Mathematical modeling forms the basis for understanding, simulating, optimizing and controlling numerous scientific phenomenon  and associated measurements.  Mathematical models frequently take the form of an ordinary or partial differential equations, and are often nonlinear. There are a few instances where analytic solutions are available and in other cases solutions can be sought through the design of efficient and accurate numerical solutions.  Analytic solutions to similar problems or numerical approximations can be used to validate mathematical models through comparison with physical experiments, measured quantities of interest, and optimized experimental design.  Methods that yield analytic solutions are often employed to reduce the complexity of mathematical models and can, in turn help describe and improve instrument behavior in physically relevant limits.  Abundant examples are found in physics, chemistry, and biology.  We are interested developing new mathematical models to simulate scientific experiments that produce measurements. This research often leads to numerical implementations and associated testing, to numerical analysis of accuracy, to optimization, optimal control or optimal design of instrument performance.