Nonlinear Optimization Algorithms for Large-Scale and Nonsmooth Applications

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The investigator, his colleagues, and his students study the development, analysis, and implementation of algorithms for large-scale PDE-constrained and nonsmooth optimization. The novelty of the work in both of these frameworks is that in each case the investigator and his group are finding powerful ways in which the most advanced methods for nonlinear programming can be enhanced and broadened to remain effective for application areas in which they have previously been inefficient or inapplicable. In the context of large-scale PDE-constrained problems, such as those in optimal design, parameter estimation, and image registration, this is being achieved by removing the need for the factorization of matrices and allowing for inexactness in the solution of large-scale linear systems, while still guaranteeing convergence to a solution point. In the context of nonsmooth applications, such as those in compressed sensing and robust stability and control, this is being achieved by enhancing leading algorithmic frameworks through a process of gradient sampling, allowing for a loosening of the assumption that the problem functions are differentiable everywhere. These works in these fields tie together algorithms and computational techniques from diverse areas, and both numerical methods and convergence theory are being provided.The broader impact of this project is that it advances pencil-and-paper engineering ideas to the point where they can be implemented in high-performance computing software and are able to solve challenging problems in the design and analysis of complex systems. For example, there is a high demand for optimization tools such as these in healthcare, particularly in the area of cancer treatment and therapy. By providing doctors and medical technicians with novel computational tools, they will be able to optimally administer hyperthermia treatment in a manner that takes into account the inner complexities of the human body, such as blood flow. They will also be able to effectively and adaptively design plans for radiation therapy that minimize damage to healthy -- and often critical -- tissue near the target area(s). Amazingly enough, these same computational tools can also be employed in medical image registration, aiding medical professionals in the detection of irregularities over time and between different (e.g., PET, CT, MRI) scans. The goal in all of these areas is to provide the user with sophisticated software that can answer, in real-time, difficult questions such as "What is the optimal way of administering this radiation?" and "Is there anything in this image that has changed or is cause for alarm?"