On 2D Inverse Problems/Harmonic functions

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Harmonic functions can be defined as solutions of differential and difference Laplace equation as follows.

A function/vector u defined on the vertices of a graph w/boundary is harmonic if its value at every interior vertex p is the average of its values at neighboring vertices. That is,

u(p) = \sum_{p\rightarrow q} \gamma(pq)u(q)/\sum_{p\rightarrow q} \gamma(pq).

Or, alternatively, u satisfies Kirchhoff's law for potential at every interior vertex p:

\sum_{p\rightarrow q} \gamma(pq)(u(p) - u (q)) = 0.

A harmonic function on a manifold M is a twice continuously differentiable function u : M → R, where u satisfies Laplace equation:

\Delta_\gamma u = \nabla\cdot(\gamma\nabla u) = 0.

A harmonic function defined on open subset of the plane satisfies the following differential equation:

(\gamma u_x)_x+(\gamma u_y)_y = 0.

The harmonic functions satisfy the following properties:

  • mean-value property

The value of a harmonic function is a weighted average of its values at the neighbor vertices,

  • maximum principle

Corollary: the maximum (and the minimum) of a harmonic functions occurs on the boundary of the graph or the manifold,

  • harmonic conjugate

One can use the system of Cauchy-Riemann equations

\gamma u_x = v_y, \\
\gamma u_y = - v_x

to define the harmonic conjugate.

Analytic/harmonic continuation is an extension of the domain of a given harmonic function.

Harmonic/analytic continuation on a square grid

Dirichlet problem[edit]

Harmonic functions minimize the energy integral or the sum

\int_{\Omega}\gamma|\nabla u|^2 \mbox{ and } \sum_{e=(p,q)\in E} \gamma(e)(u(p) - u (q))^2

if the values of the functions are fixed at the boundary of the domain or the network in the continuous and discrete models respectively. The minimizing function/vector is the solution of the Dirichlet problem with the prescribed boundary data.