How do we find meaning in an infinite dimensional continuum model of nature, such as a system of nonlinear partial differential equations?
How can predictions be made based on these models?
How can these models be used to make forecasts that help our life?

One example of research addressing these questions is the modeling of flow and transport in porous media. A porous medium consists of a solid with a large amount of connected or disconnected holes allowing the fluid to pass. Soil and human tissues are considered porous  to some extent. Applications of flow and transport include groundwater monitoring for water supply, contaminant remediation and storage of high level nuclear waste, modeling of inflammatory response due to bacterial infection for medical purposes, and production of oil and natural gas. From a humane and economical point of view, these applications are of crucial importance and the use of mathematics can help make a difference (energy can be extracted in an environmental and efficient manner..) Our group is developing state of the art numerical methods for simulating the flow and reactive transport in  porous media. The methodologies include finite element methods (continuous, discontinuous, mixed) and various tools of numerical analysis (domain decomposition, mesh and time adaptivity...)

As another example, understanding turbulent flow is central to many important problems including environmental and energy related applications (global change, mixing of fuel and oxidizer in engines and drag reduction), aerodynamics (maneuvering flight of jet aircraft) and biophysical applications (blood flow in the heart, especially the left ventricle). Turbulent flow is composed of coherent patches of swirling fluid called eddies. These range in size from large storm systems such as hurricanes to the little swirls of air shed from a butterfly's wings. Large Eddy Simulation (LES for short) seeks to predict the motion of the largest and most important eddies uncoupled from the small eddies. This uncoupling is important because the large eddies are resolvable on a computational mesh (a collection of chunks of the physical problem) which can be handled by a supercomputer. Our group's research involves modeling the large eddies (such as storm fronts, hurricanes and tornadoes in the atmosphere) in turbulent flow, predicting their motion in computational experiments and validating mathematically the large eddy models and algorithms developed. Current approaches to LES seem to be presently confronting some barriers to resolution, accuracy and predictability. It seems likely that many of these barriers can be traced to the mathematical foundation of the models used, the boundary conditions imposed and the algorithms employed for the simulations. The research undertaken is to develop these mathematical foundations as a guide for practical computation. This research promises to make it possible to extend the range of accuracy and reliability of predictions important to applications, such as those described above, where technological progress requires confronting turbulence! An important component of technology progress is the ability to make the best design choices for a device or an experiment. Choosing the shape of an air foil to minimize the fuel consumption of an airplane is one such example. Determining the protein structure that best fits a set of chrystallographic measurements is another example. Mathematically and algorithmically this is accomplished by numerical optimization, whose reach is expanded by advanced algorithms developed in our group. But optimization is useful sometimes in unexpected places. An area that has been pursued in our group has been the simulation of the collision and contact of many bodies by using time steps that are much larger (resulting in a much faster computation) compared to other algorithms. To achieve that, one has to compute one optimization problem every time the future position and velocity of the bodies needs to be predicted. Simulations of this type include simulations of granular matter, such as cereal, or powders used in drug manufacturing. We have been able to simulate with very large time steps exciting and difficult phenomena such as the Brazil nut effect (link to simulation), where the larger nuts rise to the top of a container that is shaked. But this approach is also useful in the design of interactive video games. Some of our methods have been incorporated by a commercial vendor of a physics engine for video games and is currently incorporated by many of the major commercial game developers Karma . Progress in science and technological innovation depends on simulations at the leading edge of scientific computing on high performance computers using tools developed by computational mathematicians at Pitt and elsewhere. Simulations of many important problems remain beyond the capabilities of computing capabilities for the foreseeable future. The scientific work of researchers in Computational Mathematics is essential to the United States' industrial competitiveness as well as our leadership in fundamental scientific research.