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Viscosity Solution

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Computer Vision

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Definition

Viscosity solution is a notion of weak solution for a class of partial differential equations of Hamilton-Jacobi type.

Background

A first-order partial differential equation of the type

$$ H\left(x,u(x), Du(x)\right)=0 $$
(1)

is called a Hamilton-Jacobi equation. A function u is said to be a classical solution of (Eq. 1) over a domain if u is continuous and differentiable over the entire domain and x, u(x), and Du(x) (the gradient of u at x) satisfy the above equation at every point of the domain. Consider the boundary value problem

$$ \left|{u}^{\prime }(x)\right|-1=0\;\mathrm{for}\;x\in \left(-1,1\right),u\left(\pm 1\right)=0. $$
(2)

By Rolle’s theorem, it is easily seen that classical solutions of the previous problem do not exist, whereas there exist infinite many weak solutions, that is, continuous functions which satisfy the equation at almost every point (the saw-tooth solutions, see Fig. 1a).

Viscosity Solution, Fig. 1
figure 2020figure 2020

Distance functions in 1D and 2D:(a) three...

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Author information

Authors and Affiliations

  1. Dipartimento SBAI, “Sapienza”, Università di Roma, Rome, Italy

    Fabio Camilli

  2. INRIA Rhône-Alpes, Montbonnot, France

    Emmanuel Prados

Authors
  1. Fabio Camilli
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  2. Emmanuel Prados
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Editor information

Editors and Affiliations

  1. Applied Robotics Research, Microsoft, Redmond, WA, USA

    Katsushi Ikeuchi

Section Editor information

  1. Tohoku University, Sendai, Japan

    Koichiro Deguchi

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Camilli, F., Prados, E. (2021). Viscosity Solution. In: Ikeuchi, K. (eds) Computer Vision. Springer, Cham. https://doi.org/10.1007/978-3-030-63416-2_686

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