The Monte Carlo complexity of Fredholm integral equations

Abstract

A complexity study of Monte Carlo methods for Fredholm integral equations is carried out. We analyze the problem of computing a functional μ ( y ) \mu (y) , where y is the solution of a Fredholm integral equation \[ y ( s ) = ∫ I m k ( s , t ) y ( t ) d t + f ( s ) , s ∈ I m y(s) = \int _{{I^m}} {k(s,t)y(t)\;dt + f(s),\quad s \in {I^m}} \] , on the m-dimensional unit cube I m {I^m} , where the kernel k and right-hand side f are given r times differentiable functions. We permit stochastic numerical methods which can make use of function evaluations of k and f only. All Monte Carlo methods known to the authors for solving the above problem are of the order n − 1 / 2 {n^{ - 1/2}} , while the optimal deterministic methods yield rate n − r / ( 2 m ) {n^{ - r/(2m)}} , thus taking into account the given smoothness of the data. Here, n denotes the (average) number of function evaluations performed. The optimal algorithm we present combines deterministic and stochastic methods in an optimal way. It can be seen that both rates—the standard Monte Carlo rate for general continuous data and the deterministic rate for r-smooth data—multiply. This provides the smallest error that stochastic methods of given computational cost can achieve.