This PR builds on the dual space changes recently introduced (#21, #25) and introduces the symbolic interpolation operator (ufl.Interpolate). This PR prepares the ground for operators that can be seen as operators and as forms such as Interp or ExternalOperator. Most of the bits in this PR come from the ExternalOperator dual branch which will come in another PR.
For example, let I(u, V2) be the interpolation of the coefficient u \in V1 into the function space V2, where I refers to the interpolation operator I: V1 -> V2. I can also be seen as a form using the reflexivity of V2. That is, we have I: V1 x V2^{*} -> R where I is linear with respect to both entries, we use the notation I(u, v^*) where v^* is a Coargument in V2
Main components of this PR:
Add BaseFormOperator:
We introduce the so-called BaseFormOperator class which accounts for operators than can also be seen as forms such as Interpolate or ExternalOperator.
Add Interpolate:
Add the Interpolate class which we refer to as the symbolic interpolation operator.
Add differentiation mechanisms for BaseFormOperator objects and Interpolate differentiation:
Add a BaseFormOperatorDerivative node
Add differentiation (2 stages mechanism) for base form operators:
dF(u, I(u, v^{*}); v)/du [uhat] = \partial F/\partial u + Action(dF/dI, dI/du)
which works by:
Applying differentiation by traversing the DAG and whenever we see a BaseFormOperator, we return 0 and record the operations, i.e. instead of returning dF/du we return \partial F/\partial u.
Applying chain rule on the recorded operations, i.e. add terms like Action(dF/dI, dI/du) for all the base form operators that got recorded.
This PR builds on the dual space changes recently introduced (#21, #25) and introduces the symbolic interpolation operator (
ufl.Interpolate). This PR prepares the ground for operators that can be seen as operators and as forms such asInterporExternalOperator. Most of the bits in this PR come from theExternalOperatordual branch which will come in another PR.For example, let
I(u, V2)be the interpolation of the coefficientu \in V1into the function spaceV2, whereIrefers to the interpolation operatorI: V1 -> V2.Ican also be seen as a form using the reflexivity ofV2. That is, we haveI: V1 x V2^{*} -> RwhereIis linear with respect to both entries, we use the notationI(u, v^*)wherev^*is aCoargumentinV2Main components of this PR:
Add
BaseFormOperator:We introduce the so-called
BaseFormOperatorclass which accounts for operators than can also be seen as forms such asInterpolateorExternalOperator.Add
Interpolate:Add the
Interpolateclass which we refer to as the symbolic interpolation operator.Add differentiation mechanisms for
BaseFormOperatorobjects andInterpolatedifferentiation:Add a
BaseFormOperatorDerivativenodeAdd differentiation (2 stages mechanism) for base form operators:
dF(u, I(u, v^{*}); v)/du [uhat] = \partial F/\partial u + Action(dF/dI, dI/du)which works by:Applying differentiation by traversing the DAG and whenever we see a BaseFormOperator, we return 0 and record the operations, i.e. instead of returning
dF/duwe return\partial F/\partial u.Applying chain rule on the recorded operations, i.e. add terms like
Action(dF/dI, dI/du)for all the base form operators that got recorded.Interpolatedifferentiation:dI(u, v*)/du [uhat] = I(uhat, v*)(by linearity).replace_derivative_nodesalgorithm for replacing derivative nodes.Update
analysis.py(cf. https://github.com/firedrakeproject/ufl/pull/26)