api: Symmetric interp gradients - #3001
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The symmetric placement costs an extra inline evaluation of the block, so it is wanted on the handful of equations that need it rather than on a whole operator. `Eq` takes an `interp_mode` and passes it down when it evaluates, leaving `sym_opt` as the default for everything else.
Two changes to what `interp-mode='symmetric'` re-associates. A derivative is where its `x0` puts it, not where the field it differentiates lives. `div(v)` of a staggered velocity lands on the node, and reading the operand's staggering instead made a node-centred kernel look off-node, so it was re-associated onto the operands and its compact stencil replaced by an interpolated one twice as wide. A composite is off only if one of its own operands is, which is what makes a sum of derivatives -- a divergence, a trace -- read as the node-centred quantity it is. The block is then chosen among the factors that are actually off `func`, and it takes two of them for there to be anything to re-associate. With one, the single interpolation `direct` puts on it is already the transpose-consistent form -- it is what makes the `i, j` entry of a stiffness matrix the transpose of its `j, i` entry. With two, `direct` interpolates each separately, and `I(a)*I(b)` is not `I(a*b)`: the discretized operator stops being the transpose of itself, which is invisible in a forward simulation and shows up as a first-order gradient in an adjoint one. The mode also travels down the fallback now. A product that cannot itself be re-associated is routinely wrapped around one that can -- a `dt` scaling, a sum of per-component contractions -- and dropping it there evaluated all of that in `direct`.
A zero-order derivative whose weights collapse to one is the identity, and it comes back as whatever it was applied to. When that is a sum -- the engineering shear strain of a staggered velocity, say -- the rebuild asserted rather than accepting the degenerate result it already handles for a single argument. It is reachable through the symmetric placement of a stiffness contraction.
Four of the five fail without the preceding commits. The fifth guards against over-reach and passes either way: a derivative already sitting on the target must be left alone.
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## main #3001 +/- ##
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+ Coverage 82.99% 83.02% +0.02%
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The staggered-interpolation notebook showed the two modes but not a case that needs the symmetric one. Add the gradient case it exists for: a nodal parameter read at a staggered location, and the matching accumulation back onto the node, where the standard placement is off by 4e-1 against 2e-7.
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extend interp mode to cover corner cases. Needed for the recipes