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Applied and Computational Mathematics Division

The Applied and Computational Mathematics Division (ACMD) develops mathematical and computational methods and tools that enable world-class measurement science. It collaborates with other divisions of the Information Technology Laboratory (ITL) and other laboratories across the National Institute of Standards and Technology (NIST), as well as other federal agencies to apply sophisticated mathematics, computational expertise, and tools to measurement and standards problems, with the goal of advancing trust in measurement and technology in service to the nation. It shares its results, reference data, tools, and standards with the scientific community.

Critical and emerging technologies in which the division has played a substantial part include the following:

• Advanced computing, including advanced modeling and simulation, and data processing and analysis
• Advanced engineering materials, including materials by design
• Artificial intelligence (AI), including machine learning, and AI assurance and assessment
• Biotechnologies, including biometrology, computational biology, and predictive modeling
• Semiconductors and microelectronics, including novel materials and specialized hardware
• Quantum information, including quantum computing, quantum sensing, and quantum communications
• Human–machine interfaces, including virtual reality and augmented reality

Not every project presented to the panel is discussed in the report. Only those projects about which the panel had comments are discussed.

ASSESSMENT OF TECHNICAL PROGRAMS

Overall, ACMD is both broad and deep, including state-of-the-art expertise and infrastructure for applied and computational mathematics, visualization, quantum computing and communications, and AI modeling. Their work is motivated by the needs of the nation, including international competitiveness and national security.

Accomplishments

A portfolio of projects was presented to the panel that reflected a broad spectrum of research capabilities within the mathematical modeling, analysis, and knowledge management spaces, as well as quantum information. This portfolio encompasses different types of fundamental mathematical modeling contributions (e.g. simulation, probability, machine learning, signal processing, and quantum algorithms), supporting resources and methodologies (e.g., knowledge management, visualization, and Internet of Things [IoT] devices), experimentation (with an emphasis on quantum computing and networks), and

application areas (e.g., medicine, energy, physics, and quantum computing). Several key portfolio areas and major contributions are highlighted below.

Classifying and identifying seized drugs with mass spectral library searching has demonstrated end-to-end contributions ranging from the development and maintenance of compound databases and analysis software to the highly impactful application of early detection of Fentanyl and other illegal drugs.

The NIST Digital Library of Mathematical Functions and Mathematical Knowledge Management is a large-scale library resource that has been systematically expanded and maintained for decades. This resource is widely known and used in the mathematics and scientific communities and has been cited more than 10,000 times in the past 14 years.

The Object Oriented MicroMagnetic Framework (OOMMF) forms the bedrock of many computational efforts in micromagnetics by providing a collection of portable and extensible public-domain programs. This effort has successfully built and sustained a community of users over the past 2 decades, with more than 3,700 citations since its inception. The commitment to long-term sustainability and quality control as illustrated by the OOMMF effort is a distinctive characteristic of NIST.

The Analysis of Separable Shape Ensembles project has demonstrated a deep and principled application of mathematical modeling for an underresearched but high-impact problem: shape analysis. While grounded in a concrete modeling problem (the design of wind turbines and airfoil shapes in general), the project is developing a principled representation of shapes, which can lead to a rich followup research program at the interface between shape analysis, deep mathematical modeling, and machine learning.

The Analysis of Diagnostics: Prevalence, Uncertainty Quantification, and Classification Theory research program has demonstrated the ability to attack high-complexity problems in an end-to-end fashion ranging from devices to highly informed mathematical models, and to package them within a high-impact health application area (i.e., cytometry). This project resulted in the creation of Lumos NanoLabs, a company that plans to commercialize the NIST microfluidic flowmeter.

The Computational Modeling of a Wearable System to Monitor Pulmonary Edema project models wearable networked IoT devices with supporting signal analysis and has been demonstrated within a sophisticated 3D visualization model. The group has long-term international leadership in wearable networks and IoT for health, as demonstrated by their development of high-fidelity computational models of the human lungs essential for benchmarking wearable devices, regular organization of special sessions at international conferences, and significant contributions to international standards in this area, such as Institute of Electrical and Electronics Engineers (IEEE) 802.15.6, IEEE Standard for Local and metropolitan area networks—Part 15.6: Wireless Body Area Networks.

Another set of portfolio projects centered around quantum computing and quantum networks. The Joint Center for Quantum Information and Computer Science—known as QuICS—is a collaborative center co-hosted by NIST and the University of Maryland (UMD). The research group articulated a highly successful partnership where NIST resources are augmented and complemented by UMD. This strategic partnership has demonstrated international leadership in the areas of quantum algorithms and quantum cryptography, as evidenced by their steady publications in top-tier journals, their lead in the creation of the Error Correction Zoo (an online catalog of more than 900 classical and quantum error correction codes), and more significantly, their contributions to the federal information standards for post-quantum cryptography: Federal Information Processing Standards 203, 204, and 205. This partnership allows NIST to access academic expertise, collaborative grants, and human resources as well as to benefit the UMD community and should be considered a gold standard for similar types of collaborations.

ITL researchers have world-leading expertise in quantum cryptography. In collaboration with the Computer Security Division and other NIST laboratories, ITL has made significant contributions to a Post-Quantum Cryptography Standardization program that was announced in 2016 as a competition by NIST to update the standards to include post-quantum cryptography. Final versions of the first three Post Quantum Crypto Standards were released by NIST on August 13, 2024.

One example of recent high-impact work on quantum algorithms is the understanding of the power of forgetting for quantum algorithms. Researchers at ITL and QuICS proved that, with reasonable

assumptions, to maintain an exponential speedup over any classical algorithms on a particular problem of finding an exit in a maze, the quantum walk algorithm must forget the path to get to the exit. This work sheds new light on fundamental understanding of what quantum algorithms can and cannot do.

On the experimental side, ITL has impressive laboratories for quantum components and systems development, quantum metrology, and metrology for quantum networks. ITL is part of a major effort of the NIST Gaithersburg Quantum Network (i.e., NG-QNet) Testbeds, which is a suite of testbeds being built on the NIST Gaithersburg campus to implement and characterize various aspects of quantum networks, consisting of a large collaboration with various NIST laboratories and other federal agencies. On the development of quantum systems and control, an example of a high-impact work is a publication on advances in automation of quantum dot device control in Review of Modern Physics, the most prestigious journal in physics (Zwolak and Taylor 2023).

Other portfolio projects at ITL include the advancement of Neuromorphic AI, which aims to develop computational systems that directly mimic the brain and to use these systems for AI.

The Visualization Laboratory has capabilities for both 2D and 3D visualization. Projects demonstrated within a virtual reality cave environment interacted with advanced application areas ranging from health to civil engineering. The work has had a tremendous impact on scientific visualizations through the integration of the Immersive Visualization Environment with Paraview, a widely used fully open-source software environment, and their ongoing efforts toward the development of standards for extended reality.

Opportunities and Challenges

Explicit information on any sort of a strategic plan for ACMD was lacking. A strategic vision for this division’s work was partially reflected in the presentations, but it could be communicated more explicitly and in a more coherent and systematic manner. Developing a strategic plan would serve to connect and cohere the diverse portfolio of projects in line with present and estimated future trends, link and position changes in response to external demands and mandates, and adapt to the emergence of new topics and the potential obsolescence of current topics. This would help with consolidating the project portfolio into fewer key strategic initiatives and staffing them appropriately within the current budget constraints, providing a method for preventing project fragmentation, and avoiding the creation of projects that are not aligned with, and divert resources from, the strategic goals of the division.

Recommendation 4-1: The Applied and Computational Mathematics Division should develop a strategic plan, derived from its strategic vision, to focus its efforts and resources on what have been determined to be the most important lines of work and to prevent the establishment of projects that are not aligned with the strategic vision and that would diffuse the division’s resources and reduce its impact.

AI will have numerous and perhaps significant impacts on ACMD’s work. Examples of high-impact potential opportunities include the use and development of large language models to support mathematical discovery (e.g., theorem proving), neuro-symbolic strategies, and conversely, how to deploy the expertise and resources of the division to improve the performance of large language models.

ACMD needs to develop a more ambitious AI strategy specific to ACMD’s aims. In addition to AI supporting and improving mathematical modeling, the infrastructures, tools, and methods are critical in an AI context need to be addressed. The key strategic areas of involvement and opportunities for national and international leadership within the AI space need to be identified.

Recommendation 4-2: The Applied and Computational Mathematics Division should develop a strategic plan that reflects an integrated vision of the impact of artificial intelligence (AI) on the division, both the short and long term. This plan should address critical questions such as the following:

1. How can AI support and improve productivity for mathematical modeling?
2. Which infrastructures, tools, and methods are critical within this context?
3. What are the key strategic areas of involvement and opportunities for national and international leadership within the AI space?

ASSESSMENT OF SCIENTIFIC EXPERTISE

ACMD currently has 62 federal employees. Of these, 56 hold PhDs, 52 are full-time, and 48 hold permanent appointments. There are 14 term appointees, including 6 National Research Council postdoctoral researchers. There are 46 associates, including 30 guest researchers (of whom some are postdoctoral researchers), 2 Professional Research and Experience Program postdoctoral researchers; 2

contractors; and 12 students.¹ The staffing levels have fluctuated between fiscal year (FY) 2015 and FY 2024. These levels are shown in Figure 4-1.

ACMD encompasses technical expertise in an outstandingly large number of areas of applied mathematics and computational science and has applied this expertise to an equally impressive number of application domains, balancing broader mathematical modeling areas and more specific focal areas such as quantum computing and communication, with a good balance between core mathematical modeling and applications to practical and industrial settings. The contributions within the division have a distinctive set of characteristics—namely, (1) the ability to flexibly approach a diverse spectrum of problem spaces and domains that require mathematical modeling expertise, (2) the ability to sustain long-term efforts for the construction of high-quality data and software resources, and (3) the capacity to balance a diversified portfolio of contributions as a service to other divisions across NIST while promoting its internal strategic areas. This diversity of expertise enables ACMD to engage with and support both internal and external partners in a diverse spectrum of collaboration areas, as well as to develop and sustain its research agenda.

Accomplishments

The portfolio of projects presented at the meeting showed unambiguously the technical quality of the scientific expertise within ACMD. Most notably, this is a unique team that integrates diverse and complementary scientific expertise areas such as the following:

• A set of long-term strategic research programs that have a claim to national and international research leadership, most notably in quantum algorithms, quantum cryptography, and quantum networks.
• The Digital Library of Mathematical Functions and OOMMF, which comprise a set of long-term resources and data sets that have visibility, impact, and an engaged community.
• A diverse portfolio of projects within high-impact application areas that requires a diverse and deep set of mathematical modeling skills—for example, contributions to standards for Body Area Networks in IoT-Health, analysis tools for better discrimination of drug compounds, and development of federal standards for post-quantum cryptography.

The staff collectively hold several distinctions, including

• American Association for the Advancement of Science fellow
• Association for Computing Machinery fellow
• American Statistical Association fellow
• American Physical Society fellow
• Washington Academy of Sciences fellow
• NIST fellow
• QuICS fellow

The staff collectively also hold several awards, including

• Department of Commerce Ron Brown Award
• Department of Commerce Gold Medal
• Department of Commerce Silver Medal
• Department of Commerce Bronze Medal

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¹ Associates are not NIST employees. They are outside researchers, both foreign and domestic, who collaborate with NIST researchers.

• Washington Academy of Sciences Excellence in Research in Applied Mathematics Award

Opportunities and Challenges

Given the rapid progress in generative AI and its impact on a widespread range of applications, ACMD would benefit from having more expertise in the AI area, whether through new hires or proactively upskilling its employees. Having some AI researchers and engineers on the staff will enable ACMD to adapt to the latest technology and to stay current on developments in this rapidly evolving field. While recruiting permanent staff can be a long-term option, establishing a contractor-based or visiting researcher program could be a sensible short-term option to support a more agile knowledge transfer in this domain. This would enable the division to identify new opportunities for integrating contemporary AI methods such as large language models with its existing research workflows.

Recommendation 4-3: The Applied and Computational Mathematics Division should expand the artificial intelligence (AI) expertise available to it. In the long term, it should add AI researchers and engineers. This can be accomplished through new hires, upskilling existing staff, or both. Until it can bring on permanent staff in this area, the division should establish a contractor-based or visiting researcher program to support a more agile knowledge transfer in this domain. These programs might help identify candidates for hiring.

BUDGET, FACILITIES, EQUIPMENT, AND HUMAN RESOURCES

Budget

The largest share of ACMD’s funding comes from scientific and technical research services appropriations. Other contributions include Innovations in Measurement Science—which is an internal NIST competitive grant program to fund NIST researchers to improve NIST’s capabilities, Strategic and Emerging Research Initiatives funding, and funding it receives for work done for other federal agencies. ACMD’s budget increased from approximately $15 million in FY 2015 to approximately $20 million in FY 2024. Figure 4-2 shows the ACMD budget each year between FY 2015 and FY 2024. Figure 4-3 shows the breakdown of the FY 2024 ACMD budget.

Facilities and Equipment

ACMD has access to state-of-the-art laboratories, including a data visualization laboratory with a virtual reality cave and laboratories for quantum network experimentation. A significant emphasis of the group is on mathematical modeling and software development. The environment addresses these needs with a mixture of private and shared workspaces and a portfolio of small and large meeting spaces.

Much scientific work happens in the moment, not on a schedule. Despite the general fit of the current physical infrastructure to the work being done, there is a lack of available meeting spaces for remote meetings and other conferences, whether over calls or in person, that can be used without a need for advanced scheduling. Acknowledging that office space is limited, it would be beneficial to designate rooms that researchers can reserve for remote meetings and other conferences with researchers without the need for advanced scheduling.

Recommendation 4-4: The Applied and Computational Mathematics Division should designate rooms that its staff can use for remote meetings and remote and in-person conferences with other researchers without the need to schedule them in advance.

Human Resources

It is clear that ACMD is attractive for recruiting and retaining staff. Discussions with permanent staff indicated that the reasons for this include general job security; significant research independence; diversity of technical challenges; focus on research without a substantial teaching, administration, or funding acquisition workload; and a collaborative research environment. For nonpermanent junior staff, the reported benefits include a rich technical environment and the possibility of long-term retention and mentoring.

The panel identified two opportunities for improvement. The first is anticipating the medium- and long-term strategic demands of the division that could align with future permanent positions and or contract extensions. The second is aligning the timing of the postdoctoral researcher recruitment process with the corresponding academic timelines.

EFFECTIVENESS OF DISSEMINATION EFFORTS

Between October 2022 and December 2023, ACMD staff produced 61 papers in peer-reviewed journals, 39 papers in conference proceedings, and 6 publications in various other venues. In addition, 17 papers had been accepted for publication and 40 were in review. ACMD staff gave 79 invited talks and 74 talks at conferences and workshops.

The staff are widely engaged with stakeholders, holding 11 journal editorial positions and 30 positions on various conference committees, and memberships in several organizations, including the following:

• Society for Industrial and Applied Mathematics Board of Trustees
• White House Office of Science and Technology Policy’s Working Group on Quantum Networks
• White House National Science and Technology Council’s Subcommittee on Future Advanced Computing Ecosystem
• World Wide Web Consortium Advisory Committee
• World Wide Web Consortium Math Interest Group
• Tcl Core Team
• International Federation for Information Processing WG 2.5 (Numerical Software)
• IEEE 802.15 Task Group 6ma (Body Area Networking)
• Khronos OpenXR Working Group

Accomplishments

ACMD’s software packages have had a broad reach, with evidence of having a broad impact. OOMMF is used for micromagnetic modeling. It was downloaded 5,200 times by 3,300 clients in 2023. Overall, this software package has been cited more than 3,700 times, including in 25 dissertations and 45 U.S. patent applications. There are more than 30 YouTube tutorials for this software package, and it is on nanoHub.

OOF,² used for modeling material microstructures, is also available on nanoHub. It was exercised 4,300 times in 2023 and has been exercised a total of 69,000 times since 2007. The current version is OOF2, and OOF3D is also available.

Automated Combinatorial Testing for Software, known as ACTS, has been downloaded by 4,775 distinct users since 2014.

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² This is known simply as OOF. For more information, see NIST (2023).

Other software that is in development or being distributed includes

• LaTeXML, a TeX/LaTeX to HTML/MathML converter
• Scikit-shape, a Python package for shape and image analysis
• G2Aero, a Python package for separable shape tensors
• Software for Joint Quantum State Tomography

In 2023, there were more than 14.3 million requests to the division’s web server by more than 940,000 visitors. The Digital Library of Mathematical Functions saw 5.1 million pages downloaded by 351,000 unique visitors and has been cited more than 10,000 since 2010. Other ACMD data being distributed includes Dark Solitons in Bose-Einstein Condensates data set, which supports machine learning, and QFlow, quantum dot data for machine learning.

The Handbook of Mathematical Functions has been a worthy, widely disseminated project. Its continued maintenance would be of great benefit to the broader mathematical community.

Opportunities and Challenges

The development of other reference materials like the Handbook of Mathematical Functions for the broader mathematical community might be another line of work suitable for ACMD.

Recommendation 4-5: The Applied and Computational Mathematics Division (ACMD) should maintain the Handbook of Mathematical Functions. ACMD should consider whether the division wants to develop new reference materials for the mathematical community.

A significant part of the work of the division is attractive to an audience beyond the usual stakeholders. While NIST has a Public Affairs Office and ACMD engages with it, it is unclear whether the current communication channels and processes are sufficient to provide adequate visibility of ACMD’s work to external stakeholders and others who may be interested in it.

While researchers do engage with the academic community and with the Department of Commerce more broadly, the strategy for communicating with external stakeholders is not clear. Some critical questions to address include the following: What are the other communities to which the division’s work needs to be visible (e.g., Congress, industry, education, or citizens)? What are the positive outcomes that can emerge from a broader engagement (e.g., increased funding and better access to human resources)? What communications channels are most accessible to these communities (e.g., events, videos, or textual content)? What is ACMD’s web presence strategy? What resources are required to deliver the best possible communication outcomes?

Last, dissemination metrics are based on numbers: of publications, downloads, and page views, for instance. This is a measure of reach. Metrics that can be used to measure and, critically, communicate impact to stakeholders and appropriators would be very useful but seem to be lacking.

Recommendation 4-6: The Applied and Computational Mathematics Division (ACMD) should develop a strategy for the improved communication of its work to stakeholders. ACMD should also develop additional metrics to better illustrate the impacts of its ongoing work.

REFERENCES