Question on Boundary Value Problem initial guess handling with adaptive grid refinement

[JBMcVey]

The recent PRs #500 and #511 addressed issues having to do with use of a previous solution object as the initial guess for a boundary value solution, something that is common in the continuation method for solving BVPs. The description of the fix in #511 indicated that the initial guess is stripped down to the extracted u0 vector. If this is the standard, then the user can easily do this as well by simply passing (for example) "sol.u" rather than "sol" for the initial guess. The only reason that I wanted to pass the complete solution object is to use adaptive grid refinement, as the solution object contains the non-uniform grid points in t, whereas the u0 vector assumes a uniform grid based on the specified value of "dt". The discussion in #511 seems to indicate, however, that the non-uniform grid information is never used for an initial guess. Therefore, passing the whole solution object rather than just a vector of solution values has no value that I can see.

One could still use adaptive grid refinement for continuation if one first interpolates the solution values for a non-uniform grid back to a uniform grid and then use that as the initial guess for the next call to the BVP solver. However, in my code I've found there is a significant amount of computational effort that goes into determining the optimal grid points for each solution call. This effort would then have to be replicated for each subsequent solution, which seems wasteful.

Is there any future plan to implement handling of initial guesses when adaptive grid refinement is used? If so, can you say anything about the timeline for such an enhancement?

[ErikQQY]

The only reason that I wanted to pass the complete solution object is to use adaptive grid refinement, as the solution object contains the non-uniform grid points in t, whereas the u0 vector assumes a uniform grid based on the specified value of "dt".

Just out of curiosity, so basically your initial is in the form of u being guess on uniform grid, but t being points on non-uniform grid? That does not feel right.

The discussion in #511 seems to indicate, however, that the non-uniform grid information is never used for an initial guess. Therefore, passing the whole solution object rather than just a vector of solution values has no value that I can see.

The non-uniform grid information is indeed being used as the initial grid for the solving process with a supplied solution object. Basically here is the real scenario: If only a vector of solution values are passed into the solver, the solver uses the length of vector to divide the time span and generate a uniform grid(with only solution values given, that is all it can do). But when a whole solution object are passed into the solver, the solver uses the u and t in the solution, and it does utilize the non-uniform grid information it provides.

[JBMcVey]

Just out of curiosity, so basically your initial is in the form of u being guess on uniform grid, but t being points on non-uniform grid? That does not feel right.

You are correct: it is not right. My interpretation of what Chris was saying in #511 seemed to say that it was necessary, which would be less than optimal.

The non-uniform grid information is indeed being used as the initial grid for the solving process with a supplied solution object. Basically here is the real scenario: If only a vector of solution values are passed into the solver, the solver uses the length of vector to divide the time span and generate a uniform grid(with only solution values given, that is all it can do). But when a whole solution object are passed into the solver, the solver uses the u and t in the solution, and it does utilize the non-uniform grid information it provides.

That is very good news. However, what Chris said in #511 when describing the fix was this:

Gate on AbstractVectorOfArray explicitly in MIRK, FIRK (expand + nested), and MIRKN, so initial-guess objects (previous solutions=, VectorOfArray/DiffEqArray, functions) are always stripped to the extracted u0 vector — i.e. restore the exact RAT v3-era semantics. (Ascher never had the remake and is unchanged; it predates this regression.)

To emphasize, he says that "initial-guess objects (previous solutions, VectorOfArray/DiffEqArray, functions) are always stripped to the extracted u0 vector". This sounds very much like a situation where "only a vector of solution values are passed to the solver" as you said. If I have misunderstood, please correct me. Thanks.

[ErikQQY]

To emphasize, he says that "initial-guess objects (previous solutions, VectorOfArray/DiffEqArray, functions) are always stripped to the extracted u0 vector". This sounds very much like a situation where "only a vector of solution values are passed to the solver" as you said. If I have misunderstood, please correct me. Thanks.

When the initial guess is an ODESolution/DiffEqArray, the solver will use the internal time step in it.