Forward UQ

Resources

Sullivan, T.J. 2015. Introduction to Uncertainty Quantification. Vol. 63. Texts in Applied Mathematics. Cham: Springer International Publishing. https://doi.org/10.1007/978-3-319-23395-6.

Main Idea

Let $f : X \to Y$ be our model (technically a measurable function). Consider a Random Variable $X$ with a known probability measure, $P_{μ} : σ (X) \to R^{+} .$ The goal of forward UQ is to determine the push-forward of $X$ through $f .$ That is, $f_{#} μ : σ (Y) \to R^{+}$ , but more specifically,

f_{#} μ (E) = P_{μ} ({x \in X : f (x) \in E}) .

(To be rigorous, the probability measures could map from more arbitrary sigma algebras, rather than the power set.)

Determining $f_{#} μ$ is the gold standard of forward UQ, but this often quite difficult, so there are often simplified cases of interest.

Reliability or Certification

As a starting point, for some event of failure $Y_{fail} \in σ (Y)$ , determine the failure probability $$f_{# \mu} (Y_{\text{fail}}) = \mathbb{P}\mu ({ x \in \mathbb{X} : f(x) \in Y{\text{fail}}}).$$
This is a specific class of forward UQ problems.

We often do not need the above formalism, so we will continue by taking our forward model as $f (x)$ and a random variable $X \sim D_{X}$ , in which case the goal of forward UQ is to predict the distribution of $y = f (x)$ , $D_{Y}$ .

Monte Carlo

The baseline, most common sense method is to take samples ${x_{i}}_{i = 1}^{N} \sim D_{X}$ and compute

y_{i} = f (x_{i}) \forall i \in {1, 2, . . ., N}

And use ${y_{i}}_{i = 1}^{N} \approx D_{Y}$ . The drawback is the computational expense of $f$ and the slow convergence to the distribution.

This can be paired with Generative Modeling, where we aim to generate samples from $D_{Y}$ .

Other Propagation Methods

Some approaches linearize $f$ , and then propagate the distribution through a linear function (which may not be trivial for some distributions). Other approaches approximate the distribution $D_{X}$ with a nice distribution (such as normal), and then propagate this through the nonlinear function $f$ . One such approach is the Unscented Transform, which is a specific Sigma Point Method.

When $f$ involves the solution to a differential equation, this propagation can be described through the general Fokker-Plank Equation (see also PINNs for Evolution of pdfs), which adds a dimension to the differential equation to give an equation for the pdf.

Importance Sampling

The main idea is to use a biasing distribution ${\hat{D}}_{X}$ that is different from $D_{X}$ such that predictions of moments (aka evaluations of expected value) have lower variance. Then, fewer samples need to be drawn. The difficulty is constructing the biasing distribution ${\hat{D}}_{X}$ .

Surrogate Modeling

The approach is to replace $f$ with $\tilde{f}$ that is cheaper to evaluate (where linearization is one example). The difficulty is having these functions disagree. This ties into the idea of Computational Irreducibility. If there exists a function $\tilde{f}$ that is cheaper than $f$ , then $f$ is computationally reducible.

This category includes Multi-Fidelity methods. Generative Modeling can also be considered surrogate modeling.