12.2 Uncertainty quantification and sensitivity analysis

Uncertainty quantification and sensitivity analysis are crucial tools in computational mathematics. They help us understand how input uncertainties affect model outputs and which parameters matter most. This knowledge is essential for making informed decisions and improving model reliability.

In this section, we'll explore probabilistic methods, advanced techniques, and efficient sampling strategies. We'll also dive into interpreting results through visualization and decision-making frameworks. These skills are vital for tackling real-world problems with complex uncertainties.

Quantifying uncertainty in models

Probabilistic methods for uncertainty quantification

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• Uncertainty quantification (UQ) systematically characterizes and reduces uncertainties in computational models and simulations
• Represent uncertain inputs as random variables with associated probability distributions
• Monte Carlo simulation propagates uncertainties through models by repeatedly sampling input parameters
• Bayesian inference updates model uncertainties based on observed data and prior knowledge
• Polynomial chaos expansion efficiently represents stochastic systems and quantifies output uncertainties
• Gaussian process emulation (kriging) creates surrogate models for computationally expensive simulations in UQ
• Sources of uncertainty in models include parameter uncertainty, model form uncertainty, and numerical uncertainty

Advanced UQ techniques

• Stochastic collocation methods use interpolation to approximate the relationship between inputs and outputs
• Sensitivity-driven dimension reduction identifies important parameter subspaces to simplify UQ analysis
• Multi-fidelity UQ combines models of varying accuracy and computational cost to balance efficiency and precision
• Probabilistic graphical models represent complex dependencies between uncertain variables in UQ frameworks
• Fuzzy set theory handles imprecise or linguistic uncertainties in model inputs and parameters
• Interval analysis bounds uncertain quantities without assuming specific probability distributions
• Evidence theory (Dempster-Shafer theory) quantifies uncertainties when probability distributions are not fully known

Sensitivity analysis for input parameters

Local and global sensitivity analysis methods

• Sensitivity analysis (SA) quantifies the relative importance of input parameters on model outputs and uncertainties
• Local SA methods (finite difference approximations) assess parameter importance around a specific input point
• Global SA techniques (Sobol indices, Morris screening) evaluate parameter importance across the entire input space
• Variance-based SA decomposes total output variance into contributions from individual parameters and interactions
• Derivative-based global sensitivity measures assess parameter importance in complex models
• Moment-independent SA methods (entropy-based measures) capture non-linear and non-monotonic input-output relationships
• SA results guide model simplification, experimental design, and resource allocation in uncertainty reduction efforts

Advanced sensitivity analysis techniques

• Meta-model based SA uses surrogate models to efficiently compute sensitivity indices for computationally expensive simulations
• Time-dependent SA analyzes parameter importance over different time scales in dynamic systems
• Regionalized SA identifies parameter importance in specific regions of the output space
• Copula-based SA captures complex dependencies between input parameters in sensitivity analysis
• Group SA assesses the importance of sets of parameters rather than individual inputs
• Bayesian SA incorporates parameter uncertainties and prior knowledge into sensitivity calculations
• Multi-objective SA considers multiple output quantities of interest simultaneously

Uncertainty propagation with sampling

Efficient sampling techniques

• Latin Hypercube Sampling (LHS) ensures better input space coverage compared to simple random sampling
• Importance sampling focuses computational resources on significant regions of the input space
• Quasi-Monte Carlo methods use low-discrepancy sequences to improve convergence rates
• Markov Chain Monte Carlo (MCMC) techniques (Metropolis-Hastings algorithm) sample from complex, high-dimensional distributions
• Multi-level Monte Carlo combines samples from models with different fidelity levels to reduce computational cost
• Adaptive sampling strategies dynamically adjust sampling schemes based on intermediate results
• Reliability methods (First-Order Reliability Method, Second-Order Reliability Method) estimate failure probabilities in engineering applications

Advanced uncertainty propagation methods

• Stochastic spectral methods (polynomial chaos, stochastic Galerkin) provide efficient alternatives to sampling-based approaches
• Probabilistic collocation method combines polynomial chaos expansion with deterministic quadrature for uncertainty propagation
• Moment equation methods propagate statistical moments of uncertain inputs through differential equations
• Probability boxes represent imprecise probabilities when distribution parameters are uncertain
• Optimal uncertainty quantification finds bounds on output probabilities with limited information about input distributions
• Sparse grid methods efficiently handle high-dimensional uncertainty propagation problems
• Bayesian model averaging combines predictions from multiple models to account for model form uncertainty

Interpreting uncertainty quantification results

Visualization techniques for uncertainty

• Probability density functions (PDFs) and cumulative distribution functions (CDFs) represent uncertain outcomes
• Box plots and violin plots compactly visualize output distributions (median, quartiles, outliers)
• Scatter plots and pair plots show correlations between input parameters and model outputs
• Tornado diagrams and spider plots communicate sensitivity analysis results by ranking parameter importance
• Contour plots and surface plots represent joint effects of multiple input parameters on outputs and uncertainties
• Parallel coordinates plots visualize high-dimensional data and explore parameter interactions
• Uncertainty budgets break down total output uncertainty into contributions from different sources

Advanced interpretation and decision-making

• Global sensitivity analysis visualizations (Sobol index plots, Morris screening plots) provide insights into parameter interactions
• Probabilistic risk assessment combines UQ results with consequence analysis for informed decision-making
• Robust optimization techniques use UQ results to find solutions that perform well across uncertain scenarios
• Value of information analysis quantifies the potential benefit of reducing specific uncertainties
• Uncertainty-aware machine learning incorporates UQ results into predictive models and decision support systems
• Multi-criteria decision analysis under uncertainty helps balance conflicting objectives in the presence of uncertainties
• Interactive visualization tools allow stakeholders to explore UQ results and test different scenarios dynamically