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Scientific Computing
Scientific computing, also called numerical analysis, concerns algorithms for solving mathematical problems that arise in many fields, especially in science and engineering, and their implementation on high performance computers. Researchers at Illinois are working specifically in the areas of biological molecular dynamics, materials science, semiconductor simulation, astrophysics, and the design of solid propellant rockets. Scientific computing is mathematically intense. Because computers are digital, every problem must eventually become discrete before it is fed into a computer. As a simple example, you can think of continuous and discrete mathematics in terms of a watch. Continuous mathematics is like an analog watch, with a sweep second hand, and hands for minutes and hours that move smoothly over time. Between 1:00 and 1:03, there are infinitely many possible times. Discrete mathematics is like a digital watch. The only time that can take place between 1:00 and 1:03 is 1:02. It can only show finite steps in time. (Most natural phenomena-the motion of planets, the flow of water-are continuous.) Discrete mathematical models of physical phenomena often lead to large systems of linear equations that must be solved to determine an approximate solution to the original problem. Familiar examples include various types of partial differential equations (PDEs) describing the diffusion of heat, or wave motion, or electrical potential. However, many discretization techniques for PDEs lead to very sparse linear systems (the corresponding matrix has relatively few nonzero entities). One focus is to develop new methods to deal with these systems. One of the most powerful discretization techniques for PDEs is the finite element method, which is particularly useful for problems having complex, irregular geometry. Research in this area uses the structure of the finite element mesh to make the solution process for the corresponding linear system more efficient, especially when the problem must be partitioned for parallel computation. For a linear system, effects are strictly proportional to their causes, and solutions can be constructed as a superposition of fundamental solutions. This is not the case for nonlinear systems, for which the relationship between input and output can be much more complex. Solving a nonlinear system generally requires iterative methods, and convergence cannot always be guaranteed, so robustness is an important issue. An important class of nonlinear problems arises in optimization, in which one seeks a minimum or maximum of a function of several variables, where these variables are often subject to constraints. Research in this area focuses on special nonlinear optimization problems that arise, for example, in processing surface meshes for computer graphics or numerical simulations. Numerical simulation is a technique for approximating the behavior of a physical system by solving the equations resulting from a mathematical model of the corresponding physical phenomenon. Such simulations are vital for systems that are too difficult, or expensive, or dangerous to observe or experiment with directly. Simulations can also save a great deal of time and expense in designing new products or engineering new devices. Large-scale simulations often make extreme demands on computational resources, however, and often require novel numerical techniques. Research in this area focuses on a number of applications, including astrophysical phenomena, biomolecules, semiconductors, and solid propellant rockets.
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