# Stochastic inverse problems: Models and metrics

## Abstract

In past work, we introduced model-based inverse methods, and applied them to problems in which the anomaly could be reasonably modeled by simple canonical shapes, such as rectangular solids. In these cases the parameters to be inverted would be length, width and height, as well as the occasional probe lift-off or rotation. We are now developing a formulation that allows more flexibility in modeling complex flaws. The idea consists of expanding the flaw in a sequence of basis functions, and then solving for the expansion coefficients of this sequence, which are modeled as independent random variables, uniformly distributed over their range of values. There are a number of applications of such modeling: 1. Connected cracks and multiple half-moons, which we have noted in a POD set. Ideally we would like to distinguish connected cracks from one long shallow crack. 2. Cracks of irregular profile and shape which have appeared in cold work holes during bolt-hole eddy-current inspection. One side of such cracks is much deeper than other. 3. L or C shaped crack profiles at the surface, examples of which have been seen in bolt-hole cracks. By formulating problems in a stochastic sense, we are able to leverage the stochasticmore »

- Authors:

- Victor Technologies, LLC, Bloomington, IN 47407-7706 (United States)
- Computational Tools, Gurnee, IL 60031 (United States)
- Statistical Engineering, Palm Beach Gardens, FL 33418 (United States)
- Air Force Research Laboratory (AFRL/RXCA), Wright Patterson AFB, OH 45433-7817 (United States)

- Publication Date:

- OSTI Identifier:
- 22391239

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: AIP Conference Proceedings; Journal Volume: 1650; Journal Issue: 1; Conference: 41. Annual Review of Progress in Quantitative Nondestructive Evaluation, Boise, ID (United States), 20-25 Jul 2014; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 36 MATERIALS SCIENCE; ALGORITHMS; COMPUTERIZED SIMULATION; CRACKS; EDDY CURRENTS; FLEXIBILITY; INVERSE SCATTERING PROBLEM; MATHEMATICAL MODELS; METRICS; MINIMIZATION; RANDOMNESS; ROTATION; SENSITIVITY; STOCHASTIC PROCESSES; SURFACES

### Citation Formats

```
Sabbagh, Elias H., Sabbagh, Harold A., Murphy, R. Kim, Aldrin, John C., Annis, Charles, and Knopp, Jeremy S.
```*Stochastic inverse problems: Models and metrics*. United States: N. p., 2015.
Web. doi:10.1063/1.4914812.

```
Sabbagh, Elias H., Sabbagh, Harold A., Murphy, R. Kim, Aldrin, John C., Annis, Charles, & Knopp, Jeremy S.
```*Stochastic inverse problems: Models and metrics*. United States. doi:10.1063/1.4914812.

```
Sabbagh, Elias H., Sabbagh, Harold A., Murphy, R. Kim, Aldrin, John C., Annis, Charles, and Knopp, Jeremy S. Tue .
"Stochastic inverse problems: Models and metrics". United States.
doi:10.1063/1.4914812.
```

```
@article{osti_22391239,
```

title = {Stochastic inverse problems: Models and metrics},

author = {Sabbagh, Elias H. and Sabbagh, Harold A. and Murphy, R. Kim and Aldrin, John C. and Annis, Charles and Knopp, Jeremy S.},

abstractNote = {In past work, we introduced model-based inverse methods, and applied them to problems in which the anomaly could be reasonably modeled by simple canonical shapes, such as rectangular solids. In these cases the parameters to be inverted would be length, width and height, as well as the occasional probe lift-off or rotation. We are now developing a formulation that allows more flexibility in modeling complex flaws. The idea consists of expanding the flaw in a sequence of basis functions, and then solving for the expansion coefficients of this sequence, which are modeled as independent random variables, uniformly distributed over their range of values. There are a number of applications of such modeling: 1. Connected cracks and multiple half-moons, which we have noted in a POD set. Ideally we would like to distinguish connected cracks from one long shallow crack. 2. Cracks of irregular profile and shape which have appeared in cold work holes during bolt-hole eddy-current inspection. One side of such cracks is much deeper than other. 3. L or C shaped crack profiles at the surface, examples of which have been seen in bolt-hole cracks. By formulating problems in a stochastic sense, we are able to leverage the stochastic global optimization algorithms in NLSE, which is resident in VIC-3D®, to answer questions of global minimization and to compute confidence bounds using the sensitivity coefficient that we get from NLSE. We will also address the issue of surrogate functions which are used during the inversion process, and how they contribute to the quality of the estimation of the bounds.},

doi = {10.1063/1.4914812},

journal = {AIP Conference Proceedings},

number = 1,

volume = 1650,

place = {United States},

year = {Tue Mar 31 00:00:00 EDT 2015},

month = {Tue Mar 31 00:00:00 EDT 2015}

}