1 research outputs found
Efficient finite-dimensional solution of initial value problems in infinite-dimensional Banach spaces
We deal with the approximate solution of initial value problems in
infinite-dimensional Banach spaces with a Schauder basis. We only allow
finite-dimensional algorithms acting in the spaces \rr^N, with varying .
The error of such algorithms depends on two parameters: the truncation
parameters and a discretization parameter . For a class of
right-hand side functions, we define an algorithm with varying , based on
possibly non-uniform mesh, and we analyse its error and cost. For constant ,
we show a matching (up to a constant) lower bound on the error of any algorithm
in terms of and , as . We stress that in the standard
error analysis the dimension is fixed, and the dependence on is usually
hidden in error coefficient. For a certain model of cost, for many cases of
interest, we show tight (up to a constant) upper and lower bounds on the
minimal cost of computing an \e-approximation to the solution (the
\e-complexity of the problem). The results are illustrated by an example of
the initial value problem in the weighted space ().Comment: 22 page