he’s attempting to evolve from being a dumb human imho, getting off the tds koolaid would make the most difference that stuff leads to Oxford’s Dictionary 2024 word of the year
1. Fast Marching Method (FMM) Used for: Efficient front propagation on grids (e.g., shortest paths in continuous domains). Mathematical foundation: Solves the Eikonal equation: ∣∇T(x)∣=1F(x)|\nabla T(x)| = \frac{1}{F(x)}∣∇T(x)∣=F(x)1 where T(x)T(x)T(x) is the arrival time of the front at location xxx, and F(x)F(x)F(x) is the speed function. Requires upwind finite difference schemes to ensure stability and correctness. Complexity: Uses a min-heap or priority queue (like Dijkstra’s) for ordering updates efficiently. O(NlogN)O(N \log N)O(NlogN) time where NNN is the number of grid points. 2. Level Set Methods Used for: Modeling moving interfaces (e.g., flame fronts, shape evolution). Core math: Represents surfaces as zero-level sets of higher-dimensional functions. Solves Hamilton–Jacobi equations: ∂ϕ∂t+F∣∇ϕ∣=0\frac{\partial \phi}{\partial t} + F |\nabla \phi| = 0∂t∂ϕ+F∣∇ϕ∣=0 where ϕ(x,t)\phi(x, t)ϕ(x,t) is a signed distance function. Numerical methods: High-order ENO/WENO schemes, reinitialization techniques, and curvature-dependent flow. 3. Variational Methods / Gradient Descent in Shape Optimization Used for: Finding shapes or paths that minimize a cost functional. Math tools: Calculus of variations: Derive Euler–Lagrange equations for minimal paths. Functional minimization over infinite-dimensional spaces. Might involve Riemannian manifolds for geodesics, or Sobolev space embeddings. 4. Dynamic Programming / Bellman Equations Used for: Discrete optimal control, shortest paths. Mathematics: Bellman equation: V(x)=minu{L(x,u)+V(f(x,u))}V(x) = \min_{u} \left\{ L(x, u) + V(f(x, u)) \right\}V(x)=umin{L(x,u)+V(f(x,u))} The continuous analog leads to Hamilton–Jacobi–Bellman (HJB) PDEs, solved numerically or via policy iteration.
Your inputs relate to FMM but clearly you don't ask for code... so you're trying to appear like you're not actually retarded by posting output from AI (that is not grok). Aren't you like 75 years old?
i have all the code - i originated the code from my greater than human brain and am now on 4th derivative of it with the help of ai which is actually simple but i cant type near as fast.