Chebyshev-like compression of linear and nonlinear discretized integral operators (Q2746471)

The author explores a Chebyshev-like compression, still in the direction of evaluating the action of an integral operator at a low cost, but moving at the same time from the linear to the general setting of Uryson operators: NEWLINE\[NEWLINET(u)(x_i)=\int_\Omega K(x_i,t,u(t)) dt\approx \sum_{j=1}^nw_jK(x_i,t_j,u_j), \quad 1\leq i\leq p,\quad p \geq n.NEWLINE\]NEWLINE The main qualitative ideas are presented followed by several numerical experiments, where the basic \(O(n^2)\) complexity is reduced to \(O(mn)\), with \(m\ll n\). In the case of nonsmooth kernels, a posteriori Chebyshev approximation estimates provide an indicator of the underlying discretization error.