05/2017

Abstract





.

IRSN is deeply concerned about fast and efficient understanding of complex phenomena.

Whether for the prevention, anticipation or mitigation of nuclear accidents, safety experts must be able to analyze a complex situation whose parameters are unknown or uncertain.

Moreover, during a crisis management, the decision takers are awaiting relevant informations, without prerequisite for strong scientific knowledge.

Preventing
criticality accident

Issue

One only macroscopic value to check:

reactivity aka \(k_{effective}\), to remain \(<1.0\).





Issue

One only macroscopic value to check:

reactivity aka \(k_{effective}\), to remain \(<1.0\).

Usually few control parameters to consider:

fissile mass, neutrons leakage & slow-down, …



Issue

One only macroscopic value to check:

reactivity aka \(k_{effective}\), to remain \(<1.0\).

Usually few control parameters to consider:

fissile mass, neutrons leakage & slow-down, …

But no monotonicity, interactivity

\(SlowingDown_{Neutron} \to k_{effective}\)

\(Density_{Fissile} \to k_{effective}\)

…

Issue

One only macroscopic value to check:

reactivity aka \(k_{effective}\), to remain \(<1.0\).

Usually few control parameters to consider:

fissile mass, neutrons leakage & slow-down, …



No straightforward analytical solution in general.

(even for skilled neutrons physicist)

Tools

Simulation ?

Tools

Forward simulation:

\(SlowingDown, Mass, ... \to k_{effective}\)

\(\mathbb{R}^d \mapsto \mathbb{R}^+\)

Tools

Forward simulation:

\(SlowingDown, Mass, ... \to k_{effective}\)

\(\mathbb{R}^d \mapsto \mathbb{R}^+\)

This is NOT an answer to user's needs.


  • search most penalizing points:
    \(argmax_{SlowingDown,Mass,...} \left( k_{effective} \right)\)
  • identify safety set:
    \(arg_{SlowingDown,Mass,...} \left( k_{effective} < 1.0 \right)\)

Tools

Forward simulation:

\(SlowingDown, Mass, ... \to k_{effective}\)

\(\mathbb{R}^d \mapsto \mathbb{R}^+\)

Parametric computing

\[\oplus\]

Inverse problems algorithms.




Tools

Forward simulation:

\(SlowingDown, Mass, ... \to k_{effective}\)

\(\mathbb{R}^d \mapsto \mathbb{R}^+\)

Parametric computing

\[\oplus\]

Inverse problems algorithms.

How to visualise/analyse such results ?

(assuming the algorithms returned a relevant \(\mathbb{R}^d\) sampling)

Inverse problems

(Focus on)

Optimization
\(\color{red}{\{x^*\}} = argmax_{x \in \mathbb{R}^d} \left( f(x) \right)\)

Inversion
\(\color{red}{\{x^*\}} = arg_{x \in \mathbb{R}^d} \left( f(x) = 0 \right)\)

Inverse problems

(Focus on)

Optimization
\(\color{red}{\{x^*\}} = argmax_{x \in \mathbb{R}^d} \left( f(x) \right)\)

\(\color{blue}X \in \mathbb{R}^{d \times n}, \color{blue}{f(X)} \in \mathbb{R}^n\)

Inversion
\(\color{red}{\{x^*\}} = arg_{x \in \mathbb{R}^d} \left( f(x) = 0 \right)\)

\(\color{blue}X \in \mathbb{R}^{d \times n}, \color{blue}{f(X)} \in \mathbb{R}^n\)

Summary

Standard visualisation tools:

  • relevant for any sparse sampling produced by optimization/inversion algorithm
  • to allow analysis of d-dimensional sets (\(d<10\))
    • continuous (sampled) set
    • discrete set
  • providing user interactivity for deeper/wider analysis (eg. sharpness of optimizer)
  • without requirement for any set theory skills

Practical solutions

Practical solutions

Eg. Safe design of Pu powder storage

  • Stepwise Uncertainty Reduction algorithm
  • Sample ~ 1000 points
  • \(d=5\):
    • space between tubes,
    • fog density,
    • powder wetness,
    • Pu mass,
    • powder density

Eg. Safe design of Pu powder storage

Eg. Safety design (beyond)

… stability of safety set vs. reactivity

Eg. Safety design (beyond)

… define few reliable dominance points \(x^+\)
        such as \(f(x = \{x_i < x_i^+\}_{1 \leq i \leq d}) < 0\)

Eg. Safety design (beyond)

… provide some feeling about variables interactivity

Open issues

Assuming algorithm convergence


  • Pointwise/global (or exploitation/exploration) strategy ?
    (!!!) SUR algorithms
  • How to render/select sets instead of points ?
    Collection of dominance points ?
  • How to render uncertainty ?
  • How use full surogate information ?


Need for a new/extended parset/parfun visualisation tool.

References

References