## ME 597UQ Uncertainty Quantification

Mechanical Engineering, Purdue University, West Lafayette, IN

#### Abstract

The goal of this course is to introduce the fundamentals of uncertainty quantification to advanced undergraduates or graduate engineering and science students with research interests in the field of predictive modeling. Upon completion of this course you should be able to:

• Represent mathematically the uncertainty in the parameters of physical models.
• Propagate parametric uncertainty through physical models to quantify the induced uncertainty on quantities of interest.
• Calibrate the uncertain parameters of physical models using experimental data.
• Combine multiple sources of information to enhance the predictive capabilities of models.
• Pose and solve design optimization problems under uncertainty involving expensive computer simulations.

Topics to be covered

1. Probability Theory
• Random variables, expectations, conditional probabilities.
• Common probability distributions/densities and their properties. Gaussian distribution. Multivariate Gaussian distribution.
• Random number generation. Rejection sampling. Inverse sampling.
• Bayes rule, parameter estimation, model selection.
2. Representation of Uncertainties
• Generic principles. Invariant probabilities. Maximum entropy principle.
• Dimension reduction. Principal component analysis. Non-linear dimensionality reduction methods.
• Functional uncertainty. Random fields. Gaussian processes. Covariance functions. Karhunen-Loève expansion of random fields.
3. Uncertainty Propagation
• Sampling methods. Monte Carlo, quasi-random sequences, Latin hypercube designs, multi-level Monte Carlo. Sobol sensitivity indices.
• Intrusive techniques: generalized polynomial chaos.
• Classic collocation methods: Generalized polynomial chaos, sparse grid collocation.
• Bayesian collocation methods: Bayesian linear regression, Gaussian process regression.
• Selecting optimal simulations for efficiently propagating uncertainties when simulations are expensive.
4. Calibration of Physical Models
• Classical approach via minimization of (regularized) loss functions. Statistical (Bayesian) interpretation of the model calibration problem.
• Sampling methods: Markov Chain Monte Carlo, Metropolis-Hastings, Hybrid Monte Carlo, Sequential Monte Carlo.
• Surrogate-based methods: Generalized polynomial chaos, Gaussian process regression.
• Variational methods: Relative entropy, Kullback-Leibler divergence, Automatic differentiation variational inference.
• Data assimilation: Kalman filter, ensemble Kalman filter, particle filter.
• Selecting optimal experiments measurements/sensor locations for calibrating model parameters.
• Selecting optimal simulations for efficiently calibrating model parameters when simulations are expensive.
5. Optimization Under Uncertainty
• Robust optimization, risk quantification, multi-objective optimization, Pareto-front, utility theory.
• Sampling methods: Scenario based optimization, sampling average approximation.
• Stochastic methods: Randomized search, simulated annealing, Robins-Monroe algorithm, particle swarm approximation.
• Bayesian global optimization: Probability of improvement, expected improvement, knowledge gradient.

#### Cite this work

Researchers should cite this work as follows:

• Ilias Bilionis (2018), "ME 597UQ Uncertainty Quantification," https://nanohub.org/resources/27789.

#### Location

Grissom, Room 102, Purdue University, West Lafayette, IN

#### Tags

Lecture Number/Topic Online Lecture Video Lecture Notes Supplemental Material Suggested Exercises
ME 597UQ Lecture 01: Introduction (not recorded) Notes (pdf)
ME 597UQ Lecture 02: Quantifying Uncertainties in Physical Models View HTML
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ME 597UQ Lecture 03: Introduction to Probability Theory I View HTML
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ME 597UQ Lecture 04: Introduction to Probability Theory II View HTML
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ME 597UQ Lecture 05: Common Random Variables View HTML
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ME 597UQ Lecture 06: Turning Prior Information into Probability Statements View HTML
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ME 597UQ Lecture 07: Generalized Linear Models I View HTML
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ME 597UQ Lecture 08: Generalized Linear Models II View HTML
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ME 597UQ Lecture 09: Generalized Linear Models III View HTML
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ME 597UQ Lecture 10: Priors on Functional Spaces Gaussian Processes View HTML
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ME 597UQ Lecture 11: Gaussian Process Regression View HTML
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ME 597UQ Lecture 12: Dimensionality Reduction of Gaussian Random Fields View HTML
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ME 597UQ Lecture 13: Uncertainty Propagation (Sampling Methods) I View HTML
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ME 597UQ Lecture 14: Uncertainty Propagation (Sampling Methods) II View HTML
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ME 597UQ Lecture 15: Perturbative Methods View HTML
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ME 597UQ Lecture 16: Uncertainty Propagation - Polynomial Chaos I View HTML
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ME 597UQ Lecture 17: Uncertainty Propagation - Polynomial Chaos II View HTML
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ME 597UQ Lecture 18: Uncertainty Propagation - Polynomial Chaos III View HTML
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ME 597UQ Lecture 19: Inverse Problems/Model Calibration - Classical Approaches View HTML
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ME 597UQ Lecture 20: Inverse Problems/Model Calibration - Bayesian Approach View HTML
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ME 597UQ Lecture 21: Markov Chain Monte Carlo I View HTML
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ME 597UQ Lecture 22: Markov Chain Monte Carlo II View HTML
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ME 597UQ Lecture 24: Bayesian Model Comparison using Sequential Monte Carlo View HTML
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ME 597UQ Lecture 26: Variational Interferance View HTML
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ME 597UQ Lecture 27: How to Optimize Expensive Functions View HTML
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