Calculus×Computing
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Laplacian

Level UniversityDifficulty ★★★★★Concept⌖ Open in the map

What is it?

Δf=∇⋅∇f=∑i∂2f/∂xi2\Delta f = \nabla\cdot\nabla f = \sum_i \partial^2 f/\partial x_i^2: how much ff at a point differs from the average of its neighbours. It appears in the heat, wave, Laplace, Poisson and Schrödinger equations, and its discrete version smooths meshes and detects edges.

Intuition

On a grid, Δf≈1h2(sum of the 4 neighbours−4f)\Delta f \approx \frac{1}{h^2}\big(\text{sum of the 4 neighbours} - 4f\big). If Δf<0\Delta f < 0 the point is above its neighbours' average (a local bump); heat flow ut=αΔuu_t = \alpha\Delta u flattens bumps. Functions with Δf=0\Delta f = 0 (harmonic) equal the average of their surroundings: the steady state.

Formulas

Δf=∇2f=∂2f∂x2+∂2f∂y2+∂2f∂z2\Delta f = \nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}
Δhfi,j=fi+1,j+fi−1,j+fi,j+1+fi,j−1−4fi,jh2\Delta_h f_{i,j} = \frac{f_{i+1,j} + f_{i-1,j} + f_{i,j+1} + f_{i,j-1} - 4f_{i,j}}{h^2}
five-point stencil

Where it shows up in computing

  • Heat equation and diffusion★★★★★fundamentalPhysics and simulation

    Diffusion is ut=αΔuu_t = \alpha\Delta u.

  • Wave equation★★★★★fundamentalPhysics and simulation

    Waves are utt=c2Δuu_{tt} = c^2\Delta u.

  • Mesh processing (discrete differential geometry)★★★★★fundamentalComputer graphics

    The discrete (cotangent) Laplacian is the core operator of geometry processing: smoothing, parametrization, deformation.

  • Schrödinger equation★★★★★fundamentalQuantum computing and physics

    Kinetic energy in quantum mechanics is −ℏ22mΔ-\frac{\hbar^2}{2m}\Delta.

  • Fluid dynamics and CFD★★★★★fundamentalPhysics and simulation

    The pressure projection solves a Poisson equation Δp=∇⋅u∗\Delta p = \nabla\cdot u^\ast every time step.

  • Image processing and computer vision★★★★★frequentSignals, media and vision

    Laplacian-of-Gaussian edge detection, sharpening and Poisson image blending.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

3D Computer graphics

⚛ Physics and simulation

ψ Quantum computing and physics

∿ Signals, media and vision

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