Computational simulation is essential for designing electromagnetic devices, predicting radar and sonar signatures, improving medical imaging, modeling advanced materials and larger molecular systems, and testing physical systems before they are built. These applications often require solving mathematical models that are far too large for direct calculation using standard methods, especially in realistic three-dimensional geometries. A major obstacle is that the computer representation of the underlying physical system can become an enormous array of numbers (known as a "matrix"), so large that it is impractical to store or manipulate. This project develops new mathematical and computational tools that use randomization as an algorithmic engine: instead of exhaustively examining all interactions in a simulated physical system, the methods use carefully designed random probes to discover hidden structure and build compact representations. The resulting algorithms are expected to make large-scale simulations faster, more accurate, and less costly in memory, time, and energy. Potential benefits include improved tools for electromagnetic device design, radar and sonar modeling, medical imaging, nondestructive testing, materials modeling, molecular simulation, and simulations that combine several physical models or numerical methods. By strengthening a core capability of scientific computing, the project promotes the progress of science and supports national health, prosperity, an