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# Research Programs from BAA - Mathematical Sciences

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 ARO supports and advances fundamental research and knowledge that focuses on the needs of the Army. To accomplish this objective, the Division supports extramural basic research in the four 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 subdisciplinary 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.

## 1.1. Biomathematics

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 computational biology techniques for analyzing small-scale "-omics" data to multi-compartmental modeling in physiology, epidemiology and neurobiology, to agent-based and network models involved in understanding ecosystem dynamics and human social dynamics. Beyond contributing to the understanding of 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 ultimate 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 areas of mathematics not traditionally brought to bear on biological problems, as well as innovative efforts at handling large amounts of complex data.

Technical Point of Contact: Dr. Virginia Pasour, e-mail: virginia.pasour@us.army.mil, telephone: (919) 549-4254

## 1.2. Modeling of Complex Systems

The Modeling of Complex Systems Program is a program of fundamental mathematics-oriented research the objectives of which are 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 physical phenomena, human-generated 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 is mainly mathematical analysis (not computational mathematics).

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

### 1.2.1. Network-Based Information Fusion

Information superiority is recognized as a key to success in military conflict, peacekeeping and humanitarian operations. Networkbased 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. Extraction and fusion of "soft information," that is, information from human sources, such as text/voice and 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, physics-based or humanbased, 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.

### 1.2.2. Geometric and Topological Modeling

Representation of complex, irregular geometric objects and of complicated, often high-dimensional abstract phenomena and functions is fundamental for Army, 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. Realtime representation and visualization of 3D 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 triangulated irregular networks or "TINs" and triangular mesh surfaces or "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. 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.

### 1.2.3. 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 detailed data than is realistically likely to be available."). Research on the metrics in which the accuracy of the models should be measured is needed. 3.2.4. Additional Areas of Opportunity. Analytical research that provides new ways to model and measure complex physical, informational, cognitive and social systems of Army/DoD interest can be considered for inclusion in the Program.

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

## 1.3. Numerical Analysis

Numerical computation has become an essential part of both scientific inquiry and of engineering design. 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 and memory 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 often not timely. The most recent example of this is the Corps of Engineers' inability to predict 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.

### 1.3.1. For problems that are not time-limited, research areas of interest include but are not limited to the following:

a. Advances in Numerical Analysis. New methodologies are required for solving currently intractable Army problems. Advances which reduce computer time, are amenable to implementation on advanced computer architectures, are robust and have high order accuracy are of interest. Rigorous analysis is needed to determine structure, predict performance and drive adaptivity.

b. Multi-scale methods. More and more, problems of interest to the Army are characterized by the fact that behavior at microscopic scales has a large influence on performance of systems. To solve these problems, algorithms are needed to deal with different mathematical models at different scales, interacting subsystems, and coupling between models and scales. The emphasis is on mathematical methods which have universal application rather than methods applicable only to specific problem areas.

c. Verification and Validation. Models used for simulation may not be accurate due to uncertainties in the models themselves or uncertainties in parameters or interactions among components. Likewise, analytical and computational methods are needed to quantify errors generated by the translation of a model to a computer algorithm, the choice of parameters in the algorithm and the execution of the algorithm. Systematic methods are needed to evaluate and quantify these and other sources of uncertainty. The emphasis is on determining the accuracy of the entire simulation, not just on a particular computer code.

d. Data Driven Simulations. Advances in sensors and signal processing have greatly increased the amount of data available to scientists and engineers. The type of information which can be distilled from this data is different from and complements that generated by numerical simulation. If these two modes of investigation could be combined, it might be possible to obtain information unavailable to either mode acting alone. Uncertainties and incompatibilities between data and simulation make such combinations difficult. There is considerable interest in mathematical methods for combining data with simulation.

e. Supporting Technologies. As numerical computations become larger and more complex, the non-numerical issues become more important. Computers have heterogeneous architectures, multiple processors, and complex memory hierarchies. Data is distributed among multiple computers connected to each other over networks with different bandwidths. Without mathematical tools that map algorithms to architectures with minimal input from programmers and users, computation on such systems is difficult and time consuming. In addition, large scale computations produce huge data sets. Tools are needed to extract useful information from such data sets and to present results in ways that are easily understood.

### 1.3.2. Army systems often operate under unpredictable and adverse conditions

In the face of uncertainty, it would be very useful if results could be simulated fast enough to drive decision making, exercise control, and help avoid disaster. Such simulations need to be created, run, and interpreted in better than real time. While this may not be possible at this time, we seek research directed towards making this goal achievable. Such research should include but is not limited to the following:

a. Reduced Order Models. At this time, it is not possible to carry out full scale simulations in real time. In order to investigate the behavior of systems under a variety of possible scenarios, many runs need to be made. The only economical way to do this is through reduced order models. Possible methods to create these models include adaptive simplification, methods based on singular value decompositions, and reduced order numerics. All such approaches should be investigated. To be useful, all such models should be equipped with reliable estimates of accuracy.

b. Problem Solving Environments. If decision making is to be driven by simulation, it is necessary to set up simulations very quickly and obtain results in an understandable format. Matlab is one current tool for such a problem solving environment. Are there other approaches?

c. Embedded Simulation. As the size of powerful computers decreases, it should be possible to use simulation to drive control systems. What are the advantages and disadvantages of such an idea? How accurate do such simulations need to be?

d. Decision Making. One valid criticism of numerical simulation is that it takes so long to set them up, run them, and post-process the results that they cannot be used to guide decision making. The computational mathematics program is interested in any mathematical ideas that can help address this problem.

Technical Point of Contact: Dr. Joe Myers, e-mail: josephd.myers@us.army.mil, telephone: (919) 549-4245

## 1.4. Probability and Statistics

Many Army research and development programs are directed toward system design, development, testing, and evaluation which depend on the understanding of stochastic dynamical systems, stochastic processes, and statistical data. 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.

### 1.4.1. 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:

a. 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 important role in modeling many physical and biological processes in continuum and under noisy environment. These equations are often driven by standard Brownian motion, semi-martingale (e.g. Levy processes), and/or fractional Brownian motion. To effectively describe the state processes of these equations, it is necessary that infinite dimensional Banach or Hilbert spaces be employed. The Hamilton-Jacobi-Bellman theory via dynamical programming principle and/or necessary optimality conditions in terms of maximum principles have yet to be developed for optimal control of these infinite dimensional equations. Particularly challenging problems include the optimal control of these equations under partial and/or noisy observation with applications to informational, physical, and/or biological phenomena.

b. Complex and Multi-scale Networks. Stochastic modeling, analysis, and control of complex multi-scale networks that address issues in (1) command and control of joint/combined networked forces; (2) impact of network structure on organizational behavior; and (3) relationship of network structure to scalability and reliability, and (4) reliability and survivability are among the research priorities in the Probability and Statistics Program. Mathematical studies of biological networks and/or biologically inspired networks, such as molecular motors, protein dynamics, metabolic and gene expression networks are important elements in building Army's future combat systems. The Army also has a vital interest in resource management and optimization in very large networks, especially communication networks with stochastic components. Stochastic analysis and control of high-speed wired or wireless network traffic that exhibits properties of long range dependence and selfsimilarity are important. With limited availability of bandwidths in large scale wireless communication networks, research on dynamic spectrum allocation problems is urgent for military and commercial applications. Mathematics of operations research such as scheduling, supply chain management, and manufacturing are also among the topics which will be considered under the program.

temporal event pattern recognition in nonlinear and noisy environments are considered keys to winning the war against terrorism.

d. 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. These applications are extremely important in future military operations. Quantum mechanical systems exhibit an inherently probabilistic nature upon measurement. To further understand the back action effects of measurements on quantum states and control of the system based on these measurements, mathematical development of non-commutative quantum stochastic calculus, quantum filtering and quantum control theory is necessary. Proposed mathematical research of this nature that has potential applications in quantum information and quantum computation is hereby solicited.

e. Stochastic Pursuit-Evasion Differential Games with Multi-players. Studies on multi-player stochastic pursuit-evasion differential games, hunter-prey relationships, and swarming behavior, shall be helpful in efficient operations of autonomous agents, such as UAVs and ground vehicles, in large and small scale military operations. The formulation and characterization of the value function for multi-players pursuit-evasion games with asymmetric information and/or noisy and incomplete observations require further research attention. In particular, the scenarios in which the evaders have insider or anticipated information are particularly applicable to urban warfare.

f. 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 remained unsolved. Research on the characterization and computation of the value functions and optimal control strategies for these problems are therefore solicited.

g. Other Areas that Require Stochastic Analytical Tools. Mathematics of operations research such as scheduling, supply chain management, and manufacturing are also among the topics which will be considered under the program. Other research areas of importance to the Army in stochastic analysis and applied probability include (i) stochastic fluid dynamics and turbulence; (ii) interacting particle systems and their applications to material science and nano-technology; and (iii) stochastic modeling and analysis of polymers.

### 1.4.2. Statistical Methods

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

a. 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. The two situations lead to very different statistical problems. The information available in large multidimensional data sets is frequently obscured, which suggests the application of data mining methods. Large data sets may occur in a stream, that is, they may be produced quickly and continually, so that new data compression methods are required to exact and update the relevant information for the decisionmaker. The quality of the data is often varied because environmental factors are not under the control of the individuals and systems that collect the data. The advantages associated with quantity then are superseded by the need for improved data quality. On the order hand, in many testing situations, only small amounts of data are available due to cost, time, and safety constraints. The problems to be studied are sometimes vaguely formulated and appropriate models are not developed before acquiring the data. Close collaboration with scientists who work in the field of applications is required to develop new methodologies for addressing the problem of extracting information from meager samples. To extract more information from less data, improved methods for combining information from disparate tests may be needed.

b. 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.

c. 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.

d. 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, multi-strategy learning, and multiagent learning are among the foci of statistical research in this program.

e. 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.

f. Bayesian and Non-parametric Statistics. Future emphasis in statistics on "predictive" models vice 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 which are really approximations to avoid computational issues. This suggests a need for increased emphasis on research in areas such as robust statistics, non-parametric statistics, non-linear models etc. In addition to a greater volume of data, data are increasingly messy, for example, spot reports are very free-form. More work leveraging computational capability in developing novel approaches for making sense of messy data is of interest.

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