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In the bayesian analysis of Inverse Problems most relevant cases the forwardmaps (FM, or regressor function) are defined in terms of a system of (O, P)DE'swith intractable solutions. These necessarily involve a numerical method tofind approximate versions of such solutions and lead to a numerical/approximateposterior distribution. Recently several results have been published on theregularity conditions required on such numerical methods to ensure converge ofthe numerical to the theoretical posterior. However, more practical guidelinesare needed to ensure a suitable working numerical posterior. ]Capistran2016]prove for ODE's that the Bayes Factor of the approximate vs the theoreticalmodel tends to 1 in the same order as the numerical method order. In this workwe generalize the latter paper in that we consider 1) also PDE's, 2) correlatedobservations, 3) practical guidelines in a multidimensional setting and 4)explore the use of expected Bayes Factors. This permits us to obtain bounds onthe absolute global errors to be tolerated by the FM numerical solver, which weillustrate with some examples. Since the Bayes Factor is kept above 0.95 weexpect that the resulting numerical posterior is basically indistinguishablefrom the theoretical posterior, even though we are using an approximatenumerical FM. The method is illustrated with some examples using syntheticdata.

Numerical posterior distribution error control and expected Bayes Factors in the bayesian Uncertainty Quantification of Inverse Problems