Accurate and Efficient Frequency, Time, and Hybrid Frequency-Time PDE Solvers

This research will develop new mathematical and computational tools that can significantly improve the speed, reliability, and realism of scientific simulations across a broad range of scientific and engineering applications. These capabilities are expected to support data-driven computational methods and AI-assisted engineering design and scientific discovery by enabling high-fidelity predictive models that reliably capture wave propagation, transport, and multiscale physical interactions, as well as producing simulation data that is sufficiently accurate and robust to serve as a foundation for training, validating, and benchmarking next-generation AI-based methods. The work is also relevant to wave and quantum phenomena arising in emerging sensing, photonic, and quantum-device technologies. Potential biomedical applications include improved computational approaches for medical imaging, ultrasound, diffusion and transport in biological systems, and related health technologies that depend on accurate wave and transport simulations. In manufacturing and materials engineering, the resulting methods may contribute to the design and optimization of next-generation materials, microelectronic and photonic components, and precision production processes requiring accurate modeling of complex geometries and wave interactions. Broadly, the project will provide foundational capabilities that enable faster innovation, improved predictive technologies, and efficient use of high-performance computing resources in areas of long-term national interest. This project develops new mathematical formulations and fast computational algorithms for accurate solution of partial differential equations and wave phenomena in geometrically complex, high-frequency regimes. The work includes methods for problems with geometric singularities such as edges and corners, approaches for fractional Laplacian equations on bounded domains via weakly singular integral formulations, and fast iterative eigensolvers for integral-equation models arising in electromagnetic, acoustic, quantum, and fractional-diffusion settings, enabling efficient solution of large-scale eigenproblems. The effort also extends Fourier-continuation methodologies to time-dependent and nonlocal systems, including fluid flow, reaction-diffusion dynamics, ultrasound propagation, and classical and relativistic Schrödinger equations. In addition, multidimensional Fourier continuation methods will be generalized to non-smooth domains in two and three spatial dimensions and beyond, addressing a longstanding limitation of existing Fourier extension techniques. Finally, new screened WKB methodologies will be developed for high-frequency wave propagation in heterogeneous media, including configurations with caustics, using Fourier-based decompositions on geometrically adapted screening surfaces. Collectively, these developments aim to provide scalable, highly accurate computational methods for challenging wave, transport, and nonlocal phenomena beyond the reach of many current numerical approaches. 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. NSF Award ID: 2607322 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Oscar Bruno | Institution: California Institute of Technology, PASADENA, CA | Award Amount: $450,000 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2607322 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2607322.html