On the adaptive solution of space-time inverse problems with the adjoint method

Adaptivity in space and time is ubiquitous in modern numerical simulations. The large number of unknowns associated with today’s typical inverse problem may run in the millions, or more. To capture small scale phenomena in regions of interest, adaptive mesh and temporal step refinements are required, since uniform refinements quickly make the problem computationally intractable. To date, there is still a considerable gap between the state–of–the–art techniques used in direct (forward) simulations, and those employed in the solution of inverse problems, which have traditionally relied on fixed meshes and time steps. This paper describes a framework for building a space-time consistent adjoint discretization for a general discrete forward problem, in the context of adaptive mesh, adaptive time step models. The discretize–then–differentiate approach to optimization is a very attractive approach in practice, because the adjoint model code may be generated using automatic differentiation (AD). However, several challenges are introduced when using an adaptive forward solver. First, one may have consistency problems with the adjoint of the forward numerical scheme. Similarly, intergrid transfer operators may reduce the accuracy of the discrete adjoint sensitivities. The optimization algorithm may need to be specifically tailored to handle variations in the state and gradient vector sizes. This work shows that several of these potential issues can be avoided when using the Runge–Kutta discontinuous Galerkin (DG) method, an excellent candidate method for h/p-adaptive parallel simulations. Selective application of automatic differentiation on individual numerical algorithms may simplify considerably the adjoint code development. A numerical data assimilation example illustrates the effectiveness of the primal/dual RK–DG methods when used in inverse simulations.