Library of Mathematical Functions—the work is unique and without peer in the broader community.

The work of the division is accomplished primarily through a culture of small “atomic” collaborations common in the work of applied mathematics. Impressively, members of the division have established effective collaborations with materials science and physics researchers.

On the downside, however, the culture of small, atomic collaborations may be a hindrance to attacking important areas that require the formation of collaborative teams from across many disciplines. However, the DLMF work demonstrated a capacity to form a larger team of collaborators than is typical of this division. This ability to form larger, effective collaborations may, in the long run, bode well for ACMD’s ability to cope with future multidisciplinary problem areas in which the ACMD will be called on to participate.

In regard to the charge of the Director of NIST to assess the technical merits and scientific caliber of the current laboratory programs relative to comparable programs worldwide: There are areas in which the accomplishments and capabilities of the ACMD are excellent and surpass the capabilities of comparable programs worldwide. The ability of staff to engage with physicists and materials scientists to use mathematics and computing to provide insight on experiments is among the most successful such efforts in the world. The Digital Library of Mathematical Functions is a unique and enduring accomplishment. The Quantum Communications Program seems to be among the best in the field.

In other areas the level of technology in the division work, while still excellent, is below the state-of-the-art level. That is, the work is excellent by academic standards in terms of the development of novel mathematical methods but is below the state of the art when measured against the work of the division’s peers in other mission agencies in meeting the goals of providing mathematical modeling and simulation expertise to the rest of their organizations. For example, in the area of large-scale partial differential equation (PDE) simulation, capabilities are mostly (but not exclusively) two-dimensional. In addition, the level of mathematical rigor varies widely across the division. Code verification is now viewed as an essential component for any large-scale simulation project. Very systematic convergence studies similar to those done in the Parallel Hierarchical Adaptive MultiLevel (PHAML) project might be adopted in other areas as a method of code verification.

As to the charge to assess the alignment between laboratory R&D efforts and those services and other mission-critical deliverables for which that laboratory is responsible: While the level of technical expertise in the ACMD is certainly sufficient to meet upcoming challenges, some significant cultural problems in the program composition and scientific culture in the division could inhibit the response of the ACMD to the challenges that the division views as critical to meeting its future needs.

The challenges facing the ACMD are multiscale and multiphysics, involve complex geometries and new applications, and are of necessity multidisciplinary. In addition, computational science is facing a major disruptive change in technology. This change is the transition to multiple computing cores on a single processor, for which many of the methods for parallelization and software engineering developed over the past 15 years lead to very low performance. Since this is an architectural change on the level of a single processor, it will affect scientific computing at all scales.

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