Overview

Mathematical language, theory, and methods pervade research, development, testing, and evaluation encountered by the Army and the academic disciplines in science, engineering, and technology. Furthermore, increased demands are being placed on the mathematical sciences because of its role in building a foundation for emerging sciences and technologies in the information, network, life, decision, and social sciences. Although these problems are often naturally stated in terms of their disciplinary context, their solutions are often dependent on new mathematical results and theories. For example, promising approaches to computer vision for automatic target recognition (ATR) require research in a wide range of mathematics including constructive geometry, numerical methods for stochastic differential equations, Bayesian statistics, probabilistic algorithms, and distributed parallel computation. In the area of modeling and simulation of large-scale systems (systems-of-systems approach), improvements in model fidelity and capacity depend on the mathematics of optimization, stochastic methods, large-scale scientific computing and real-time computing for embedded systems. Similarly, advances in robotic and sensor systems depend on mathematics of dynamics, control, communication, logic, cooperation, and complexity.

In order to respond to these increasing demands on the mathematical sciences, the Army Research Office (ARO) supports and advances fundamental research and knowledge that focuses on the needs of the Army. To accomplish this objective, the Mathematics Division supports extramural basic research in the five areas that follow. The research supported by the Division does not cover all the topics in these areas, only those areas that are of strategic importance for the Army. The sub-disciplinary boundaries within the Division and the disciplinary boundaries in the ARO are not rigidly drawn and there is strong interest in and appreciation for multidisciplinary research in which the mathematical sciences play a major role.

 1.  Modeling of Complex Systems

The Modeling of Complex Systems Program involves fundamental mathematics-oriented research with objectives to develop quantitative models of complex phenomena of interest to the Army, especially those for which current models 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 Program are mainly human-made phenomena (information, wireless networks, geometric modeling), and human cognitive and behavioral phenomena. Complete and consistent mathematical analytical frameworks for the modeling effort are the preferred context for the research, but research that does not take place in such frameworks can be considered if the phenomena are so complex that such frameworks are not feasible. Metrics are part of the mathematical framework and are of great interest. Traditional metrics, when they exist, often do not measure the characteristics in which observers in general and the Army in particular are interested. For many complex phenomena, new metrics need to be developed at the same time as new models. Just as is the case for the modeling effort, these metrics should preferably be in a complete mathematical analytical framework. The research in modeling of and metrics for complex phenomena supported by the Modeling of Complex Systems Program may include numerical/computational work as a subordinate component. However, research that focuses mainly on numerical/computational issues should be directed to the Computational Mathematics Program in the Mathematics Division of the ARO. The investment in the Modeling of Complex Systems Program is in the following five areas.

1.1.  Information Fusion in Complex Networks

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.) has been under investigation, with considerable success, but with many questions still unanswered, for over a decade. Physics-based sensors are important, but they can't be in place "everywhere" and, especially in urban conflict, there are often not a lot of them in places where needed. Operations depend not only on information from physics-based sensors but also on "soft information," which includes signals intelligence (SIGINT─information from intercepted communications, radar, and other forms of electromagnetic transmissions), communications intelligence (COMINT─intercepted messages or voice information), open-source intelligence (OSINT─newspapers, radio, and TV programs), human intelligence (HUMINT), and databases. Extraction and fusion of soft information from text/voice and from databases has been extensively investigated, but not in the context of fusion with "hard information" from physics-based sensors. Full understanding in operational situations is provided by information from all sensors, where "sensor" now has a wider definition of "any source that provides relevant information." The information produced by the sensors has to be transmitted and fused in a fashion that provides reliable summary information with low error rates while using the minimum amount of network resources. Basic research in network-based fusion of hard information from physics-based sensors with soft information, as well as in network-based fusion of hard information and soft information separately, is needed. The Information and Signal Processing Program in the Computing and Information Sciences Division of the ARO supports related engineering research.

1.2.  Modeling of Wireless Communication, Sensor and Actuator Networks

Military wireless communication, sensor, and actuator networks are relatively flat and often have strong power and bandwidth constraints. Fundamental principles and metrics for wireless communication, sensor, and actuator networks are needed so that networks can be designed to fit operational requirements. In situations where a completely flat network is not acceptable, one has to identify which hierarchical structure should be used. If a minimal hierarchical structure is deemed advisable, for example, for scalability, or a "good" hierarchical structure is proposed, the metric measuring hierarchical structure has to be stated. Global or semi-global optimization of networks is of interest. Pricing frameworks adopted from economics and biologically inspired frameworks are options, but need to be justified based on principles from inside wireless networks, not by analogy with situations in economics and biology. Decomposition of (semi-)global network optimization into distributed components is of interest. For many situations, optimal performance under high load is important. (The network is most needed in crisis situations when "everyone" needs it.) Little is known about how to achieve optimal system performance under heavy load in relatively flat, dynamic wireless systems or even about what "optimal" means in such circumstances. Considerable evidence has accumulated that the properties of the traffic often follow heavy-tailed, rather than Gaussian or Poisson, distributions. Using statistical assumptions and error measures that correspond to properties of traffic in real networks is essential. Discovering and implementing quality-of-service (QoS) criteria that are mathematically sound (consistent with empirical observations and theoretical knowledge, as sparse as they are), practically useful, and computationally feasible in a distributed implementation are of interest. Of interest are obtaining asymptotic properties of QoS for large networks, discovering differences between the QoS resulting from different classes of routing policies, and obtaining conditions under which QoS-dependent routing policies exist and are unique. The efforts supported by the Modeling of Complex Program take a step back from typical engineering efforts and consider fundamental mathematical principles of network theory. The Mobile Wireless Communication and Networks Program in the Computing and Information Sciences Division and the Stochastic Analysis, Applied Probability, and Statistics Program in the Mathematics Division of the ARO support related engineering and statistical/stochastic research.

1.3.  Modeling of Irregular Objects and Functions

Representation of complex, irregular geometric objects and of complicated, often high-dimensional abstract phenomena and functions is fundamental for Army, Department of Defense (DoD), and civilian needs in modeling of urban and natural terrain, geophysical features, biological objects (including humans and their clothing), effectiveness of military training, and many other objects and functions. Real-time representation and visualization of three-dimensional (3-D) terrain (not just as a height field but with multivalent height functions and non-genus-0 topology) directly from real-time or stored point-cloud data cannot be achieved with current techniques. A key to achieving this goal is data compression at ratios and with accuracy that strongly exceed what is currently available. A multitude of variants of piecewise planar surfaces (including those on TINs and TMSs), splines, multiquadrics, kriging, wavelets, neural nets, and many other techniques developed in the past perform well on many types of data. However, none of these procedures are able to provide, without human intervention, representation of irregular objects and functions with the accuracy and compression that is needed. A new approximation theory that does not require the assumptions (primarily smoothness) of classical approximation theory and that provides structure for the many new non-smooth approximation techniques currently under investigation is required. Research on the metrics in which approximation should take place is needed. Approximation theory for information flow and other abstract phenomena in large wireless communication and sensor networks is of interest. The approximation theory developed under support of this program is expected to provide building blocks for computational geometry, pattern recognition, automatic target recognition, visualization systems, information processing, and network information flow. The Discrete Mathematics and Computer Science Program in the Mathematics Division and Signal Processing Program in the Computing and Information Sciences Division of the ARO support related computational and engineering research.

1.4.  Human Cognitive and Behavioral Modeling

Quantitative, analytical models of cognition and behavior are required for training, simulation (computer generated forces) and mission planning. One of the most challenging areas of cognitive and behavioral research has been the creation of these models. Models that do exist are often time consuming to build, require large amounts of data as input and have limited accuracy. Research focused on mathematically justified, practically useful, computationally tractable and data-tractable models is needed. ("Data-tractable" means "does not require more data or more intricate data than is realistically likely to be available."). Research on the metrics in which the accuracy of the models should be measured is needed. In modeling of training, new research is needed, particularly on new types of training such as distance learning, artificially intelligent trainers and virtual environments. The Stochastic Analysis, Applied Probability, and Statistics Program in the Mathematics Division of the Army Research Office and the Human Research and Engineering Directorate of the Army Research Laboratory support related statistical-modeling, empirical and experimental research.

1.5..  Additional Areas of Opportunity

Analytical procedures that provide new ways to "image" networks, such as "network tomography" (deduction of network topology or other network properties from measurements at a limited number of nodes and/or over a limited number of paths), will be required for the maintenance and protection of networks. The analysis and design of advanced complex materials for structures, armor, and sensors is an interdisciplinary area in which some basic principles are known but many more remain to be discovered. The interests of the Modeling of Complex Systems Program include these areas and may include mathematical-analysis-oriented research for other (non-biomedical) complex phenomena of interest to the Army that may be proposed by researchers.

Technical Point of Contact:  Dr. John Lavery, e-mail: john.lavery2@us.army.mil, (919) 549-4253.

2.  Computational Mathematics

Numerical computation has become an essential part of engineering design and scientific investigation. It is now possible to simulate potential designs and analyze failures after they have occurred. Such simulations often require considerable effort to set up, considerable computer time on large scale parallel systems and considerable effort to distill useful information from the massive data sets which result. In addition, it is not often possible to quantify how well the models simulate the real problem or how accurate the simulation is. This problem is especially acute for simulations of failure processes. Data has become ubiquitous but mathematically sound methods for incorporating the data into accurate simulations are lacking. Finally, simulations are not timely. The most recent example of this was the fact that the Corps of Engineers was not able to determine with enough reliability that the levees in New Orleans would fail before
they did. The emphasis in the Computational Mathematics program is on mathematical research directed towards overcoming these and related shortcomings.

Technical Point of Contact: Dr. Janet Spoonamore, e-mail: janet.spoonamore@us.army.mil, 919-549-4284.

3.  Stochastic Analysis, Applied Probability, and Statistics

The Stochastic Analysis, Applied Probability, and Statistics Program supports critical Army needs in decision making under uncertainty.

3.1.  Statistical Methods

There is great interest in statistical methods for very large data sets or very small data sets, sampled from nonstandard, poorly understood distributions. The extraction of more information from small data sets requires improved methods for combining information from disparate sets, as in meta-analysis. Useful statistical models should be based on a thorough understanding of physical processes combined with sound statistical theory. Thus, it is important to integrate statistical procedures with scientific and engineering information about mechanisms as exemplified by a probabilistic methodology that describes the nature of the growth of cracks in different media and the associated statistical analysis. More research is required in several statistical areas including text data mining, Bayesian methods, Markov random fields, cluster analysis, change point methods, and Markov chain Monte Carlo methods. It is important to bring novel statistical thinking into resource management and optimization in very large communication and logistics networks.

Army research and development (R&D) programs directed toward system design, development, testing, and evaluation problems generate a need for research in the field of stochastic processes, including stochastic differential equations. Special emphasis is placed on research into methods for the analysis of observations from phenomena modeled by such processes and to numerical methods for stochastic delay and partial differential equations that have Army relevant applications in life, physical, and information sciences. Research areas of importance to the Army in probability and its applications include 1) stochastic modeling, analysis, and control of complex multi-scale networks; 2) theory of spatial-temporal random fields and nonlinear filtering that are applicable to ATR, information assurance, anomaly detection, and antiterrorism in real time; 3) stochastic optimization and approximation in mathematics of operations research, manufacturing systems, and supply chain management; 4) optimal control of stochastic delay and partial differential equations driven by Levy processes or fractional Brownian motion; and 5) stochastic analysis and control of fluid turbulence (especially turbulence in helicopter aerodynamics) and complex physical and biological systems that exhibit the properties of long range dependence and self-similarity. Ideas are needed from Markov random fields, stochastic systems with memory, nonlinear stochastic analysis, and infinite-dimensional stochastic differential equations. The techniques required include large deviations, heavy traffic analysis, interacting particle systems, Pontryagin maximum principle for non-Markovian processes, infinite-dimensional Hamilton-Jacobi-Bellman equations and inequalities, stochastic attractors, and semi-flows.

Technical Point of Contact:  Dr. Mou-Hsiung Harry Chang, e-mail: mouhsiung.chang@us.army.mil  919-549-4229.

4.  Discrete Mathematics and Computer Science

Discrete mathematics and computer science play key roles in the effective implementation of the digital battlefield. Research in these areas is also crucial in providing tools to dominate information warfare and determine and analyze alternatives for battle strategies. The main goals of this program are to enhance the understanding of discrete phenomena and digital information environments, provide rigorous algorithmic foundations and better modeling tools, as well as advance the underpinnings of the mathematics and enabling technology for distributed interactive simulation for both physically and non-physically based models. (Training simulation is an example of a non-physically based model. Imagery for automatic target recognition is a physically based model.) The major thrusts in this program are targeted to address the Army and DoD's interests in training, war-gaming, reconnaissance, surveillance imaging, battlefield management, large scale information and data management, and virtual environments and prototyping.

4.1.  Discrete Mathematics

The foci of the research in discrete mathematics are the development and analysis of solution procedures for discrete problems in computational geometry, computational algebra, robust geometric computing, logic, network flows, graph theory, and combinatorics. Research in these areas offers powerful tools for a number of Army applications including robotics; autonomous navigation; battle management; command, control, and communications (C3); virtual prototyping; and manufacturing; and computational modeling; and simulation. In addition this subarea supports Army interests in Soldier systems and vulnerability and lethality analysis which may require geometric and solid modeling, interactive graphics, and 3-D visualization tools, as well as physically based modeling.

4.2.  Theoretical Computer Science

Advances in computer hardware and architecture continue to outpace development in algorithms and software for the solution of applied physical and biological problems, such as terrain modeling and human dynamics. Many computationally intensive problems, such as those encountered in advanced distributed simulation, require rapid information processing and manipulation of extremely large and often heterogeneous data sets. Of interest is research on fundamental issues in parallel computing and algorithms; distributed computation; models and algorithms for the control of heterogeneous concurrent computing; input/output (I/O) communication and large‑scale memory management; human‑computer interface and synthetic environments, etc. Exploring fundamental techniques that advance computational algorithms and analytical tools to enable battlefield digitization is a research area of great strategic importance to the Army.

Technical Point of Contact Dr. Janet Spoonamore, e-mail: janet.spoonamore@us.army.mil 919-549-4284.

5.  Cooperative Systems

The goal of this work package is to exploit the power of collaboration and cooperation in complex intelligent systems to enhance Army systems and operations. Research areas include the mathematical foundations of system theory, communication, networking, language, learning, swarming, game theory, decision-making, and information processing related to autonomous intelligent systems. This program involves innovative, mathematics-based research to study and advance the understanding and utility of multi-component, adaptive intelligent systems (e.g., multi-robot systems, human-machine systems, groups of pursuers and evaders, self-optimizing communication or transportation systems, sensor networks, and expert artificial intelligence (AI) systems). These systems can originate from applications in any form (i.e., physical, informational, social, behavioral, or life sciences) and are often multi-scaled and complex (systems of systems). While multi-component, information-rich systems can utilize centralized management, this research program seeks to replace centralized organization with distributed cooperation (component collaboration) through the development of structures and processes for communication, adaptation, learning, reasoning, and decision making by many, if not all, components of the system. Principal research areas include the mathematical foundations for and the qualitative and quantitative analysis of distributed system theory; measures of system complexity; measures of the value of system information and intelligence; models for the transfer of data into information and information into intelligence (text exploitation); interoperability and connectivity of communication/transportation/logistics nets; power and limitation of swarming phenomena; multi-player/multi-objective game theory; information processing and data fusion for decision making; adaptive data structures for intelligent and dynamic systems; applications of cellular automata to problems in distributed control; development of methodology for language (formal and natural) and cognition for autonomous systems; metrics and theories related to networks of various types; and new architectures and processes for streamlined command, control, computer, communication, intelligence, surveillance, and reconnaissance (C4ISR) systems. Major objectives of the program are to develop new mathematical language, theories, structures, and processes for the development of fundamental principles to understand information science, network science, cognitive science, decision science, and intelligent cooperation and to apply the principles to science and technology for effective C4ISR systems. The program also supports mathematical development to enhance and employ cooperation into existing systems with naturally different or conflicting components or structures (e.g., hybrid, multidisciplinary, multi-scale, or multi-perspective mathematics such as discrete/continuous, linear/nonlinear, and deterministic/stochastic).

Technical Point of Contact:  Dr. Chris Arney, e-mail: david.arney1@us.army.mil, 919-549-4254