[petsc-users] Converting complex PDE to real for KNL performance ?

[Sajid Ali]

 Hi everyone,

As Hong pointed out the optimization variable and gradient are both complex
in my use case. Just to give some context, the TS solves the IVP with the
parameters representing the refractive indices of the object at a given
orientation (Ni orientations in total). The optimization problem to solve
is :
obtain F such that for each θi ⊂ (0; π), obtain  yθi = TS(Aθi ∗ F)
(where Aθi represents a sparse matrix that rotates the F vector by angle
θi.)

Thus, a naive implementation for the same would be :
for i ⊂ (0; Ni) :
- obtain parameters for this orientation by MatMult( Aθi*F )
- obtain yθi = TS(Aθi ∗ F) and y'θi = TSAdjointSolve(Aθi ∗ F) (cost
function being L2 norm of yθi and actual data)
- rotate gradient back by MatMultTranspose(Aθi*F) and update F.

But in the future I'd have preferred to bunch the Ni misfits (with bounds
and regularizers) together as a multi-objective cost function and let TAO
handle the parallelization (whereby TAO is initialized with
`mpi_comm_world` but each PDE evaluation happens in it's own `sub-comm` and
TAO handles the synchronization for updates) and the order of estimations
(instead of naive sequential).

While I don't know the optimization theory behind it, the current practise
in the x-ray community is to model the forward solve using FFT's instead
and use algorithmic differentiation to obtain the gradients. My motivation
for exploring the use of PDE's is due to (a) Adjoint solves being faster
when compared to algorithmic differentiation (b) Multigrid solvers being
fast/optimal (c) PDE models being more accurate on downsampled data.

PS : @Alp : Could you share the slides/manuscript from the siam pp20
meeting that describes the new multi-objective minimization features in TAO
?


Thank You,
Sajid Ali | PhD Candidate
Applied Physics
Northwestern University
s-sajid-ali.github.io
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