OK, here goes. For the benefit of those who have plain text mail readers, I
will use a LaTex kind of notation for the math. I hope it's not too
cumbersome to decode.
I am debugging an algorithm I wrote for an adaptive meshless method. My
question is related to how to take the derivative of an integral with
variable bounds.
Ordinarily, if the integral looks like $\frac{d}{dx} \int_{u_1(x)}^{u_2(x)}
f(t) dt$ then the result is $f(u_2(x)) \grad u_2 - f(u_1(x))\grad u_1$. So
this is the case if the bounds are functions of the integration variable.
My case is more complicated; I have partial derivatives. Specifically, I
have an integral that looks something like this:
$I =
\sum_{j=1}^{i-1}
\int_{a_j(p_j)}^{b_j(p_j)}\phi_j(x,p_j)\phi_i(x,\vec{p_i})dx +
\int_{a_i(p_i)}^{b_i(p_i)}\phi_i(x,p_i)\phi_i(x,\vec{p_i})dx $ where the
$p_j$ are vectors of parameters for the basis functions $\Phi_j$. The
variable $x$ is a spatial variable, 1, 2, or 3-D. So in this case, the
bounds are functions of indexed parameter vectors $p_i$ as well as of the
integration variable.
Now, the point is I want to take the grad of $I$ with respect to the
specific parameter vector $\vec{p}_i$ (the other vectors
$p_j,~,j=1,2,\ldots,p_{i-1}$ are held constant).
How should I proceed? The more I look at it, the more it looks like I can't
really do this without more information.
One added wrinkle - the functions $a$ and $b$ use things like maxval(),
whose derivative is a little complicated; so taking the partial derivatives
of the bounds is not that straightforward.
My apologies if this appears disjointed, but this is as much as I can
simplify without possibly over-simplifying.
Alvaro Fernandez
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