Verification and validation in computational engineering and science: basic concepts

摘要

Some essential points presented thus far are summarized as follows: 1. Validation involves comparison of observed physical events with those predicted by mathematical models of the component events (validation problems). 2. The prediction can be only verified to certain tolerances. 3. By physical event, we mean a specific physical entity, a quantity (or list of quantities) of interest that is (are) generally specified a priori, before the computer simulation. 4. The accuracy of a model's prediction can be judged only relative to tolerance supplied by the analyst and then only for a limited number of observations, never for all possible situations covered by the model; these tolerances are arbitrary, in the sense that different analysts may choose different tolerances for different quantities of interest for different purposes. 5. The tolerances for validation are statistical in nature, generally given in terms of a significance level. 6. Verification involves two basic components, code verification, which encompasses the software engineering processes of determining if the code faithfully implements the computational model, and solution verification, which is concerned with numerical accuracy with which the mathematical model is approximated by the computational model. 7. Solution verification involves assessing the accuracy of computed results for both the global and the validation component models compared with those capable of being predicted by the mathematical models selected to depict the physical event of interest. Solution verification involves a posteriori estimation of a numerical or discretization error. 8. Experimental results used in the validation process can themselves be in error. The reproducibility of experimental results for validation component models is essential. The impossibility of total and absolute verification and validation of computer models, and the dependence of V&V on subjective processes based on human experience, in no way make the subject inferior or different from any other scientific endeavor. Ernest Nagel noted that "the daily affairs of men are carried out within a framework of steady habits and confident beliefs, on the one hand, and of unpredictable strokes of fortune and precarious judgments on the other... In spite of such uncertainties, we manage to order our Jives with some measure of satisfaction; and we learn, though not always easily, that, even when grounds for our belief are not conclusive, some beliefs can be better grounded that others." And, finally, ".. .the methods of the natural sciences are the most reliable instruments men have thus far devised for ascertaining matters of fact, but that withal the conclusions reached by them are only probable because they rest on evidence which is formally incomplete."