Residual Versus Error in Transport Problems
The relationship between error and residual is very important in many applications. In most cases, the residual — or equation error — is much easier to estimate than the difference between the exact and approximate solutions and we have often used residua
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Abstract . The relationship between error and residual is very important in many applications . In most cases, the residual - or equation error - is much easier to estimate than the difference between the exact and approximate solutions and we have often used residual as a measure of the error in our work with transport problems. In this paper, we will attempt to justify this by applying theoretical relationships between error and residual for some fairly simple, but illustrative, transport problems.
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Introduction
When solving particle transport problems by Monte Carlo methods, it is usual to use the variance of the Monte Carlo estimator to evaluate the error. Recently, however , adaptive Monte Carlo algorithms have been developed that are capable, in principle, of providing zero variance estimates of the global solution of the problem after sufficiently many adaptive stages have been run [5], [4], [6] . Such methods operate by processing a fixed number of ordinary (i.e., sampled from unchanging density functions throughout) Monte Carlo random walks in each adaptive stage and using information derived from these to estimate the inner products of the global solution with finitely many functions chosen from some convenient basis set . When processing the next stage in the adaptive algorithm, either the sampling densities or the values of the estimators used or both are altered to obtained improved estimates of these expansion coefficients. For such an algorithm, the variance in the estimate of the coefficients provides no definitive information about how close the approximate global solution is to the exact solution. Methods capable of providing such information are , therefore, of great interest, especially in conjunction with such adaptive algorithms. In this paper we make use of the fact that the residual - that is, the error made when substituting the approximate solution into the governing transport equation - may be used to provide error estimates. This is especially useful because our adaptive method based on sequential application of correlated sampling [4] - in which successive stages estimate corrections to the K.‒T. Fang et al. (eds.)., Monte Carlo and Quasi-Monte Carlo Methods 2000 © Springer-Verlag Berlin Heidelberg 2002
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solution by solving a problem in which the residual is the source - must therefore calculate the residual for each successive stage. Thus, error estimates obtained from the residual are especially attractive. Although relationships involving the error and residual are known from the general theory of linear integral equations, we have slightly improved one of these for use in this paper. We stress that our primary interest, however, is in the role that might be played by the residual in assessing the accuracy of an approximate solution of the transport equation as obtained, for example, by adaptive Monte Carlo algorithms. We will discuss these relationships in the case of the following class of transport equations: (P) = K(P) + S(P) (1) where
K(P) ==
l
K(P, Q)(Q)dQ,
PEr th
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