Applying the Checkpointing Routine treeverse to Discretizations of Burgers' Equation?

We consider the inviscid Burgers' equation enlarged with a control function and its adjoint equation. For several discretization schemes of the inviscid Burgers' equation the \adjoint" nite diier-ence methods are derived. Applying these discretization methods and the checkpointing routine treeverse, approximations of the solution of both diierential equations are calculated and compared. 1 The Checkpointing Routine treeverse In its basic form the reverse mode of computational diierentiation yields the gradient of a scalar-valued function at a small multiple of the computational work needed to evaluate the function itself. However, the corresponding memory requirement is proportional to the run-time of the evaluation program. Therefore the practical applicability of the reverse mode is limited despite the availability of ever larger memory systems. This observation leads to the development of checkpointing schemes. Here, the key feature is a tradeoo between spatial and temporal complexity. Under ideal-ized assumptions it was shown in 2] that this tradeoo can be chosen such that both temporal and spatial complexity grow logarithmically relative to the time required to evaluate the function itself. The routine treeverse implements a reened version of the algorithm described in 2]. To apply treeverse, the user must be able to break the function evaluation into a sequence of time-steps that are roughly of equal size in terms of computational work. Furthermore procedures for the following tasks have to be available: 1. Advancing the modeled system to a certain state 2. Saving the current state onto a stack of checkpoints 3. Restoring the most recently saved program state 4. Performing one combined forward and adjoint step (with special provisions for the very rst time this happens) Naturally, the question arises where the checkpoints should be set to. The answer is given in the proofs of the following two propositions from 3]. They constructively yield checkpoint schedules that are optimal with respect to the number of forward steps and to the number of checkpoints that have to be set.