Homework 5

Theory

Suppose that we use the gradient operator for data interpolation:

$$\min\,\left\vert\nabla \mathbf{m}\right\vert^2\;.$$ (1)

This approach roughly corresponds to minimizing the surface area and represents the behavior of a soap film or a thin rubber sheet.

The corresponding inverse model covariance operator is the negative Laplacian $\mathbf{C}_m^{-1}=\nabla^T\,\nabla=-\nabla^2$ . The corresponding covariance operator corresponds to the Green's function $G(\mathbf{x})$ that solves

$$-\nabla^2 G = \delta(\mathbf{x}-\mathbf{x}_0)\;.$$ (2)

In 2-D, the Green's function has the form

$$G(\mathbf{x}) = \displaystyle A - \frac{\ln \vert\mathbf{x}-\mathbf{x}_0\vert}{2\pi}$$ (3)

with some constant $A$ .

To derive equation (3), we can introduce polar coordinates around $\mathbf{x}_0$ with the radius $r= \vert\mathbf{x}-\mathbf{x}_0\vert$ and note that the Laplacian operator for a radially-symmetric function $\phi(r)$ in polar coordinates takes the form

$$\nabla^2 \phi = \displaystyle \frac{1}{r}\,\frac{d}{dr}\,\left(r\,\frac{d \phi}{dr}\right)$$ (4)

Away from the point $\mathbf{x}_0$ , solving

$$\frac{1}{r}\,\frac{d}{dr}\,\left(r\,\frac{d G}{dr}\right) = 0$$ (5)

leads to $G(r) = A + B\,\ln r$ . To find the constant $B$ , we can integrate $\nabla^2 G$ over a circle with some small radius $\epsilon$ around the origin and apply the Green's theorem

$$-1 = \iint \nabla^2 G dx\,dy = \oint \nabla G \cdot \vec{ds} = \i... ...partial G}{\partial r}\right\vert _{r=\epsilon}\,\epsilon\,d\theta = 2\pi\,B\;.$$ (6)

Derive the model covariance function $G(\mathbf{x})$ which corresponds to replacing equation (1) with equation

$$\min\,\left\vert\nabla^2 \mathbf{m}\right\vert^2$$ (7)

and approximates the behavior of a thin elastic plate.

Homework 5