Efficient finite-dimensional solution of initial value problems in infinite-dimensional Banach spaces

TL;DR

This paper develops finite-dimensional algorithms for solving initial value problems in infinite-dimensional Banach spaces, providing error bounds, cost analysis, and demonstrating the approach with examples in weighted b1 spaces.

Contribution

It introduces a new finite-dimensional algorithm with error and cost analysis for initial value problems in Banach spaces with Schauder bases, including lower bounds and complexity estimates.

Findings

Derived error bounds depending on truncation and discretization parameters.

Established tight bounds on the minimal computational cost for b5-approximations.

Illustrated results with examples in weighted b1 spaces.

Abstract

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 $N$. The error of such algorithms depends on two parameters: the truncation parameters $N$ and a discretization parameter $n$. For a class of $C^r$ right-hand side functions, we define an algorithm with varying $N$, based on possibly non-uniform mesh, and we analyse its error and cost. For constant $N$, we show a matching (up to a constant) lower bound on the error of any algorithm in terms of $N$ and $n$, as $N,n\to \infty$. We stress that in the standard error analysis the dimension $N$ is fixed, and the dependence on $N$ 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 $\ell_p$ space ($1\leq p<\infty$).