Add note on non negligible cost of finite differences to tutorial on …

…shape derivatives (#2883)

* add note on non negligible cost of finite differences to tutorial on shape derivatives

* mention that material parameters must be smoothed over the design grid

* fix typo

doc/docs/Python_Tutorials/Adjoint_Solver.md

@@ -856,7 +856,7 @@ Calculating the derivative of the diffraction efficiency ($F$) with respect to t
 
 ![](../images/levelset_gradient_backpropagation.png#center)
 
-The derivative $\partial \rho / \partial h$ can be approximated by a finite difference using a one-pixel perturbation applied to the grating height; this is computationally convenient because it greatly simplifies the construction of $\rho(h)$ as explained below. (A finite difference involves two function evaluations of $\rho(h)$, but the cost for this is negligible since it involves no Meep simulations.) The construction of $\rho(h)$ involves two steps: first, constructing a simple binary image $b(h)$ at a high resolution; and second, smoothing $b(h)$ into a continuous level-set function $\rho(h)$. This smoothing of the image can be performed using a number of different methods including a [signed-distance function](https://en.wikipedia.org/wiki/Signed_distance_function) or convolution filter. In this example, the smoothing is based simply on downsampling the image from a high-resolution grid (10$\times$ the resolution of the simulation grid) to the lower-resolution simulation grid using bilinear interpolation, which leads to "gray" pixels at the boundaries between materials in a way that changes continuously with $h$. Only these boundary pixels have nonzero derivatives in the Jacobian $\partial\rho/\partial h$ in this case. This calculation is summarized in the schematic below.
+The derivative $\partial \rho / \partial h$ can be approximated by a finite difference using a one-pixel perturbation applied to the grating height; this is computationally convenient because it greatly simplifies the construction of $\rho(h)$ as explained below. A finite difference involves two function evaluations of $\rho(h)$, but the cost for this is usually negligible since it involves no Meep simulations. However, in cases when there are a large number of shape parameters and a dense design grid, the cost of the finite differences can be non negligible: $M$ shape parameters require $M$ finite differences each involving $N$ grid points. The construction of $\rho(h)$ involves two steps: first, constructing a simple binary image $b(h)$ at a high resolution; and second, smoothing $b(h)$ into a continuous level-set function $\rho(h)$. This smoothing of the image can be performed using a number of different methods including a [signed-distance function](https://en.wikipedia.org/wiki/Signed_distance_function) or convolution filter. In this example, the smoothing is based simply on downsampling the image from a high-resolution grid (10$\times$ the resolution of the simulation grid) to the lower-resolution simulation grid using bilinear interpolation, which leads to "gray" pixels at the boundaries between materials in a way that changes continuously with $h$. Only these boundary pixels have nonzero derivatives in the Jacobian $\partial\rho/\partial h$ in this case. This calculation is summarized in the schematic below.
 
 ![](../images/levelset_jacobian_matrix.png#center)