A class of implicit-explicit two-step Runge-Kutta methods

This work develops implicit-explicit time integrators based on two-step Runge-Kutta methods. The class of schemes of interest is characterized by linear invariant preservation and high stage orders. Theoretical consistency and stability analyses are performed to reveal the properties of these methods. The new framework offers extreme flexibility in the construction of partitioned integrators, since no coupling conditions are necessary. Moreover, the methods are not plagued by severe order reduction, due to their high stage orders. Two practical schemes of orders four and six are constructed, and are used to solve several test problems. Numerical results confirm the theoretical findings

A fully discrete framework for the adaptive solution of inverse problems

We investigate and contrast the differences between the discretize-then-differentiate and differentiate-then-discretize approaches to the numerical solution of parameter estimation problems. The former approach is attractive in practice due to the use of automatic differentiation for the generation of the dual and optimality equations in the first-order KKT system. The latter strategy is more versatile, in that it allows one to formulate efficient mesh-independent algorithms over suitably chosen function spaces. However, it is significantly more difficult to implement, since automatic code generation is no longer an option. The starting point is a classical elliptic inverse problem. An a priori error analysis for the discrete optimality equation shows consistency and stability are not inherited automatically from the primal discretization. Similar to the concept of dual consistency, We introduce the concept of optimality consistency. However, the convergence properties can be restored through suitable consistent modifications of the target functional. Numerical tests confirm the theoretical convergence order for the optimal solution. We then derive a posteriori error estimates for the infinite dimensional optimal solution error, through a suitably chosen error functional. This estimates are constructed using second order derivative information for the target functional. For computational efficiency, the Hessian is replaced by a low order BFGS approximation. The efficiency of the error estimator is confirmed by a numerical experiment with multigrid optimization

Space-time adaptive solution of inverse problems with the discrete adjoint method

Adaptivity in both space and time has become the norm for solving problems modeled by partial differential equations. The size of the discretized problem makes uniformly refined grids computationally prohibitive. Adaptive refinement of meshes and time steps allows to capture the phenomena of interest while keeping the cost of a simulation tractable on the current hardware. Many fields in science and engineering require the solution of inverse problems where parameters for a given model are estimated based on available measurement information. In contrast to forward (regular) simulations, inverse problems have not extensively benefited from the adaptive solver technology. Previous research in inverse problems has focused mainly on the continuous approach to calculate sensitivities, and has typically employed fixed time and space meshes in the solution process. Inverse problem solvers that make exclusive use of uniform or static meshes avoid complications such as the differentiation of mesh motion equations, or inconsistencies in the sensitivity equations between subdomains with different refinement levels. However, this comes at the cost of low computational efficiency. More efficient computations are possible through judicious use of adaptive mesh refinement, adaptive time steps, and the discrete adjoint method. This paper develops a framework for the construction and analysis of discrete adjoint sensitivities in the context of time dependent, adaptive grid, adaptive step models. Discrete adjoints are attractive in practice since they can be generated with low effort using automatic differentiation. However, this approach brings several important challenges. The adjoint of the forward numerical scheme may be inconsistent with the continuous adjoint equations. A reduction in accuracy of the discrete adjoint sensitivities may appear due to the intergrid transfer operators. Moreover, the optimization algorithm may need to accommodate state and gradient vectors whose dimensions change between iterations. This work shows that several of these potential issues can be avoided for the discontinuous Galerkin (DG) method. The adjoint model development is considerably simplified by decoupling the adaptive mesh refinement mechanism from the forward model solver, and by selectively applying automatic differentiation on individual algorithms. In forward models discontinuous Galerkin discretizations can efficiently handle high orders of accuracy, $h/p$-refinement, and parallel computation. The analysis reveals that this approach, paired with Runge Kutta time stepping, is well suited for the adaptive solutions of inverse problems. The usefulness of discrete discontinuous Galerkin adjoints is illustrated on a two-dimensional adaptive data assimilation problem