Multilevel methods in PDE constrained optimization

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The objective of this project is to develop efficient multilevelalgorithms for large-scale optimization problems constrained by partialdifferential equations (PDEs) with additional inequality constraints (ICs)on the controls and states. The computational revolution of the last twodecades has fostered not only high-resolution numerical computations basedon PDE models, but also a shift from model based simulation to model baseddesign. The latter translates into the question of solving optimizationproblems in order to identify initial and/or boundary values, materialproperties, sources, and other parameters for which the PDE models behavein a desired way. However, in general, by increasing resolution not onlydo optimization problems get larger, but they also become more difficultto solve, thus rendering an ever widening gap between the resolution ofPDEs and that of associated parameter identification problems that can besolved using state of the art resources; in order to take full advantageof these resources, highly efficient algorithms are critical. While suchefficient algorithms have been developed over the past few years, they aremostly restricted to problems without ICs. The addition of ICs on thecontrols and/or states normally increases the difficulty of the problemdue to the presence of Lagrange multipliers that have lower regularitythan the solution. Recent years have witnessed a sensible progress in theoptimization algorithms that target such problems, however, it is expectedthat significant efficiency can further be gained by improvements in thelinear algebra technology needed during the optimization process. In thisproject the PI specifically aims to develop optimal order multilevelpreconditioners for the linear systems arising in the interior pointmethod and semismooth Newton method solution processes of optimizationproblems constrained by linear and semilinear elliptic or parabolic PDEswith ICs on the controls and/or states. For the more difficult problem ofstate ICs, both Lavrentiev and Moreau-Yosida regularizations will beconsidered. The long term goal is to develop efficient multilevelalgorithms for large-scale control problems for fluid flows (Stokes, andNavier-Stokes systems).The results of this project are expected to enable end users of the software - engineers, applied scientists - to solvehigh-resolution, relevant optimization problems at a cost that iscomparable (a small multiple of) to that of performing a singlesimulation. Long-term targeted applications include data assimilation forweather prediction and air contamination modeling. Fast data assimilationfor high resolution models would enable, for example, gaining in a timelymanner a better quantitative understanding of the current state of theatmosphere around a hurricane, thus potentially improving the currentpredictive capabilities. From an educational perspective, the successfulproject will help the PI's efforts in promoting this field of research atUniversity of Maryland Baltimore County (UMBC), and it will allow graduate and undergraduate UMBC students to gainexperience in a research area of strategic interest, which is likely toincrease their opportunities of finding a good position in a researchuniversity or laboratory.