,neural networks, computational
chemistry and many other problems is the geometrical characterization of high
dimensional functions. In inverse calculations one aims to characterize the set
of models that fit the data (among other constraints). If the data misfit
function is unimodal then one can find its peak by local optimization methods
and characterize its width (related to the range of data-fitting models) by
estimating derivatives at this peak. On the other hand, if there are local
extrema, then a number of interesting and difficult problems arise. Are the
local extrema important compared to the global or can they be eliminated (e.g.,
by smoothing) without significant loss of information? Is there a sufficiently
small number of local extrema that they can be enumerated via local
optimization? What are the basins of attraction of these local extrema? Can two
extrema be joined by a path that never goes uphill? Can the whole problem be
reduced to one of enumerating the local extrema and their basins of attraction?
For locally ill-conditioned functions, premature convergence of local
optimization can be confused with the presence of local extrema. Addressing any
of these issues requires topographic information about the functions under
study. But in many applications these functions may have hundreds or thousands
of variables and can only be evaluated pointwise (by some numerical method for
instance). In this paper we describe systematic (but generic) methods of
analysing the topography of high dimensional functions using local optimization
methods applied to randomly chosen starting models. We provide a number of
quantitative measures of function topography that have proven to be useful in
practical problems along with error estimates.
