- Home
- About ARL
- ARL S&T Campaign
- Extramural Basic Research
- Information Sciences
- Mathematical Sciences

# Mathematical Sciences

U.S. Army Research Office

ATTN: RDRL-ROI-M

P.O. Box 12211

Research Triangle Park, N.C. 27709-2211

Commercial: (919) 549-4321

DSN: 832-4321

Fax: (919) 549-4354

The mathematical sciences have great impact on a wide range of Army systems and doctrine. The objective of the programs of the Mathematical Sciences program is to respond to the quantitative requirements for enhanced capabilities of the Army in the twenty-first century in technologies related to the physical, informational, biological, and behavioral sciences and engineering. Mathematics plays an essential role in measuring, modeling, analyzing, predicting, and controlling complex phenomena and in understanding the performance of systems and organizations of critical interest to the Army. In particular, mathematical sciences are integral parts of research in network science, decision science, intelligent systems, and computational science. Mathematical Sciences also play an important role in solving Army issues related to materials, information, robotics, networks, C4ISR, testing, evaluation, decision-making, acquisition, training, and logistics. With the advent and subsequent refinement of high-performance computing and large-scale simulation, the mathematical sciences have become an integral part of every scientific and engineering discipline and the foundation for many interdisciplinary research projects. Computing and simulation now complement analysis and experimentation as fundamental means to understand informational, physical, biological, and behavioral phenomena. High-performance computing and advanced simulation have become enabling tools for the Army of the future. Real-time acquisition, representation, reduction, and distribution of vast amounts of battlefield information are key ingredients of the network-centric nature of the modern digital battlefield. Management and understanding of modern information-dominated battlefields and complex, inter-related networks provide significant motivation for fundamental research in the design and development of intelligent and cooperative systems.

The Mathematical Sciences Division seeks to create coherent basic research programs that are revolutionary, innovative and yet responsive to the future needs of the Army. Research supported within this Division falls into four areas or programs:

- Probability and Statistics
- Modeling of Complex Systems
- Numerical Analysis
- Biomathematics

The Mathematical Sciences Division supports the following research areas:

### Division Chief

Dr. Joseph Myers

(919) 549-4245

joseph.d.myers8.civ@mail.mil

## Biomathematics

Dr. Virginia Pasour

(919) 549-4254

virginia.b.pasour.civ@mail.mil

Biomathematics is an exciting and important new area of activity for ARO. The introduction of Biomathematics as a separate area of basic research recognizes the importance and specialized nature of quantitative methods in the biological sciences. Biology involves a large number of entities that interact with each other and their environment in complex ways and at multiple scales. This complexity makes Biomathematics a highly interdisciplinary field that requires unique and highly specialized mathematical competencies to quantify structure in these relationships. Mathematical techniques currently utilized in the field range from bioinformatics and computational biology techniques for analyzing small-scale -omics data to multicompartmental modeling in physiology, epidemiology and neurobiology, to agent-based and network models involved in understanding ecosystem dynamics and human social dynamics. Beyond understanding biological systems, research in control techniques is also valuable for its potential application in militarily important areas such as microbial biowarfare and disease spread.

The goal of the Biomathematics program focuses on using existing mathematics and creating new mathematical techniques to uncover fundamental relationships in biology, spanning different biological systems as well as multiple spatial and temporal scales. Of special interest are high-risk attempts to use techniques in fields not traditionally brought to bear on biological problems, as well as innovative efforts at handling large amounts of complex data.

## Modeling of Complex Systems

Dr. Joe Myers

(919) 549-4245

joseph.d.myers8.civ@mail.mil

The Modeling of Complex Systems program is fundamental mathematical-analysis-oriented research with the objectives to develop quantitative models of complex phenomena of interest to the Army, especially for those current models that are not based on first/basic principles, and to develop new metrics, preferably those based on first/basic principles, for these models. The complex phenomena of interest to the Modeling of Complex Systems are mainly human-made phenomena (information, wireless networks, geometric modeling) and human behavioral, cognitive and social phenomena. The basic research carried out by this program contributes to the Future Force Technologies Command, Control, Communication, Computers, Information, Surveillance and Reconnaissance (C4ISR), lethality, survivability and mobility and to the Future Combat Systems characteristics comprehensive situational awareness, networked fires—extended range lethality, survival of first engagement, manned/unmanned integration and reduced logistics.

The complex systems of interest to the Modeling of Complex Systems program include those in the following four areas:

- Network-Based Information Fusion—Information superiority is recognized as a key to success in military conflict, peacekeeping and humanitarian operations. Network-based sensing by organized or self-organizing networks of large numbers of geographically dispersed physics-based sensors of various modalities (optical, IR, acoustic, electromagnetic, etc.), information-based sources and human-based sources is an area of prime interest. Targets of interest are physical, informational, cognitive and social targets.
- Geometric and Topological Modeling—Representations of complex, irregular objects and of complicated, often high-dimensional abstract phenomena and functions are fundamental for Army and other DOD agencies in modeling 3-D urban and natural terrain, geophysical features, biological objects (including humans and their clothing), effectiveness of military training and many other objects and functions.
- Human Cognitive and Behavioral Modeling—Quantitative, analytical models of cognition and behavior are required for training, simulation (computer generated forces) and mission planning.
- Human Cognitive and Behavioral Modeling—Quantitative, analytical models of cognition and behavior are required for training, simulation (computer generated forces) and mission planning.

### More information:

## Probability & Statistics

Dr. Joseph Myers

(919) 549-4245

joseph.d.myers8.civ@mail.mil

The Probability and Statistics program supports research in stochastic analysis, applied probability, and statistical methods in response to the Army's need for real-time decision making under uncertainty and for the test and evaluation of systems in development. Special emphasis is placed on methods for analyzing data obtained from phenomena modeled by such processes. The two major areas of research are described below.

### Stochastic Analysis and Applied Probability

Many Army research and development programs are directed toward modeling, analysis, and control of stochastic dynamical systems. Such problems generate a need for research in stochastic processes, random fields, and/or stochastic differential equations in finite or infinite dimensions. The thrust research areas in stochastic analysis and applied probability include but are not limited to the following:

- Stochastic Delay and Partial Differential Equations—Research on analytical and numerical methods for solving stochastic delay and partial differential equations and their related nonlinear filtering and control problems is one of the program objectives. These equations play an important role in modeling many physical and biological processes in continuum and under noisy environment.
- Complex and Multiscale Networks—Stochastic modeling, analysis, and control of complex multiscale networks that address issues in (i) command and control of joint/combined networked forces; (ii) impact of network structure on organizational behavior; and (iii) relationship of network structure to scalability and reliability; and (iv) reliability and survivability are among the research priorities in the Probability and Statistics Program.
- Spatial-Temporal Event Pattern Recognition—Developments of theoretical foundation and efficient algorithms for spatial-temporal event pattern recognition in nonlinear and noisy environments are considered keys to winning the war against terrorism.
- Quantum Stochastics and Quantum Control—With technological advances now allowing the possibility of continuous monitoring and rapid manipulations of system at quantum level, there is an increasing awareness of the applications and importance of quantum filtering and quantum control in engineering of quantum states, quantum error correction, quantum information, and quantum computation.
- Stochastic Pursuit-Evasion Differential Games with Multiplayers—Studies on multiplayer stochastic pursuit-evasion differential games, hunter-prey relationships, and swarming behavior, will be helpful in efficient operations of autonomous agents, such as UAVs and ground vehicles, in large- and small-scale military operations.
- Stochastic Control of Systems Driven by Fractional Processes—Stochastic systems driven by fractional processes such as fractional Brownian motion and fractional Levy processes have wide-range applications in many areas of science and engineering. However, optimal control problems described by these systems remain unsolved. Research on the characterization and computation of the value functions and optimal control strategies for these problems are therefore solicited.
- Other Areas that Require Stochastic Analytical Tools—Mathematics of operations research such as scheduling, supply chain management, and manufacturing are also among the topics that will be considered under the program. Other research areas of importance to the Army in stochastic analysis and applied probability include (1) stochastic fluid dynamics and turbulence; (2) interacting particle systems and their applications to material science and nanotechnology; and (3) stochastic modeling and analysis of polymers.

### Statistical Methods

The following research areas are of interest to the Army and are important in providing solutions to Army problems.

- Analysis of Very Large or Very Small Datasets—The state of the art in statistical methods is well adapted to elicit information from medium-size data sets collected under reasonable conditions from moderately well understood statistical distributions. However, Army analysts frequently have very large or very small data sets sampled from nonstandard, poorly understood distributions.
- Reliability and Survivability—This research area is dedicated to the study of the performance and cost of engineered systems. Many of the models and methods developed will have immediate application to problems that face the military. For example, reliability and life-length methodologies are needed for analyzing mechanical and electrical systems, especially those with extremely low-failure rates. To support future network-centric operations, the Army needs novel and efficient statistical tools for improving network reliability and survivability, and for analyzing data collected from sensor networks.
- Data, Text, and Image Mining—Analysis of data stream in real time as well as cluster analysis and their applications to data, text, and image mining are important tools for anomaly detections in the global war against terrorism. New and unifying methodologies are needed in order to provide efficient search for patterns or meaning from the analysis of usually huge data sets that consist of multivariate measurements. Developments of mathematical theory for data, text, and image mining techniques are also highly desirable.
- Statistical Learning—Theoretical developments and computational approaches to statistical learning that are applicable to problems such as classification, regression, recognition, and prediction are crucial in making good and timely military decisions under uncertainty at all levels. Supervised and unsupervised learning methods (including learning decision and regression trees, rules, connectionist and probabilistic networks), visualization of patterns in data, automated knowledge acquisition, learning in integrated architectures, multistrategy learning, and multiagent learning are among the foci of statistical research in this program.
- Data Stream—The Army has pressing research needs in the area of streaming data. Especially, sampling theory methodology or the consideration of data epochs with meta-analysis relating findings across epochs may reduce the need to retain the entire stream of information. Since the information sought may be contained in a very small fraction of the data, useful methods for data reduction may depend on effective modeling of the data stream and the relationship of the relevant information to the overall stream.
- Bayesian and Nonparametric Statistics—Future emphasis in statistics on "predictive" models verses explanatory models is important. Military operations call for predictive models based on a growing base of sensor-fueled data stores. Increased computational capability is also leading statistics in a new direction, away from using "classical" results that are really approximations to avoid computational issues.

## Numerical Analysis

Dr. Joe Myers

(919) 549-4245

joseph.d.myers8.civ@mail.mil

The Numerical Analysis program supports the strategic themes of the Mathematics Sciences program by developing innovative, efficient, and accurate numerical methods and their implementations in scalable scientific software tools. Such methods and tools assure that mathematical models can be translated into realistic simulations. The quantitative predictions of many modern theories can only be derived from extensive computations. As Army problems become more complex, new and better approaches are needed to understand and develop their solutions. The focus of the Numerical Analysis program is on developing algorithmic methods to model new applications and discover general solution methods for large classes of problems.

Numerical computation and simulation have become an essential part of scientific investigation and engineering design. In science, it has become the accepted third component of the scientific method, complementing theory and experiment. In engineering, it enables us to simulate potential designs over wide parameter ranges and over a wide range of operating conditions, and to virtually autopsy failures after they have occurred. Such simulations often require considerable effort to set up, require considerable run time on large-scale parallel systems, and require considerable effort to distill useful information from the resulting massive data sets. In addition, we often operate our simulations in regimes or on scales at which we cannot collect reliable experimental data, and so, it is often not possible to directly quantify the fidelity of these models in all the regimes or scales in which we employ them. This problem is especially acute for simulations of failure processes. Data has become ubiquitous, but mathematically sound methods for incorporating these into simulations are incomplete. Research to improve validation, relevance and completeness in modeling are critical. The Numerical Analysis program enables mathematical research directed toward empowering the Army in these and related areas.