The investigator and his colleagues organize a series of annual Copper Mountain Conferences. The subject of these meetings alternates between multigrid methods in odd-numbered years and iterative methods in even-numbered years. The Copper Mountain Conferences provide a forum for the exchange of ideas in these two closely related fields. The program for a conference in this series consists of tutorials, invited lectures, and contributed papers, as well as time for scientific interaction among the participants. Topics of emphasis for the conferences include advanced architectures, algebraic-type methods, and nonsymmetric linear systems. This grant provides support for students to participate in the conferences.
The mathematical description of real-world problems leads inevitably to equations to solve. Many times the equations are nonlinear ones, arising because the underlying problem is nonlinear. Often, however, the original problem is linear and the mathematical equations are too. Such problems arise in all areas of science and technology, and are of special interest in biology, materials, environmental studies, and manufacturing. Progress in these areas requires solution of more comprehensive modeling equations --- such models are usually more nonlinear than simpler models ---, or more accurate solution of existing models --- increasing the size of the numerical problem to be solved. Many problems lead to equations that are not symmetric; computational methods for such problems offer special difficulties. Numerical methods for solving a differential equation usually begin by imposing a grid on the region where the equation holds. From the differential equation, algebraic equations are then developed; their solution represents the solution of the differential equation. The accuracy of the approximate solution commonly is measured by the fineness of the grid. Multigrid methods are numerical methods for solving partial differential equations that systematically exploit the relationship between approximate solutions on different grids to arrive at a solution whose accuracy is consistent with the finest grid but for considerably less work. The methods are often dramatically more efficient than others. The conferences address advances in iterative methods and in multigrid methods to deal with larger numerical systems, nonsymmetric and nonlinear systems, and the use of multiprocessor computers. The methods are of great practical import in engineering, manufacturing, materials, physics, and fluid dynamics. This project supports student participants at the Copper Mountain conferences on multigrid methods and on iterative methods. The students present a talk on their research in the regular sessions of the conference. The primary objective here is to encourage student participation in these rapidly evolving areas, and to provide an excellent opportunity for these students to demonstrate their new results, to learn more about the field from its experts, and to become a more integral part of the discipline. Supporting student participation is critical for developing the next generation of scientists and engineers.