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Exascale computing systems are essential for the scientific fields that will transform the 21st century global economy, including energy, biotechnology, nanotechnology, and materials science. Progress in these fields is predicated on the ability to perform advanced scientific and engineering simulations, and analyze the deluge of data. On July 29, 2013, ASCAC was charged by Patricia Dehmer, the Acting Director of the Office of Science, to assemble a subcommittee to provide advice on exascale computing. This subcommittee was directed to return a list of no more than ten technical approaches (hardware and software) that will enable the development of a system that achieves the Department's goals for exascale computing. Numerous reports over the past few years have documented the technical challenges and the non¬-viability of simply scaling existing computer designs to reach exascale. The technical challenges revolve around energy consumption, memory performance, resilience, extreme concurrency, and big data. Drawing from these reports and more recent experience, this ASCAC subcommittee has identified the top ten computing technology advancements that are critical to making a capable, economically viable, exascale system.

An a posteriori error formula is described when a statistical measurement of the solution to a hyperbolic conservation law in 1D is estimated by finite volume approximations. This is accomplished using adjoint error estimation. In contrast to previously studied methods, the adjoint problem is divorced from the finite volume method used to approximate the forward solution variables. An exact error formula and computable error estimate are derived based on an abstractly defined approximation of the adjoint solution. This framework allows the error to be computed to an arbitrary accuracy given a sufficiently well resolved approximation of the adjoint solution. The accuracy of the computable error estimate provably satisfies an a priori error bound for sufficiently smooth solutions of the forward and adjoint problems. The theory does not currently account for discontinuities. Computational examples are provided that show support of the theory for smooth solutions. The application to problems with discontinuities is also investigated computationally.