53 | | The variational problem then leads to the equation $$ \rho\Delta^2 f= \sum (f_i-f) \delta ({\bf r-r_i}) $$ The solution is still obtained as a sum of radial basis functions $\phi({\bf|r-r_i}|)$, whose weight $S_i$ now satisfies $M_\rho*S=F$, with the matrix $$ M_\rho=M+\rho I_{000} $$ where $I_{000}$ is the NxN identity matrix extended by three columns of 0.^ |
| 53 | $ E = \sum(|f({\bf r_i})-f_i|^2)+ \rho\int\left[\left(\frac{\partial^2 f}{\partial x^2}\right)^2 + 2\left(\frac{\partial^2 f}{\partial xy}\right)^2 + \left(\frac{\partial^2 f}{\partial y^2}\right)^2 \right] \textrm{d} x \, \textrm{d}y $ |
| 54 | |
| 55 | The variational problem then leads to the equation |
| 56 | |
| 57 | $ \rho\Delta^2 f= \sum (f_i-f) \delta ({\bf r-r_i}) $ |
| 58 | |
| 59 | The solution is still obtained as a sum of radial basis functions $\phi({\bf|r-r_i}|)$, whose weight $S_i$ now satisfies $M_\rho*S=F$, with the matrix |
| 60 | |
| 61 | $ M_\rho=M+\rho I_{000} $ |
| 62 | |
| 63 | where $I_{000}$ is the NxN identity matrix extended by three columns of 0.^ |
56 | | Spatial derivatives of the interpolated quantity $f$ can be obtained by direct differentiation of the result. For any function $\phi(r)$, with radial distance $r=|{\bf r-r_i}|$, $r^2=X^2+Y^2$, $\partial_X \phi=(d\phi/dr) \partial_X r$, and $\partial_X r=X/r$. This yields $\partial_X \phi=X (2 \log(r) +1)$, so that,^ |
| 66 | Spatial derivatives of the interpolated quantity $f$ can be obtained by direct differentiation of the result. For any function $\phi(r)$, with radial distance $r=|{\bf r-r_i}|$, $r^2=X^2+Y^2$, $\partial_X \phi=(d\phi/dr) \partial_X r$, and $\partial_X r=X/r$. This yields $\partial_X \phi=X (2 \log(r) +1)$, so that, |