Vanderbilt Risk and Reliability Studies Brief: Uncertainty Assessment and Management in Inverse Problems

Overview: Computational models are being increasingly used by the scientific community to study many complicated and scientific engineering problems. The problem of predicting the outcome of a particular phenomenon is called the modeling problem or the forward problem. The inverse problem consists of using the observed results of a particular phenomenon to infer the properties of the corresponding computational model. In deterministic analysis, the forward problem usually has a unique solution, whereas the inverse problem does not. Hence it is necessary to assess the confidence in the solutions of inverse problems. The situation is further complicated when the computational model is stochastic, i.e. the model inputs, the model parameters and the model form are not deterministic, which is often the case in several engineering applications. In such scenarios, it is not only essential to identify all the plausible solutions of the inverse problem, but also necessary to quantify the uncertainty associated with each of these solutions.

There may be different types of uncertainty associated with engineering problems. These sources of uncertainty can be classified into two types: aleatory uncertainty (inherent or natural variability) and epistemic uncertainty (reducible uncertainty). Epistemic uncertainty may arise from two different sources: data uncertainty and model uncertainty. This research focuses on the development of a unified framework for overall management of uncertainty in inverse problems.

The proposed research work classifies inverse problems into two major kinds: inference and decision making. Three types of inference problems are considered inference on model inputs, inference on model parameters and inference on model form. The various sources of aleatory uncertainty and epistemic uncertainty are included in a systematic manner. A probabilistic methodology is used to represent epistemic uncertainty using probability distributions, thereby facilitating the combined treatment of aleatory and epistemic uncertainty.

In general, inverse problems have several unknown quantities and a global sensitivity analysis of the forward problem is recommended to select those that need to be calibrated. Having calibrated these quantities, an inverse global sensitivity analysis method is developed to apportion the different sources of uncertainty to the results of the inverse problem. The sensitivity of model inputs, model parameters, and model form is quantified using a systematic procedure.

The methods developed for uncertainty analysis are extended to practical applications involving multi-level problems and time-dependent variables. The domains of application include fault diagnosis in structural health monitoring, and calibration and prediction for fatigue crack growth analysis.

Finally, decision making under uncertainty in studied. Four different aspects of decision making are considered: decision making with respect to (i) model outputs, (ii) model inputs, (iii) model parameters, and (iv) model form. Several problems such as design, model validation, risk assessment, life cycle management etc. are considered and a robust framework for decision making under uncertainty is developed.

Applications: There are several applications in science and engineering that require inverse analysis. A few applications will be considered in detail. (1) Structural Frames (2) Pump Actuation Systems (3) Fatigue Crack Growth in Mechanical Components (4) Thermally Induced Vibrations in Aerospace Structures

Vanderbilt University Engineering Capability Brief

Uncertainty Assessment and Management in Inverse Problems

Vanderbilt University
Engineering Capability Brief