Global sensitivity analysis for cardiovascular models

Motivation, concepts, and intuition – Seminar at PoliMi 3–4 February 2026

Leif Rune Hellevik

NTNU and Politecnico di Milano (PoliMi)

Welcome & framing

Format

• Short conceptual introductions
• Interactive notebook demonstrations
• Discussion-oriented, not tool-heavy
• Slides: https://lrhgit.github.io/uqsa2025/

Welcome everyone, both students and faculty in Gliwice Poland, my postdoc Rahul in Trondheim, and to the group here at PoliMi (students, postdocs, and faculty), and in particular Francesco who is hosting me.

and I am Leif Rune Hellevik, professor in biomechanics at NTNU, Trondheim and currently at PoliMi at a sabbatical.

This seminar on GSA for cardiovascular models is meant to be practical and concept-driven: not a full course, not a software tutorial, but a guided exploration of how GSA helps us reason about uncertainty in models.

Emphasize: notebooks are used as a medium for explanation and exploration, not something participants are expected to run line-by-line during the seminar.

For you as attendants I encourage and hope for interaction and questions throughout the sessions.

Before we go any further, please open the seminar slides in your browser. https://lrhgit.github.io/uqsa2025/

That way you can follow links and jump to notebooks when we get to the interactive parts.

What we intended to do yesterday

• Frame and motivate the need for GSA
• Revisit basic statistics
• Introduce variance-based SA
• Explore interactive notebooks

Summary from day one

• Frame and motivate the need for GSA
• Revisit basic statistics
  • Expectation \(\mathbb{E}(X) = \int_{x_{\min}}^{x_{\max}} x\,p(x)\,\mathrm{d}x\)
  • Variance \(\operatorname{Var}(X) = \mathbb{E}\!\left[(X-\mathbb{E}(X))^2\right] =\operatorname{Var}(X)= \mathbb{E}(X^2) - \mathbb{E}(X)^2\)
  • Total law of variance \(\operatorname{Var}(Y) = \operatorname{Var}\left(\mathbb{E}(Y \mid X)\right) + \mathbb{E}\left(\operatorname{Var}(Y \mid X)\right)\)
• OAT vs GSA
• Scatterplots and normalized derivatives
• Demonstration of conditional variances with sliced scatterplots
• Sobol indices as a variance based means to rank parameters according to their influence on output uncertainty
• MC and PCE as computational methods to quantify the Sobol indices