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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 sub-disciplinary boundaries within the Division and the disciplinary boundaries within 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. Modeling of Complex Systems

The Modeling of Complex Systems Program conducts fundamental mathematical-analysis-oriented research, the objectives of which are to develop quantitative and qualitative 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 for those models, preferably those based on first/basic principles. The complex phenomena of interest to the Modeling of Complex Systems Program are largely physical and geometric phenomena (topological data analysis, mathematical information and signal theory, dynamical systems on complex structures, etc.) and human cognitive and social phenomena (strategic group formation, behavioral and cultural modeling, human-computer interaction and social informatics, etc.) 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 welcome if the phenomena are so complex that such frameworks are not feasible. Metrics for describing system parameters and model performance measures are a vital 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 will preferably be part of a complete mathematical analytical framework, but that is not a requirement for consideration. The research in modeling of complex phenomena supported by the Modeling of Complex Systems Program is primarily mathematical and geometric analysis (not computational mathematics).

The complex systems of interest to the Modeling of Complex Systems Program include, but are not limited to, those in the following four areas:

### 1.1.1. Information and Data

Information superiority is recognized as a key to success in defense efforts, peacekeeping, and humanitarian operations. Areas of prime interest include – but are not limited to – information sensing and collection over geographically large or dispersed domains, multimodal and hard-soft data fusion (optical, infrared, acoustic, human, etc.), analysis and modeling of organized or dynamic information networks, and topological data analysis of large, noisy data sets and under-sampled or corrupted signals.

### 1.1.2. Geometric and Topological Modeling

Representations of complex, irregular geometric objects and of complicated, often high-dimensional abstract phenomena and functions are fundamental for Army and other DOD agencies in modeling geophysical features and terrains, urban and man-made settings with vision, mobility, or other topological obstructions, complex data structures and information flows, and biological objects. Extending classical mathematical modeling and geometric analysis techniques to spaces with irregular (i.e. non-smooth) fine structure is also of key interest.

### 1.1.3. Human Cognitive and Behavioral Modeling

Quantitative, analytical models of cognition, behavior, sociolinguistics, and group/network formation are required for training, simulation, mission planning, predictive and causal modeling, and intelligence analysis.

### 1.1.4 Human-Machine Interaction

Models of human-computer interaction at both the small scale (individual/small group and machine) and the large scale (social informatics, large social information networks) are of interest, particularly for information gathering and supervised machine learning on the small scale, and for the influence of social information networks on strategic group formation and social action on the large scale.

Technical Point of Contact: Dr. Jay Wilkins, e-mail: leonard.d.wilkins3.civ@mail.mil, (919) 549-4334

## 1.2. 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.

### 1.2.1. 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, stochastic differential equations in finite or infinite dimensions, and quantum stochastic differential equations. 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 characterizing and 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. These equations are often driven by standard Brownian motion, semi-martingale (e.g. Levy processes), or other fractional processes, or other distribution-free noise. 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.
- Quantum Stochastics and Quantum Control. With technological advances now allowing the possibility of continuous monitoring and rapid manipulations of a system at the 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.
- 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 nanotechnology, and (iii) stochastic modeling and analysis of polymers.

### 1.2.2. 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. 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 decision maker. 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.
- 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 operations, the Army needs novel and efficient statistical tools for improving system reliability and survivability.
- 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 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. Joseph Myers, e-mail: joseph.d.myers8.civ@mail.mil, telephone: (919) 549-4225

## 1.3. 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 multicompartmental differential equations modeling in physiology, microbiology, epidemiology, and neurobiology, to agent-based and network models involved in understanding ecosystem and human dynamics, to innovative applications of traditionally pure mathematics to a wide range of biological systems.

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.

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

## 1.4 Computational Mathematics

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 that 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.

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

- Advances in Numerical Analysis. New methodologies are required for solving currently intractable Army problems. Advances that 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.
- Multiscale 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 that have universal application rather than methods applicable only to specific problem areas.
- 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.
- 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 that can be distilled from this data is different from and complements that which is 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.
- 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.

### 1.4.2. 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 toward making this goal achievable. Such research should include but is not limited to the following:

- 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.
- 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?
- 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?
- 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: joseph.d.myers8.civ@mail.mil, telephone: (919) 549-4245