Perspective on the Relation of Current Engineering Practice to Inverse Problems

This contribution has two purposes. One is to stimulate conversation among engineers and others who perform experiments and estimate parameters. It gives a general framework which I see many engineers (and others) working. The second purpose is to give mathematicians and others an understanding of the common experimental-analytical paradigms for unknown processes and their relationship to the study of inverse problems. I would be happy to hear from anyone who would like to discuss these ideas further.

Below are some thoughts on two current research paradigms in engineering. These paradigms are contrasted with what I consider to be a more powerful paradigm - which is actually part of the subject of inverse problems. This third type is familar to the inverse problems community but it is not widely known or practiced in engineering.

Common Research Paradigms in Engineering

Two types of paradigms in engineering research are commonly used. Type A involves investigating a "simple" phenomena and a single parameter is found using a simple algebraic equation. Type B has its objective to verify that the model is satisfactory to describe a certain phenomena.

Common Paradigm of Type A

In the type A paradigm, a process has an unknown such as thermal conductivity, heat transfer coefficient, diffusion coefficient, Young's modulus, or friction coefficient. Although the mathematical model for the phenomena may be complex, the final equation for finding the parameter of interest is usually quite simple, frequently as an algebraic equation.

The other part of the type A procedure involves an experiment. The experiments is selected to produce measurements that are compatible with the model. From these measurements and the model, the parameter is determined.

Common Paradigm of Type B

In the type B of the common paradigm, an incompletely understood engineering process is investigated in two distinct and complementary ways: one uses experiments and the other uses analytical or computer modeling. The first part involves an analytical model. This can involve the solution of ordinary or partial differential equations. Any needed constants are found from the literature or completely separate experiments of Type A which are found by breaking the problem into several independent parts. After all the parts are found, they are assembled into one large model and a prediction is made for some experimental conditions.

An experimental effort produces measurements for the same process. No interaction between the analysis and the experiment for the complete process is allowed. The experimental group in effect "throws over the wall" the data and description of the experiment to the analytical group.

Then a figure of overall results is produced, comparing those from the model and the experiment. Characteristically, the comparison of the graphical results is visual and not quantitative. Instead the agreement is usually simply said to be "satisfactory" or even "excellent," showing that the model is also satisfactory. An important point is that the results of the experiment and analysis are purposely kept apart until the last possible moment, and then compared only on same plot. The intent is to avoid any "knobs" to turn to get agreement between the model and the measurements. Results of the model may be not used to modify and improve the experiment; similarly the model may not be modified based on the experiment.

New Research Paradigm in Engineering
- Involving Inverse Problems: Type C

In the "new research paradigm," Type C paradigm, the emphasis is upon combined and interactive experiments and analysis. The concepts of experiment design and "stretching and straining" the model enters. Computers are used both in the experiments, modeling and estimating of parameters or determining better models. The paradigm is now described in more detail.

The paradigm is directed toward understanding some physical engineering process that has some unknown aspects. A first objective is to identify what is unknown. This in turn leads to the design of an experiment that will provide measurements that can be used to determine what is unknown. Two aspects should be considered at this point. First, the errors (or uncertainty) of the measuring devices(s) should be understood and quantified. The second aspect is that the experiment should be optimally designed, as much as possible without precisely knowing all the parameters or possibly the correct model. A simulation should be performed to see if the experiment will reveal what is thought to be unknown. This then requires some interaction with the analysis/modeling group in the beginning of the investigation. The purpose is to reveal if the experiment has the potential to determine the unknowns.

Then the experiment is performed. After that, the analysis is performed (possibly involving finite differences or elements). Instead of simply performing a direct calculation and comparing the results in a graphical fashion, the analysis now includes an inverse algorithm for estimating some parameters or functions. This estimation algorithm may be nonlinear and involve iteration. The residual principle may be used in which the estimated standard deviation between the measurements and the estimated values are made to be about equal to the expected measurement errors. The residuals are examined to determine any systematic trends or signatures. Confidence regions are constructed.

After the experiment has been analyzed, it may be possible to improve the experiment using optimality concepts. Furthermore the residuals might give some insight for improving the model.

An important point is that this Type C paradigm does not require breaking the problem into a number of parts (Type A experiments). In some cases it may still be very wise to do that. However, there are cases in which the individual parts are not independent. For example, some materials change (dry, burn, ablate, cure, etc.) during the process; in such cases the Type B paradigm is not adequate. In other cases, the desired result is a function of time, such as a time-dependent heating condition, which cannot be found by the Type B paradigm.

I would appreciate any comments. James V. Beck, Professor (
Department of Mechanical Engineering
A231 Engineering Building
Michigan State University
East Lansing, MI 48824
Tel no. 517-355-8487, Fax: 517-353-1750

From IPnet Vol 2, No 9
Translated to HTML by Keith A. Woodbury>