Fast volumetric integral equation solvers and applications

The ability to efficiently conduct high-fidelity simulations of complex physical phenomena simultaneously reflects our increased understanding of the underlying physics and enables future technological developments based on rapid iterative/inverse design. This project concerns a class of simulation techniques that rely on fundamental solutions (that is, by expressing solutions as a complicated superposition of `point sources' of light, sound, etc.) which have been highly effective when applicable as they have enabled transformational simulations of problems in electrostatics, wave phenomena as well as in human blood flow contexts. But more complicated phenomena (e.g. featuring nonlinearities or spatially-varying media), which are increasingly relevant in applications in medical imaging and also have long-standing intrinsic importance in geophysical exploration, have posed a substantial barrier to this class of methods---which thus have significant untapped potential in these application domains. This project will develop numerical methods with rigorous approximation guarantees to solve these physical problems and enable new scientific questions to be answered / for new technology to be designed, thereby strengthening the U.S. competitiveness and national defense. On the education front, the project will involve training in modern scientific computing generally and for their use in integral equation methods and applied to wave propagation particularly, all rare skills highly valued by industry. Integral equation formulations of nonlinear and/or variable-coefficient problems typically involve one or more volume integral operators (VIOs) featuring the free-space Green's function over the (generally complicated) domain of interest. This project proposes a novel class of volume regularization methods for accurately discretizing any of these VIOs, and if necessary to allow for their simultaneous computation. By suitably exploiting Green's identity the proposed methods allow for singularity-oblivious quadrature to nevertheless be used for these erstwhile singular integrals; accompanying error analysis in two and three dimensional contexts and in curvilinear domains will place the proposed methods on a firm theoretical foundation, and they will be integrated with fast N-body interaction methods. These methods will be coupled into advanced solvers that enable treatment of complex nonlinearities that arise in acoustic problems found in medical imaging. More generally, the methods will allow the extension of the integral equation methods to nonlinear and possibly time-dependent PDEs, as VIOs reduce inhomogeneous linear PDEs to corresponding homogeneous problems---for which surface integral methods have always been highly advantageous. Implementation of the numerical methods will be publicly distributed via the open-source integral equations software project Inti.jl. 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.