The Monte Carlo complexity of Fredholm integral equations

Authors:
Stefan Heinrich and Peter Mathé

Journal:
Math. Comp. **60** (1993), 257-278

MSC:
Primary 65C05; Secondary 45L10, 60J10, 65R20

MathSciNet review:
1153164

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Abstract: A complexity study of Monte Carlo methods for Fredholm integral equations is carried out. We analyze the problem of computing a functional , where *y* is the solution of a Fredholm integral equation

*m*-dimensional unit cube , 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 , while the optimal deterministic methods yield rate , 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.

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DOI:
http://dx.doi.org/10.1090/S0025-5718-1993-1153164-3

Article copyright:
© Copyright 1993
American Mathematical Society