What is uncertainty quantification?
UQ involves studying all sources of error and uncertainty in models and experiments, such as measurement errors, theoretical model limitations, numerical approximations, and human errors
What are the two types of uncertainty?
Aleatoric:
Inherent randomness in a system
Irreducible: doesn’t diminish with increase in knowledge
Epistemic:
Due to we don’t know / understand: uncertain information, measurement errors, simplifications etc.
Reducible: diminish with increase in knowledge
What are type 1 and 2 errors?
Type I error Mistaken rejection of actually true null hypothesis („false positive“)
Type II error reject null hypothesis of actually false („false negative“)
What are forward and inverse UQ?
Forward UQ
▪ Determine output uncertainty given the uncertainty in the input parameters
▪ Propagate uncertainties in inputs through model & quantify uncertainty in outputs
▪ For understanding range of possible outcomes and robustness of mode
Inverse UQ:
▪ Determine input uncertainties consistent with data
▪ Impose allowable / desirable uncertainty distribution on outputs (or deterministic output) and determine input uncertainty distributions yielding the defined outcome
What is Polynomial Chaos Expansion?
It’s a mathematical technique used for representing and quantifying uncertainty
in systems governed by stochastic processes. It can be used for uncertainty propagation in model based computations under uncertainty
What is a Confidence Interval?
CI is a range of value likely to contain the true, unknown parameter
What is the UQ procedure?
- Determine uncertainty specifics in inputs
- Uncertainty propagation
- Analysis of uncertainty in output
What are the steps for UQ with MC simulation?
- Define the probability distributions for the uncertain input parameters.
- Generate a large number of random samples from these distributions.
- Evaluate the model for each set of input samples.
- Analyze the distribution of the output results.
What are the pros and cons of UQ with MC simulation?
Pros:
- Simple
- Unbiased Estimator
- Highly Parallelizable
Cons:
- Statistical Uncertainty
- slow convergence rate
- If sample number is too high MC is infeasible
What is the concept behind Polynomial Chas Expansion?
Build a meta-model of the system which behaves like the original model but yields a deterministic PCE version of the otherwise uncertain output. Then perform moment analysis on the PCE representation of the model.
What is Sensitivity Analysis?
- The study of how small variations in the model inputs affect the model output
Why do we need Sensitivity analysis?
- Parameter Prioritization
- Parameter fixing
- Parameter Screening
- Parameter Mapping
- What-if-analysis
What are the types of Sensitivity Analysis (SA) techniques?
- Linear Regression Based
- Morris One-At-A-Time
- Derivative Based
- Variance Based
- Moment-free
- Monte-Carlo Filtering
What is the difference between local and global SA?
Local: Evaluate sensitivity of output to small variations in
inputs
Global: Evaluate overall sensitivity over entire uncertainty ranges
What is One-at-a-time approach for local SA?
- considers a small neighborhood of the data
- involves varying one parameter at a time
- Sensitivity parameters are found by taking partial derivatives of the output with respect to the inputs
What are variance- based methods for global SA?
- Typically based on conditional variance
- Use sensitivity index to measure importance of input factor
What is Sobol’s method?
- Global SA method
- Variance Based
- Computes First-order effects
- Based on sensitivity indices
- Idea is to decompose model into a sum of variances using combinations of input parameters with increasing dimensionality
What are some considerations with regard to Global SA?
- Non-linearity
- Exponential computational cost with increasing inputs
What are surrogate models?
- black box models
- computationally affordable
- sufficiently accurate
- replace original model
What is the process for creating a surrogate model?
- Select design variables to consider, and appropriate
surrogate modeling technique - Select appropriate data sample points (or collect data via experiments / simulations at chosen design space points)
- Construct surrogate model over the chosen design space using the selected samples
- Assess accuracy of surrogate with respect to the original model
- Iterate if necessary
What are some typical surrogate methods?
▪ Nonintrusive Polynomial Chaos Expansion (PCE)
▪ Multivariate Adaptive Regression Splines (MARS)
▪ Response Surface Models (RSM)
▪ Radial Basis Functions (RBF)
▪ Kriging
