Quantifying Uncertainty and Sensitivity in Mathematics through Designed Experiments
Quantifying uncertainty and understanding sensitivity are foundational for engineering research, especially when mathematical models are used in the analysis and design process. In this chapter, we will explore methodologies for rigorously quantifying uncertainty and performing sensitivity analysis using designed experiments, focusing on five critical sub-topics: sources and characterization of uncertainty, experimental design for uncertainty quantification, statistical modeling and Bayesian inference, sensitivity analysis techniques, and global versus local sensitivity analysis. Each section will delve deeply into theory, mathematical formulations, and practical implementation for engineering graduate students.
1. Sources and Characterization of Uncertainty in Mathematical Models
Uncertainty in engineering models arises from various sources, each of which must be identified and characterized to enable robust analysis and decision-making. The primary sources include:
- Parameter Uncertainty: Variability in physical or empirical parameters (e.g., material properties, environmental conditions) due to measurement errors, manufacturing variability, or incomplete knowledge.
- Model Structure Uncertainty: Imperfect representation of real-world systems due to simplifying assumptions, unmodeled dynamics, or incomplete physics.
- Input/Boundary Condition Uncertainty: Variability in initial or boundary conditions that serve as inputs to the mathematical model.
- Computational/Numerical Uncertainty: Errors introduced by numerical algorithms, discretization, and finite precision in computations.
Characterization of uncertainty requires both qualitative identification and quantitative description. Typically, uncertainties are modeled as random variables with specific probability distributions. For example, if a parameter is known to have Gaussian variability, it is modeled as ( X sim N(mu, sigma^2) ), where ( mu ) is the mean and ( sigma^2 ) is the variance. The stochastic dimension of a model is defined by the number of independent uncertain variables considered[4][5].
2. Experimental Design for Uncertainty Quantification (UQ)
Designed experiments are critical for systematically probing the response of mathematical models to variations in uncertain parameters. The goals are to efficiently collect data that maximizes information gain about the sources of uncertainty and to facilitate accurate parameter estimation.
- Design of Experiments (DoE): Classical DoE strategies include factorial designs, fractional factorial designs, and response surface methodologies. These techniques allow for systematic variation of multiple parameters to study their individual and interactive effects on model output.
- Sampling Methods: In UQ, common sampling approaches include Monte Carlo sampling, Latin Hypercube Sampling (LHS), and stratified sampling. These are used to generate representative distributions of uncertain parameters for simulation studies[3][6].
- Sequential Experimentation: Adaptive or sequential designs use information from earlier experiments to refine the experimental plan—improving efficiency in characterizing uncertainty[1].
An illustrative example is the use of designed experiments to estimate the uncertainty in the Helmholtz energy model, where parameters are systematically varied, and the resulting model outputs are analyzed to quantify the impact of input variability[3].
3. Statistical Modeling and Bayesian Inference for Uncertainty Quantification
Statistical modeling forms the backbone of uncertainty quantification. The two principal approaches are frequentist and Bayesian inference, with the latter being particularly powerful for integrating prior knowledge and updating beliefs in light of new data.
- Likelihood Function: The likelihood describes the probability of observing the experimental data given specific parameter values. For a set of observations ( y_{exp} ) and parameters ( theta ), the likelihood is ( f(y_{exp}|theta) ).
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Prior and Posterior Distributions: The prior ( f(theta) ) represents pre-experimental knowledge. Bayes’ theorem is used to update this with experimental data, yielding the posterior distribution ( f(theta|y_{exp}) ):
[
f(theta|y_{exp}) = frac{f(y_{exp}|theta)f(theta)}{f(y_{exp})}
] - Approximate Bayesian Computation (ABC): ABC methods are used when the likelihood function is intractable or highly complex. They approximate the posterior by simulating from the model and accepting parameter sets that produce outputs close to the observed data. ABC combined with Markov Chain Monte Carlo (MCMC) methods enables efficient exploration of high-dimensional parameter spaces[1].
Bayesian methods also allow sequential updating: posterior distributions from one dataset can serve as priors for subsequent experiments, integrating multiple sources of data for refined uncertainty estimates[1].
4. Sensitivity Analysis Techniques
Sensitivity analysis (SA) quantifies the effect of uncertainty in input parameters on model outputs, identifying which parameters most influence the system. SA is essential for prioritizing data collection, model simplification, and risk assessment.
- Local Sensitivity Analysis: Examines the effect of small perturbations around nominal parameter values using derivatives (e.g., partial derivatives or Jacobian matrices). For a model output ( y = f(x) ), the local sensitivity with respect to parameter ( x_i ) is calculated as ( frac{partial y}{partial x_i} )[2].
- Global Sensitivity Analysis: Considers variations over the entire parameter space. Techniques include variance decomposition (e.g., Sobol indices), which partition the output variance among input parameters, and screening methods (e.g., Morris method)[4][8].
- Adjoint Sensitivity Analysis: For large systems, adjoint methods efficiently compute sensitivities with respect to many parameters by solving an adjoint system of equations, rather than recalculating the full model for each parameter perturbation[2].
Sensitivity analysis can reveal model robustness, highlight dominant sources of uncertainty, and suggest which parameters should be targeted in future experiments or refined models[2][4].
5. Global versus Local Sensitivity Analysis: Theory and Implementation
Both global and local sensitivity analysis have distinct advantages and are chosen based on model complexity and the nature of uncertainty.
| Aspect | Local Sensitivity Analysis | Global Sensitivity Analysis |
|---|---|---|
| Scope | Small perturbations near a nominal operating point | Entire parameter space, accounts for nonlinearities and interactions |
| Computation | Uses derivatives, often analytical or numerical | Requires repeated evaluation (e.g., Monte Carlo), statistical decomposition |
| Interpretation | Parameter importance at a specific point | Parameter importance integrated over all possible inputs |
| Typical Methods | Finite differences, adjoint sensitivity | Sobol indices, ANOVA decomposition, Morris method |
In practice, global sensitivity analysis is often preferred for models with strong nonlinearities or interactions among parameters, as it provides a more comprehensive picture of parameter importance and model behavior under uncertainty. Modern computational techniques, such as surrogate modeling (e.g., polynomial chaos expansions, Gaussian process emulators), are frequently used to reduce computational costs in global analysis[4][8].
A well-structured approach to uncertainty and sensitivity analysis leverages designed experiments, rigorous statistical inference, and both local and global sensitivity analysis to yield robust, interpretable insights for engineering systems.