README.md

BOUNTY

Bounded Uncertainty Numerical Toolbox.

BOUNTY is an interactive, web-based research tool for exploring invariant objects in set-valued dynamical systems with additive bounded noise. It was developed as part of the Applied Computing Project (ACP2) research course at the University of Oulu.

Live at: https://namvdo.github.io/set-valued-viz

Technical report: https://namvdo.github.io/bist_technical_report_24042026.pdf

Mathematical Background

In classical analysis, a single-valued function (or simply a function) $f: X \to Y$ assigns each point $x \in X$ to exactly one point $y \in Y$, written $y = f(x)$. Traditional dynamical systems using single-valued maps to describe deterministic evolution: given initial state $x_0$, the trajectory is uniquuely determined as $x_1 = f(x_0), x_2 = f(x_1), x_3= f(x_2)$ and so forth.

In contrast, a set-valued function (or multivalued map) $F: X \to \mathcal{(Y)}$ assigns to each point $x \in X$ a subset $F(x) \subseteq Y$ where $\mathcal{P}(Y)$ denotes the power set of $Y$. Rather than producing a single output, set-valued functions produce a set of possible outputs:

$F(A) = \bigcup_{x \in A} F(x)$ In our setting, we model bounded additive noise through set-valued map: $F(x) = B_\epsilon(f(x)) = {f(x) + \xi : |\xi| \leq \epsilon}$ where $f: \mathbb{R}^n \to \mathbb{R}^n$ is the underlying single-valued deterministic map (the Hénon map in our case), and $B_\epsilon(f(x))$ represent all possible perturbed states within distance $\epsilon$ of the deterministic image.

Rather than tracking every possible point within the noise ball $B_\epsilon(f(x))$ which would be computationally expensive to compute as the noise balls grow, we instead track the boundary evolution through an extended boundary map $F(x,y,nx,ny)=\bigl(f(x,y)+\varepsilon,\mathbf{nx}',\mathbf{ny}'\bigr)$. Since the maximum uncertainty occurs at the boundary $\partial B_\epsilon(f(x))$ (points at distance exactly $\epsilon$ from the deterministic image), we focus exclusively on tracking how these boundary points evolve.

Forward invariant-set approximation

For the Hénon map, Invariant-set propagation starts from either a random seed in the current computation domain or the position in Extended start state. BIST maps that seed deterministically, samples the ε-circle around its image as extended states, transports every outward normal with the inverse-transpose Jacobian, and repeats the forward boundary step for a user-selected number of iterations. The viewport can show all forward fronts or only the latest curve, with optional boundary samples and outward normals. Small transversal self-intersection loops are pruned before a curve is reused.

This is a finite sampled boundary approximation, not a proof that the final curve is invariant. Check convergence by increasing both the boundary point count and the number of forward iterations; the workspace research notes record the loop-erasure assumptions and domain stopping rules in study_docs/docs/forward_invariant_set_approximation.md.

Bifurcation of the Hénon boundary map with $a \in [0.594, 0.600]$, $b=0.3$, and $\epsilon=0.0625$

Getting Started

1. Clone the Repository

git clone <repository-url>
cd set-valued-viz

2. Build WebAssembly Module

cd frontend && npm install && npm run build-wasm

This creates the WebAssembly module in frontend/pkg/ so the worker and UI use the current Rust implementation.

3. Start the frontend server

cd frontend && npm install && npm run dev

License

MIT License — see LICENSE for the full text.

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.