Numerical Analysis of Partial Differential Equation Constrained Optimization Problems

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Every process in nature tries to optimize certain quantities under the available resources. The limited available resources act as constraints and, for this reason, constrained optimization problems are ubiquitous in science and engineering. Such optimization problems are inherently nonlinear due to the presence of constraints. This makes the development and analysis of algorithms and computational techniques to simulate the relevant phenomena extremely challenging and of practical relevance. A systematic approach is needed to allocate resources, without violating constraints, and remain effective in minimizing cost. This project will lead to development of advanced, efficient algorithms to design micro- and nano-scale lab-on-chip devices, as well as novel materials with desired configuration and user specified properties. As an outcome of shape optimization tools, robust algorithms for image processing and structural design will be developed. A significant amount of effort will be devoted to develop new models for cardiac electrical response and diffusion of water molecules in brain tissue. Many of the partial differential equations (PDEs) that describe the constraints are geometric, nonlinear, and multiscale, with an unknown domain, i.e. free boundary problems (FBPs). This makes the development and analysis of numerical techniques for the solution of these problems challenging. This project focuses on constrained optimization problems with PDE constraints -- the so-called PDE constrained optimization problem. In fact, it aims to develop and study numerical techniques for a new collection of such problems for which classical ideas and methods do not work. They are: optimization with Stokes FBP (including surface tension) as constraint, problems where the Hessian appears in the cost functional, multilevel techniques for shape optimization and image segmentation problems, optimization with nonlocal (fractional) operators, and degenerate parabolic equations as constraints. The applications range from micro- to nano-scales, where surface effects dominate bulk effects. This will have impact in designing next generation lab-on-chip devices. By manipulating curvature, one can drive particles towards a desired assembly or a specific configuration. This will lead to highly sophisticated materials with user specified properties. The multilevel techniques developed to solve shape optimization problems will provide a general tool to study inverse problems (acoustic scattering, for example), image processing, structural design, and fluid dynamics (for example, in aircraft design).