Generalizing interval matrix operations for design: Fusing the labeled interval calculus and interval matrix arithmetic.

The mathematical models of physical systems may be only approximations, because the characteristics of the systems are known or can be measured only approximately. Thus, if designs are based on the assumption that the model is exact, the design performance will normally not reach the desired goal. Therefore, these design problems should be described by intervals of values, rather than nominal values. Interval constraint propagation, which is based on the interval analysis, is often used in parametric design to refine parameter values through a set of constraints. However, the conventional interval propagations are inadequate for design purposes, but must be extended, a task partially accomplished in the Labeled Interval Calculus (LIC). The LIC is a formal system that performs quantitative inferences about sets of artifacts under sets of operating conditions. It refines and extends the idea of interval constraint propagation and is distinguished from other scalar interval mathematics in part by having a richer set of propagation operations. However, the LIC has been restricted to monotonic scalar functions: it cannot reason using simultaneous linear equations, Ax = b, which are often encountered in engineering. On the other hand, the study of interval methods for solving linear interval systems of equations has focused on the development of algorithms for narrower enclosure or shorter computing time; it has not aimed at design, and therefore has omitted a number of important issues. This thesis partially fuses these two classes of work: (1) It extends the current LIC to interval matrix operations. (2) It unites the design-oriented work with a substantial branch of applied mathematics, the interval matrix arithmetic, organizing that work to expose an underlying, previously unseen, unity; provides methods for computing a number of cases not previously considered in the interval matrix literature; and demonstrates by examples the utility of the operations in design problems. It thereby both extends interval matrix arithmetic, and connects it to the design inference procedures provided by the still evolving LIC.PhDApplied SciencesMathematicsMechanical engineeringPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/128925/2/9303709.pd