We reflect on some theoretical aspects of gradient-only optimization for
the unconstrained optimization of objective functions containing non-physical step or
jump discontinuities. This kind of discontinuity arises when the optimization problem
is based on the solutions of systems of partial differential equations, in combination
with variable discretization techniques (e.g. remeshing in spatial domains, and/or
variable time stepping in temporal domains). These discontinuities, which may cause
local minima, are artifacts of the numerical strategies used and should not influence
the solution to the optimization problem. Although the discontinuities imply that the
gradient field is not defined everywhere, the gradient field associated with the computational
scheme can nevertheless be computed everywhere; this field is denoted the
associated gradient field.
We demonstrate that it is possible to overcome attraction to the local minima if
only associated gradient information is used. Various gradient-only algorithmic options
are discussed. A salient feature of our approach is that variable discretization
strategies, so important in the numerical solution of partial differential equations, can
be combined with efficient local optimization algorithms.