Variational Method Based Optimal Basis And Its Application To Fredholm Integral Equation Of The Second Kind

We introduce a variational framework based on Bayesian numerical homogenization^[1]to identify optimal basis functions for integro-differential equation.Rough polyharmonic splines^[2]are developed especially for equation with rough coefficients as a type of technique of numerical homogenization.In this paper we apply this framework to Fredholm integral equations of the second kind.After imposing appropriate constraints to the interpolation basis functions and defining inner product associated with the integral equation,the residual of the interpolation attains its minimum with respect to the induced norm.And the residual is located in the space V₀,which is an orthogonal complement space to the space spanned by the interpolation basis function.Then we show how to apply the method to Fredholm integral equations of the second kind.Assuming the integral operator is compact and the kernel K（s,t）satisfies uniform Lipschitz conditon,the iterated colloca-tion method has a convergence rate of H³2.Compared with the traditional polynomial basis,the method has weak requirement on the smoothness of the solution.We also check the decay property of the basis and its behavior with respect to different kind of kernels numerically.