CAREER: Integrated Approaches for Fast and Accurate Large-Scale Inversion

The ability to compute solutions to inverse problems is essential in various scientific applications (e.g., for cancer diagnosis or for crack detection in underground mines), but computing real-time solutions to large nonlinear problems that incorporate physics- or data-informed constraints is not feasible with current inversion algorithms. Moreover, as numerical solutions to inverse problems are increasingly being used for data analysis and to aid in decision-making, these computational limitations pose significant bottlenecks in algorithms for uncertainty quantification (e.g., for estimating solution variances). The overarching goal of this project is to significantly reduce the costs of numerical inversion and to enable statistical tools to aid scientists in making informed decisions. These developments will lead to scientific advancement in many important fields. For example, existing collaborations with biomedical and mining engineers will ensure that the proposed research can result in improved medical diagnosis via advanced point-of-care imaging technologies, fewer injuries due to improved ground control monitoring of underground mines, and advanced signal estimation for real-time analysis of physiological systems. Moreover, the PI will continue to actively engage in activities that encourage students from historically under-represented groups. The PI's focus on upper elementary to high school girls and on outreach that will feed back into the greater research and teaching communities (e.g., K-12 teachers) will contribute to the recruitment, training, and retention of a diverse next generation of computational scientists. This research will advance knowledge in the field of computational inverse problems by developing faster methodologies and more robust frameworks for the design, computation, and analysis of solutions to inverse problems. An integrated framework will be adopted, where the main research thrusts are (i) to develop novel regularization methods and implementations to handle application-specific constraints, while simultaneously incorporating robust parameter selection methods; (ii) to advance technologies for real-time computation of solutions to large, nonlinear inverse problems (e.g., by integrating stochastic methods and update approaches); and (iii) to enable critical, yet previously unobtainable, quantitative diagnostics for complex, nonlinear systems by developing efficient error estimation methods.