Mathematics plays an important role in the science of metrology. Mathematical models are needed to understand how to design effective measurement systems, and to analyze the results they produce. Mathematical techniques are used to develop and analyze idealized models of physical phenomena to be measured, and mathematical algorithms are necessary to find optimal system parameters. Finally, mathematical and statistical techniques are needed to transform the resulting data into useful information. In physical metrology it is often necessary to fit a mathematical model to experimental results in order to recover the quantities being measured. In some cases the desired variables can be measured more or less directly, but the measuring instruments distort the measured function so much that mathematical modeling is required to recover it. In other cases the desired quantities cannot be measured directly and must be inferred by applying a model to the measured variables. In many cases the mathematical problems that need to be solved are ill-posed, that is, small perturbations in the input data can lead to very large variation in the results. Such problems are among the most challenging to solve in applied mathematics, often requiring highly sophisticated and specialized techniques.

We focus our energies on mathematical problems that recur in measurement science. The broad topic areas here include (a) inverse and ill-posed problems, (b) data modeling, (c) nonlinear dynamical systems, and (d) mathematical optimization. Success comes from the deep understanding of such problems of our staff, obtained from independent research, and our willingness to work closely with collaborators in the NIST Measurement and Standards Laboratories.