On Apr 18, 2018, at 15:06, Munson, Todd <[email protected]> wrote:

In the master branch, setting a H0 matrix and the gradient norm appear to be correct only for the quasi-Newton code. Those notions have not been propagated anywhere else.

Yes, that’s my understanding as well.

The quasi-Newton approximation, for example, can be used as a preconditioner in the Newton methods. If the H0 matrix is set and it uses a KSP solver to apply the "inverse", then do we need to add flexible variants of NASH, STCG, and GLTR for correctness?

Yes, or disallow quasi-Newton preconditioning for non-identity H0 matrices for these methods.

The H0 also need to be symmetric or act like a symmetric operator for the quasi-Newton approximation to be reasonable, and certainly when it is used as a preconditioner in NASH, STCG, and GLTR. Does this restrict the KSP solvers that can be applied when we need a inv(H0)*vector product?

I don’t see why it would restrict you to methods for symmetric operators.

At the end go the day, the question becomes how much thought and effort do we want to spend in order to support the H0 matrix? Or do we want to leave this as supported only for the TAO quasi-Newton method?

I think that’s up to you and the TAO developers. It’s helpful for mesh-independent convergence — it corresponds to a natural preconditioner — but if that feature isn’t critical for you guys in terms of funding or customers, I understand not implementing it. The abstraction also makes sensible the idea of mesh refinement or coarsening during the optimization algorithm because these operations correspond to mapping between different subsets of the same Hilbert space, and should be endowed with the same norms and inner products. These operations don’t really make sense if the decision variables are an n-tuple of reals with no additional structure. Geoff

Todd.

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On Apr 13, 2018, at 9:11 PM, Oxberry, Geoffrey Malcolm <[email protected]> wrote:

On 4/12/18, 15:06, "petsc-dev on behalf of Munson, Todd" <[email protected] on behalf of [email protected]> wrote:

I am not looking at Geoff's pull request right now.

Let me try to be clearer, in the master branch, the TaoGradientNorm() function is only used for termination tests inside the optimization methods. It does not change anything else that goes on inside of the methods.

Gradients and Hessians also depend on duality pairings, which in turn affects how users write these quantities, and how Hessians are approximated in quasi-Newton methods. Related code for the Hilbert space case was included in PR#347 (h/t Patrick Farrell).

A user-defined convergence test (presuming we can get the callbacks right) would suffice. As all norms are equivalent in finite dimensions, a user could also scale the standard termination tolerance by the correct constant.

This decision makes tolerances discretization-dependent, which is a leaky abstraction, and an unnatural way to encode algorithm convergence criteria. Scaling the $\ell_{2}$-norm in place of using the correct primal and dual norms also ignores the metric geometry of the primal and dual spaces, and will affect algorithm convergence negatively. A callback for a user-defined convergence test would be preferable.

If you need to live in function spaces, which seems to be the argument, then it seems that PETSc needs to be changed by more than just a single termination test.

A similar discussion can be found in the discussion of PR#347.

Thanks, Todd.

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On Apr 12, 2018, at 3:27 PM, Stefano Zampini <[email protected]> wrote:

The gradient norm is the one induced by the mass matrix of the DM associated with the control. In principle, TaoGradientNorm() can be replaced by DMCreateMassMatrix() + solve with the mass matrix.

For PDE constrained optimization, the “gradient norm” is crucial, since we consider optimization problems in Banach spaces. We should keep supporting it, maybe differently than as it is now, but keep it.

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On Apr 12, 2018, at 11:21 PM, Jed Brown <[email protected]> wrote:

Are you thinking about this PR again?

https://bitbucket.org/petsc/petsc/pull-requests/506

There's an issue here that Krylov methods operate in the discrete inner product while some higher level operations are of interest in (approximations of) continuous inner products (or norms). The object in PETSc that endows continuous attributes (like a hierarchy, subdomains, fields) on discrete quantities is DM, so my first inclination is that any continuous interpretation of vectors, including inner products and norms, belongs in DM.

"Munson, Todd" <[email protected]> writes:

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There is a bit of code in TAO that allows the user to change the norm to a matrix norm. This was introduced to get some mesh independent behavior in one example (tao/examples/tutorials/ex3.c). That norm, however, does not propagate down into the KSP methods and is only used for testing convergence of the nonlinear problem.

A few questions then: Is similar functionality needed in SNES? Are TAO and SNES even the right place for this functionality? Should it belong to the Vector class so that you can change the inner products and have all the KSP methods (hopefully) work correctly?

Note: that this discussion brings us to the brink of supporting an optimize-then-discretize approach. I am not convinced we should go down that rabbit hole.

Thanks, Todd.