Collaborative Research: MATH-DT: Reduced order modeling and data assimilation for wave-dominated systems with applications to digital twins in ultrasonics

Millions of patients each year rely on medical procedures that use sound waves to break apart kidney stones or target diseased tissues without surgery. As these technologies advance, so does the need for greater precision, safety, and adaptability in treatment. This project contributes to the development of digital twins — real-time computer models that simulate how sound waves interact with the body and, based on imaging as the procedures unfold, help guide the process. By enabling more accurate and responsive treatments, such models could significantly improve clinical outcomes and reduce costs in noninvasive medicine. Beyond the immediate medical application, the project addresses a broader national need: the development of fast, reliable computational tools for predicting and controlling complex physical systems in real time. These methods are broadly applicable to technologies that rely on wave-based sensing and actuation, including systems in biotechnology, nondestructive materials testing, environmental monitoring, and national defense. The project will also serve as a multidisciplinary training ground for graduate students working at the intersection of applied mathematics, computation, and engineering. The research aims to develop fast, accurate algorithms that simulate how high-frequency waves, such as shock waves or focused ultrasound, propagate through complex, heterogeneous materials. The key technical challenge is that, in many real-world systems, waves must be tracked over hundreds of wavelengths, making traditional computational methods too slow for real-time use, especially when repeated simulations are needed to account for uncertainty or to estimate unknown parameters. To address this, the project will construct reduced-order models that capture the essential dynamics of wave propagation between source and target, while drastically reducing computational cost. These models will be designed to adapt in real time based on measurements from the system, enabling rapid prediction, control, and decision-making. The algorithms will be validated through comparison with full-scale simulations and, where possible, experimental data from laboratory-scale models of therapeutic ultrasound. The work is expected to lead to new strategies for real-time modeling and control in complex wave-driven systems. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.