Uncertainty Quantification

Цели урока

• Propagate uncertainties through nonlinear models using Taylor expansion and Monte Carlo methods
• Compute Sobol sensitivity indices to rank input uncertainty contributions
• Build polynomial chaos expansion (PCE) surrogate models and use them for efficient UQ
• Apply interval arithmetic and verified computing to obtain guaranteed error bounds

Предварительные знания

• nm-07
• nm-10

• nm-07
• nm-10

Simulations are only as good as their error characterization. Engineering and scientific decisions based on models require understanding not just the nominal answer but the full distribution of possible answers given uncertain inputs. UQ is what separates trustworthy predictive models from ones that cannot be relied upon.

• NASA uncertainty quantification for re-entry vehicles: PCE and Monte Carlo combined to bound aerodynamic heating loads - directly impacts astronaut safety margins
• Climate model ensembles (CMIP6) run hundreds of model variants to characterize uncertainty in projected temperature rise; Sobol analysis identifies which parameterizations drive the spread
• Nuclear reactor safety analysis (NUREG-2191) mandates statistical UQ on thermal-hydraulic simulations before regulatory approval of reactor designs
• Financial risk management: Monte Carlo VaR (Value at Risk) and stressed-scenario analysis quantify portfolio loss under uncertain market conditions

History of Uncertainty Quantification

1947: Ulam and Metropolis develop Monte Carlo method at Los Alamos for neutron diffusion in nuclear weapons design. 1990: Ghanem and Spanos publish polynomial chaos expansion in stochastic finite elements - spectral UQ for computational mechanics. 2001: Saltelli, Tarantola, Campolongo publish practical Sobol index estimators - global sensitivity analysis becomes accessible

Error Propagation: Taylor, Monte Carlo, and PCE

Uncertainty quantification asks: given uncertain inputs X ~ p(X), what is the distribution of model output Y = f(X)? Three approaches span the trade-off between cost and generality: linear (Taylor) propagation, sampling-based Monte Carlo, and spectral polynomial chaos expansion.

Sobol Sensitivity Indices

Sensitivity analysis asks: which uncertain inputs contribute most to output variance? Sobol indices provide a variance-based global answer. The first-order index S_i measures the fraction of output variance explained by input X_i alone; the total-order index T_i measures the fraction including all interactions involving X_i.

Polynomial Chaos Expansion Surrogates

Polynomial chaos expansion (PCE) represents a random output Y = f(X) as a series in orthogonal polynomials of the input random variables: Y = sum_alpha c_alpha * Psi_alpha(X). For Gaussian inputs, the Psi_alpha are Hermite polynomials; for uniform inputs, Legendre polynomials (Wiener-Askey scheme). Once the coefficients c_alpha are found, the surrogate replaces f(X) for cheap repeated evaluations.

Interval Arithmetic and Verified Computing

Monte Carlo and PCE provide statistical characterizations of uncertainty. Sometimes guaranteed, worst-case bounds are needed - for safety-critical systems (aircraft, nuclear reactors) where a 99.9% confidence bound is insufficient. Interval arithmetic replaces floating-point numbers with intervals [a, b] and propagates these through computations, yielding rigorous enclosures of the true result.

Interval arithmetic is appropriate when uncertain inputs have known hard bounds (manufacturing tolerances, sensor resolution) and worst-case guarantees are required. Probabilistic UQ (MC, PCE) is appropriate when inputs have known probability distributions and statistical moments or confidence intervals suffice.

Related Topics

PCE surrogate models are used in optimization under uncertainty, replacing expensive simulation calls with cheap polynomial evaluations for robust design

• PCE Surrogate Models for Robust Design — Related topic
• Ensemble Kalman Filter — Related topic
• Interval Arithmetic in Safety Systems — Related topic

Key Takeaways

• Taylor propagation is fast but valid only for small perturbations and mildly nonlinear models; Monte Carlo handles arbitrary nonlinearity at the cost of N model evaluations
• Sobol indices decompose output variance by input contributions; T_i > S_i reveals interaction effects that first-order analysis misses
• PCE surrogates give mean, variance, and Sobol indices analytically once built; sparse PCE via LASSO extends this to high-dimensional input spaces
• Interval and affine arithmetic give rigorous worst-case bounds, necessary for safety-critical certification where probabilistic guarantees are insufficient

Вопросы для размышления

• A structural FEM simulation has 50 uncertain material parameters. Monte Carlo requires 10,000 evaluations for 1% accuracy. How many evaluations does Saltelli's Sobol index estimator require, and at what point does a PCE surrogate become more economical?
• A function f(x1, x2) = exp(x1 + x2) has x1, x2 ~ N(0, 0.1^2). Compute the first-order Taylor approximation of Var[f] and compare it to the exact value. For what parameter values does the Taylor approximation fail significantly?
• Describe why interval arithmetic applied to x^2 - x^2 gives a non-zero interval, and explain how affine arithmetic resolves this. What is the computational cost penalty of affine arithmetic over plain interval arithmetic?

Связанные уроки

• nm-07
• nm-10
• nm-28
• nm-26-quantum-algorithms
• nm-29-scientific-ml-pinns