A Posteriori Error Estimates for Energy-Based Quasicontinuum Approximations of a Periodic Chain

We present a posteriori error estimates for a recently developed atomistic/continuum coupling method, the Consistent Energy-Based QC Coupling method. The error estimate of the deformation gradient combines a residual estimate and an a posteriori stability analysis. The residual is decomposed into the residual due to the approximation of the stored energy and that due to the approximation of the external force, and are bounded in negative Sobolev norms. In addition, the error estimate of the total energy using the error estimate of the deformation gradient is also presented. Finally, numerical experiments are provided to illustrate our analysis

All-at-Once Solution if Time-Dependent PDE-Constrained Optimisation Problems

Time-dependent partial differential equations (PDEs) play an important role in applied mathematics and many other areas of science. One-shot methods try to compute the solution to these problems in a single iteration that solves for all time-steps at the same time. In this paper, we look at one-shot approaches for the optimal control of time-dependent PDEs and focus on the fast solution of these problems. The use of Krylov subspace solvers together with an efficient preconditioner allows for minimal storage requirements. We solve only approximate time-evolutions for both forward and adjoint problem and compute accurate solutions of a given control problem only at convergence of the overall Krylov subspace iteration. We show that our approach can give competitive results for a variety of problem formulations