Direct solution of Poisson's equation

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

$$\nabla^2 u = -f(x,y,z)$$ (13)

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 thesis.

To solve Poisson's equation with constant coefficients on a regular grid Claerbout (1997), we can approximate the differential operator, $-\nabla^2$, by a convolution with a small finite-difference filter, d. Taking the operator, ${\bf D}$, to represent convolution with filter, d, Poisson's equation becomes

$${\bf D \; u}={\bf f}.$$ (14)

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  

$${\bf D \; u}= {\bf L} \; {\bf L}^T \; {\bf u}={\bf f}.$$ (15)

The operator, ${\bf L}$, is the helical derivative, discussed in more detail by Claerbout (1998a). We can calculate u directly from equation () since ${\bf L}$ and its transpose are easily invertible by recursive polynomial division:

$${\bf u} = ({\bf L}^T)^{-1} \; {\bf L}^{-1} \; {\bf f}.$$ (16)

The right panel of Figure  shows the solution of the Poisson's equation with the single source and sink shown in the left panel. The center panel shows the intermediate result, ${\bf L}^{-1} \; {\bf f}$.

 

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).