Poisson's equation

As a simple illustration of how helical boundary conditions can lead to recursive solutions to partial differential equations, we consider Poisson's equation, which relates potential, u, to source density, f, through the Laplacian operator:

$$\nabla^2 u = f(x,y,z)$$ (1)

Poisson's equation crops up in many different branches of physics: for example, in electrostatics, gravity, fluid dynamics (where the fluids are incompressible and irrotational), and steady-state temperature studies. It also serves as a simple analogue to the wave-propagation equations which provide the main interest of this paper.

To solve Poisson's equation on a regular grid Claerbout (1997), we can approximate the Laplacian by a convolution with a small finite-difference filter. Taking the operator, ${\bf D}$, to represent convolution with filter, d, Poisson's equation becomes

$${\bf D \; u}={\bf f}.$$ (2)

Although ${\bf D}$ itself is a multi-dimensional convolution operator that is not easily invertible, helical boundary conditions Claerbout (1997) allow us to reduce the dimensionality of the convolution to an equivalent one-dimensional filter, which we can factor into the product of a lower-triangular matrix, ${\bf L}$, and its transpose, ${\bf L}^T$. These triangular matrices represent causal and anti-causal convolution with a minimum-phase filter, in the form

$${\bf D \; u}={\bf L}^T \; {\bf L} \; {\bf u}={\bf f}.$$ (3)

We can then calculate u directly since ${\bf L}$ and its transpose are easily invertible by recursive polynomial division:

$${\bf u} = {\bf L}^{-1} \; ({\bf L}^T)^{-1} \; {\bf f}.$$ (4)

 

lapfac
Figure 1 Deconvolution by a filter whose autocorrelation is the two-dimensional Laplacian operator. This amounts to solving the Poisson equation. After Claerbout (1997).