Accuracy of approximations of solutions to Fredholm equations by kernel methods

Keywords: Approximate solutions to integral equations Radial and kernel-based networks Gaussian kernels Model complexity Analysis of algorithms a b s t r a c t Approximate solutions to inhomogeneous Fredholm integral equations of the second kind by radial and kernel networks are investigated. Upper bounds are derived on errors in approximation of solutions of these equations by networks with increasing model complexity. The bounds are obtained using results from nonlinear approximation theory. The results are applied to networks with Gaussian and kernel units and illustrated by numerical simulations. Fredholm integral equations play an important role in many problems in applied science and engineering. They arise in image restoration [1], differential problems with auxiliary boundary conditions, potential theory and elasticity [2, Chapter IV], and many other problems (see, e.g., [3]). Solving an inhomogeneous Fredholm integral equation of the second kind is an inverse problem of finding for a function f representing measured data a function / which is mapped to f by a linear operator of the form I À kT K , where I is the identity operator, T K is an integral operator with a kernel K defined as