Since the stepped profile does have corners, serious problems arise in trying to adapt traditional codes. An IMA team has found a new approach to solving the Maxwell equations, which leads to a very promising numerical method. Electrophotography is a process in which pictures are made of light and electricity.

The common example is photocopying documents. One of the steps involved is creating a visual image from the electric image. Here the toner ink accumulates near the electric image of the dark spots of the document. The boundary of the toner's region is a "free boundary. The potential is continuous with its first derivative across the free boundary, and its normal derivative vanishes on the free boundary.

The mathematical problem represents an entirely new kind of free-boundary problem. An IMA team has shown that for some range of parameters, the problem has a unique solution and for another range of parameters, it has an infinite number of solutions. There are still many open questions regarding this problem. However, already at this stage certain important constants have been computed that may help the designer to improve the image of the photocopy. Growth of crystals in solution. A large number of crystals lie in a solution within a photographic film.

To achieve the best size distribution for a specific function of the film, one has to study the evolution of the crystals in time. This problem can be viewed as a dynamical system that approximates a conservation law with nonlinear nonlocal terms. The problem was considered by people at the IMA. Their analysis discovered the asymptotic size of the crystal grains; it also explained in what sense the dynamical system is a good approximation to the conservation law. So far only the case of crystals that are cube-like bodies has been considered. The next step will be to study a more realistic model where the crystals are cylinder-like.

Industrial mathematics conferences have been organized annually at the Rensselaer Polytechnic Institute. These conferences have an unusual format.

The speaker, an industrial participant, presents a prob-. They try to determine the basic variables and equations, the essential features of the problem, and the acceptable approximations. They discuss how to describe the problem through mathematical formulas. If this stage is successful, then discussion might continue on methods for the solution of the model equations.

Several such problems will be considered in the course of the conference. A similar industrial clinic is organized at the Claremont Colleges; an instructor and a team of students spend typically one year working on a specific problem. The Center for Quality and Productivity Improvement is an interdisciplinary center located at the University of Wisconsin. The staff of the center is evenly divided between statisticians and engineers.

The center supports a large range of technology transfer activities, from conferences to guest lectures to consulting, and conducts a research program in quality and productivity improvement on which the technology transfer is based. A consortium of industrial sponsors has been organized by the Institute for Oil Recovery and the Department of Mathematics at the University of Wyoming. The scientific program of the institute follows the research interests of its faculty in petroleum reservoir modeling and numerical simulation.

Computational ideas and algorithms developed within the research programs of the institute are available to the industrial sponsors, including dynamically adaptive grids and characteristic methods of differencing. At Duke University, a program was initiated in the modeling of granular flow. This problem had not been attempted previously by the mathematical community and looked rather disordered at the outset.

Design engineers for grain silos had encountered various problems important to the design process that they did not understand. After some effort, it was discovered that the mathematical problems were quite interesting and were illustrative of the class of problems that change type from elliptic to hyperbolic. The change of type was associated with the formation of shear bands in the granular material, which was just the problem that had puzzled the design engineers. A number of large industrial laboratories maintain in-house mathematics groups. These groups have similar technology transfer problems.

Similarly, the national laboratories have developed algorithmic, computational, and software capabilities and technology, which have been transferred to U. A very effective method of technology transfer is to educate students who will later find employment in industrial or national laboratories. Meetings of the mathematical sciences professional societies provide a forum for technology transfer, and in some cases attract engineers and mathematical scientists from industry. Because of the central role of computing, software is an increasingly important mechanism for technology transfer.

Well-designed software allows immediate application of new advanced algorithms and techniques in disparate fields. A common route for technology transfer involves implementation of a high-level computational algorithm in a computer code. Mathematica and Nastran provide examples.

Argonne National Laboratory has pioneered with the establishment of a software library, Netlib, for the electronic exchange of software. Often, research groups in industry pick up on academic research codes and incorporate the ideas into their in-house production and simulation codes. A portable translator was distributed within the Bell Laboratories and to universities for a nominal cost, thereby encouraging experimentation by users and feedback to the designer.

Similarly the statistical language S has grown from a research tool to a commercial product that is today the de facto standard among statisticians for both research and student training. These examples show that it is possible for technology transfer to succeed. There are a variety of ways to transfer technology.

Technology users can be canvassed, to determine their needs and interests; the problem can be selected in an area where users are known to be interested; and novel areas can be found, for which users and their prob-. To ensure that technology transfer occurs, the mathematical scientists, engineers, manufacturers, and business leaders must accept the task to be accomplished and plan for the result. Mathematical and computational analysis is an essential tool in product design and system development.

Oil exploration, automotive engine design, wing and fuselage design for aircraft, circuitry components for computers, finance, robotic control, the design of novel composite materials, and construction design provide only a few examples of this fact. Simulation provides such information more quickly and cheaply than the classic construction and experimentation still commonplace in many industries. Similar design speeds have been obtained by other manufacturers. Another example is the positioning of the engine nacelle on the Boeing to increase lift significantly.

The manufacturer of the was able to obtain substantial improvements in performance while reducing the number of wind tunnel tests from more than 60 to about Highly efficient methods of computational fluid dynamics lead to airplane geometries with optimal flight characteristics and lower fuel consumption see Figure 4. Complex processes are characterized by their many interacting sub-processes.

They must be efficiently designed, built, modified, and maintained with sufficient flexibility to be viable in new, flexible manufacturing environments. These goals cannot be achieved without detailed analysis and simulation of the entire system to indicate sensitivities of process output to changes in interacting component subsystems. Although engineering and scientific computing have become central tools of engineers and scientists over the past decades, there is a potential among U. These composite overlapping grids are used in modeling fluid flow around objects with complex geometrical structure.

Some of their advantages over other methods are their smoothness and their ability to provide high resolution where it is needed, both of which are important for accurate modeling. The composite grid shown in this figure is used for modeling air flow around a wing. Reprinted, by permisssion, from [ 16 ]. The importance of scientific and engineering computing has been confirmed by numerous U. Simulation and modeling have a critical role among the remaining fifteen identified technologies. Simulation has been at the heart of progress in technology and science for many reasons, including the following:.

The cutting-edge problems that challenge engineers and scientists typically cannot be solved by other methods. During the past several decades, there have been substantial advances in the mathematical methods and algorithmic development that unite science and technology with the computer. Simulation technology has enriched the knowledge base and benefited the intuitive problem-solving approach used by practicing engineers. Absent such simulation, the available tools would be rather inadequate for the type of problems that are being addressed today.

Aerospace and petroleum examples were mentioned earlier. In the microelectronics industries, the design of new semiconductor devices and the circuitry employing them can be carried out only through simulation.

In the pharmaceutical industry, computational methods for understanding the structure of molecules see Figure 4. The design of new drugs is widely expected to benefit greatly from a systematic use of simulation. Quantum chemistry depends heavily on large-scale high-performance computing, including simulation. Computational modeling in quantum chemistry will provide the scientific basis for new advances in pharmacology.

In the textile industry, the computerized layout of apparel cutting patterns to minimize waste is a problem in integer programming and optimization.

## Introduction to Methods of Applied Mathematics or Advanced Mathematical Methods for Scientists ...

Only a decade ago, computational power was measured in megaflops millions of floating point arithmetic operations per second. It is now measured in gigaflops billions of floating point arithmetic operations per second and will evolve to the multiteraflop range as powerful parallel computers come on line in the years ahead. Advances in graphics enable the user to comprehend pictorially massive amounts of data and results. Wideband networks are making supercomputing widely available to engineers and scientists at geographically remote locations.

At the same time, powerful workstations provide desktop computational capability formerly available only on mainframes at a limited number of central locations. Of equal importance to the raw computing power are advances being made in mathematical sciences, including the development of algorithms for parallel processing. In the past decade, knowledge of the behavior of the equations governing such vital areas as fluid dynamics and transport phenomena has increased dramatically.

New algorithms are significantly improving the stability, accuracy, and speed of solutions of such equations. The computation is based on the principle of minimization of an effective free energy. Computer simulation is an increasingly important method for determining the shape and structure of biological molecules and will be an increasingly powerful tool in biotechnology. Reprinted, by permission, from [ 13 ], Figure Effective simulation depends on modeling, algorithms, and analytic understanding, as well as validation against reality.

Modeling involves setting up mathematical equations whose solutions describe the behavior of the process to be modeled. The solutions must incorporate enough of the underlying science to ensure that results will be meaningful. The parameters in the equations must be observable or deducible from measurements and simple enough so their behavior can be understood. Finally, efficient and effective numerical methods for solving the equations must be developed and tested in each case.

- Asymptotic Methods and Perturbation Theory.
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Modeling is more than mathematical and numerical analysisâ€”it must of necessity be an interdisciplinary effort requiring the cooperation of engineers and scientists who understand the problems and mathematicians who understand the computational and mathematical modeling process. As technology advances and there is increased understanding, the mathematical model must be improved to represent more accurately the physical phenomena, with increases in complexity.

For example, the so-called drift-diffusion equations for modeling the behavior of semiconductor devices have been very useful. However, as technology advances to the regime of submicron devices, those equations may cease to be accurate. Revision of the model to incorporate more details of the transport of electrons through a version of the Boltzmann equation or by Monte Carlo simulation is progressing. Radically new algorithms are needed at all levels to use high-performance parallel computers efficiently as well as to deal with problems of ever increasing complexity.

Three-dimensional problems are orders of magnitude more complicated than two-dimensional ones. Simulation of a whole airplane is orders of magnitude more complicated than simulation of the wings. Understanding the structure and interactions of large organic molecules requires computational capability orders of magnitude larger than that required for simple molecules. System complexity will continue to increase as the underlying equations describing them incorporate more of the underlying science.

New numerical methods inherently suitable for parallel computing are needed to accommodate increasing computational demands. Input and output constitute another important area in which algorithmic advances are needed. The time it takes to input the geometry of the problem and generate the mesh on which many solution approaches depend is measured in weeks, whereas the time to perform the computing is measured in hours or minutes.

To achieve large-scale simulation, such bottlenecks must be overcome. New methods for describing the geometry of the problem must be found. Better automatic methods to generate acceptable meshes for efficient and accurate numerical integration are urgently needed. The output of the results is equally important. Graphic representation of the results of the computations is mandatory if engineers and scientists are to make sense of them.

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This area of research is in its infancy, but the results are encouraging and give rise to realistic expectations that current obstacles will be overcome. Large computer models are costly to run. The need to obtain information concerning the many parameters of the model requires efficient selection of parameter settings inputs.

This problem can be phrased as one of statistical experimental planning. The methods and concepts of quality control and statistical design of experiments began in the s. Quality control began as a way to monitor or test output and thus to discard or repair defects. Statistical design of experiments in industrial contexts started as a way to identify causes of defects. The two areas have since been merged in many of their aspects. They have been transformed into a system for the building of quality into the design of products, the control of manufacturing processes to assure quality, and the installation of simple statistical tools at all stages of production to permit early detection and diagnosis of problems.

This change of emphasis was stressed by Deming in his now famous 14 points for creating quality products. Improvement is achieved by a careful study of processes and by finding and removing root causes for defects.

Quality improvement is not a one-step process. It is an. Statistical methods of experimental design are used in a trouble-shooting mode. They are not restricted to a postmanufacture testing phase, but are used by engineers, foremen, and workers on the factory floor. Those closest to the problems are directly involved in their solution. Quality improvement results in reduced wastage, loss, and scrap and, in contrast to traditional quality control measures, is generally a cost-reducing measure.

These methods are not tied to unique cultural differences between national work forces. For example, a U. The facility had a product failure rate of percent, meaning that most television sets required repair, and some required multiple repairs, before manufacture was complete.

## APPLIED MATHEMATICS

After introduction of quality improvement methods, the failure rate was reduced to 2 percent, with an increase in product quality and a decrease in manufacturing costs. Statistical methods, to be used by factory workers, must be simple and robust. The methods are not a complex set of deductive rules, but rather are a simple set of tools, to be applied experimentally to diagnose problems.

Technology transfer is here a central concern. Moreover, development of appropriate statistical tools for this context is a research question currently engaging U. Statistical methods for design of experiments, such as factorial designs, blocking, and randomization, are well established in agriculture but less widely used in manufacturing. The selection of significant variables from among the less important ones, the reduction in the effective dimension of large or high dimensional data sets, and response surface methods are useful in data analysis.

The value of these methods is greatly enhanced when developed into convenient and robust computer software, and supported by good graphical representations. Quality improvement in manufacturing is not the end of the story. Quality is carried upstream to the design of products and the design of the manufacturing process. Quality by design, as this is called, requires collaboration among statisticians and engineers working with design, manufacturing, and quality. An example of an issue that arises in quality by design is the reduction of the variability of certain attributes of the product, as a function of the corresponding variability of the components.

The manufacturing process provides an enormous wealth of. Automated control provides a method for the use of this information in a self-learning or machine intelligence mode. Design variables for control of a chemical process for example, temperature and pressure might be specified initially through the solution of some model equation, which approximates the true manufacturing process.

The role of automated control is to observe these control variables and ensure that they attain their desired values. However, one could also monitor the output for some measure of manufacturing quality and force the control variables to search in a small neighborhood of their specified design values for the optimum values, which give the best output.

Today, simulation models are prevalent and increasingly used to provide the data needed to achieve quality by design. Use of computer models in engineering design requires determination of many design parameters, often interacting in complicated ways. Statistical methods of experimental design; fitting of response surfaces in high dimensions, often with limited data; handling of large data sets; and statistical methods of data reduction are examples of the ideas and tools coming from statistics that will aid the design by simulation process, just as they have traditionally aided the experimental design process.

What can be done for the design of products can also be transferred further "upstream" to the design of the manufacturing process. Quality improvement is not the unique province of statistics. All the areas of the technology base for example, simulation, modeling, and theory in engineering design contribute to manufacturing quality.

Differential equations are widely used in the modeling of natural phenomena. They are the basis of every one of the physical sciences and of the associated technology. Thus it is no surprise that they play a central role in the technology base required for economic competitiveness. Early in this century, boundary layer theory was developed as a powerful tool to attack nonlinear flow problems in a realistic way.

Asymptotic methods and singular perturbation theory provide insights into many critical phenomena in chemistry and physics, from shock waves to phase diagrams. Asymptotic theories depend on a small or large parameter and the possibly singular behavior that can result from small changes in a system.

Usually, problems with this character are hard to handle, numerically, and special insight can be derived from analytic treatment. Stiff differential equations are one of the most successful theories of asymptotics. Prerequisite: permission of instructor. AMATH Vector Calculus and Complex Variables 5 Emphasizes acquisition of solution techniques; illustrates ideas with specific example problems arising in science and engineering.

Prerequisite: either a course in vector calculus or permission of instructor. Prerequisite: either a course in differential equations or permission of instructor. AMATH Introduction to Fluid Dynamics 4 Eulerian equations for mass-motion; Navier-Stokes equation for viscous fluids, Cartesion tensors, stress-strain relations; Kelvin's theorem, vortex dynamics; potential flows, flows with high-low Reynolds numbers; boundary layers, introduction to singular perturbation techniques; water waves; linear instability theory.

Prerequisite: either a course in partial differential equations or permission of instructor. AMATH Applied Probability and Statistics 4 Discrete and continuous random variables, independence and conditional probability, central limit theorem, elementary statistical estimation and inference, linear regression.

Emphasis on physical applications. Prerequisite: some advanced calculus and linear algebra. Offered: jointly with STAT Legendre transformation, Hamiltonian systems. Constraints and Lagrange multipliers. Space-time problems with examples from elasticity, electromagnetics, and fluid mechanics. Sturm-Liouville problems.

Approximate methods. Offered: W, odd years. AMATH Networks and Combinatorial Optimization 3 Mathematical foundations of combinatorial and network optimization with an emphasis on structure and algorithms with proofs. Topics include combinatorial and geometric methods for optimization of network flows, matching, traveling salesmen problem, cuts, and stable sets on graphs.

Special emphasis on connections to linear and integer programming, duality theory, total unimodularity, and matroids. Offered: jointly with MATH Basic problem types and examples of applications; linear, convex, smooth, and nonsmooth programming. Optimality conditions. Saddlepoints and dual problems. Penalties, decomposition. Overview of computational approaches. Desirable: optimization, e. Math , and scientific programming experience in Matlab, Julia or Python. Steepest descent, quasi-Newton methods.

Quadratic programming and complementarity. Exact penalty methods, multiplier methods. Sequential quadratic programming. Cutting planes and nonsmooth optimization. Controllability, optimality, maximum principle. Relaxation and existence of solutions. Techniques of nonsmooth analysis. AMATH Mathematical Analysis in Biology and Medicine 5 Focuses on developing and analyzing mechanistic, dynamic models of biological systems and processes, to better understand their behavior and function. Prerequisite: either courses in differential equations and statistics and probability, or permission of instructor.

Draws examples from molecular and cell biology, ecology, epidemiology, and neurobiology. Topics include reaction-diffusion equations for biochemical reactions, calcium wave propagation in excitable medium, and models for invading biological populations. Focuses on analyzing dynamics leading to functions of cellular components gene regulation, signaling biochemistry, metabolic networks, cytoskeletal biomechanics, and epigenetic inheritance using deterministic and stochastic models. Prerequisite: either courses in dynamical systems, partial differential equations, and probability, or permission of instructor.

AMATH Neural Control of Movement: A Computational Perspective 3 Systematic overview of sensorimotor function on multiple levels of analysis, with emphasis on the phenomenology amenable to computational modeling. Topics include musculoskeletal mechanics, neural networks, optimal control and Bayesian inference, learning and adaptation, internal models, and neural coding and decoding.

The book is a comprehensive, self-contained introduction to the mathematical modeling and analysis of infectious diseases. It includes model building, fitting to data, local and global analysis techniques. Various types of deterministic dynamical. Matrix algorithms are at the core of scientific computing and are indispensable tools in most applications in engineering. This book offers a comprehensive and up-to-date treatment of modern methods in matrix computation. It uses a unified approac.

## 585.615 - Mathematical Methods for Applied Biomedical Engineering

This book provides a systematic treatment of the mathematical underpinnings of work in data assimilation, covering both theoretical and computational approaches. Specifically the authors develop a unified mathematical framework in which a B. This text provides a framework in which the main objectives of the field of uncertainty quantification UQ are defined and an overview of the range of mathematical methods by which they can be achieved.

Complete with exercises throughout, the boo. This book presents various results and techniques from the theory of stochastic processes that are useful in the study of stochastic problems in the natural sciences. The main focus is analytical methods, although numerical methods and statistical.

Computational Electromagnetics is a young and growing discipline, expanding as a result of the steadily increasing demand for software for the design and analysis of electrical devices.