In recognition of the increasing importance of computational simulation and the increasing need to assess uncertainties in computational results, the National Research Council (NRC) was asked to study the mathematical foundations of VVUQ and to recommend steps that will ultimately lead to improved processes. The specific tasking to the Committee on Mathematical Foundations of Verification, Validation, and Uncertainty Quantification is as follows:

•  A committee of the National Research Council will examine practices for VVUQ of large-scale computational simulations in several research communities.

•  The committee will identify common concepts, terms, approaches, tools, and best practices of VVUQ.

•  The committee will identify mathematical sciences research needed to establish a foundation for building a science of verification and validation (V&V) and for improving the practice of VVUQ.

•  The committee will recommend educational changes needed in the mathematical sciences community and mathematical sciences education needed by other scientific communities to most effectively use VVUQ.


Figure 1.1 illustrates the different elements of VVUQ and their relationships to the true, physical system, the mathematical model, and the computational model. Uncertainty quantification does not appear explicitly in the figure, but it plays important roles in the processes of validation and prediction.

There is general agreement about the purposes of verification, validation, and uncertainty quantification, but different groups can differ on the details of each term’s definition. For purposes of this report the committee adopts the following definitions:

•  Verification. The process of determining how accurately a computer program (“code”) correctly solves the equations of the mathematical model. This includes code verification (determining whether the code correctly implements the intended algorithms) and solution verification (determining the accuracy with which the algorithms solve the mathematical model’s equations for specified QOIs).

•  Validation. The process of determining the degree to which a model is an accurate representation of the real world from the perspective of the intended uses of the model (taken from AIAA, 1998).

•  Uncertainty quantification (UQ). The process of quantifying uncertainties associated with model calculations of true, physical QOIs, with the goals of accounting for all sources of uncertainty and quantifying the contributions of specific sources to the overall uncertainty.


FIGURE 1.1 Verification, validation, and prediction as they relate to the true, physical system, the mathematical model, and the computational model. (Adapted from AIAA [1998].)

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