A fast Nyström-Broyden solver by Chebyshev compression (Q1773070)

This paper is concerned with solving integral equations, as for instance the nonlinear Urysohn integral equation \[ u(x)=\mathcal K [u](x)+g(x)=\int _a^bK\left (x,t,u(t)\right )dt+g(x), \quad x\in [a,b] , \] at a low cost. The Nyström method computes \(u(x)\) by means of the approximating integral equation \[ u\,_l(x)=\sum _ {j=1}^ {n\, _l}u_ {j,n\, _l}K\left (x,x_ {j,n\, _l},u\,_l(x_ {j,n\, _l})\right )+g(x), \quad x\in [a,b], \] obtained by discretization with a numerical convergent scheme \[ \int _a^b f(x)dx \approx \sum _{j=1}^{n_l} w_{j,n_l}f(x_{j,n_l}). \] Now, fixed \(z\in \mathbb R ^{n_l}\) denote by \(\mathcal K\,_l\) the discrete integral operator defined by \[ \mathcal K\,_l[z ](x)=\sum _{j=1}^{n_l}u_{j,n_l}K(x,x_{j,n_l},z_j),\quad x\in [a,b]. \] The author, based on the results obtained by \textit{M. Vianello} [Neural Parallel Sci. Comput. 8, 327-354 (2000; Zbl 0986.65141)], in the framework of nonlinear operators, suggests to approximate the action of \(\mathcal K\,_l \), i.e. the function \(\mathcal K\, _l[z ]\) by truncated Chebyshev series and shows how to use such compression to implement a fast Nyström-Broyden solver. Several interesting linear and nonlinear examples are presented.