Multilevel methods in PDE constrained optimization

该项目的目标是开发有效的多层算法，用于受控制和状态上附加不等式约束的偏微分方程（PDEs）约束的大规模优化问题。过去二十年的计算革命不仅促进了基于PDE模型的高分辨率数值计算，而且促进了从基于模型的模拟到基于模型的设计的转变。后者转化为解决优化问题的问题，以确定初始值和/或边界值，材料属性，来源和PDE模型行为所需的其他参数。然而，一般来说，随着分辨率的增加，优化问题不仅变得更大，而且也变得更难以解决，从而使得偏微分方程的分辨率与可以使用最先进资源解决的相关参数识别问题的分辨率之间的差距越来越大；为了充分利用这些资源，高效的算法至关重要。虽然这种高效的算法在过去几年里已经被开发出来，但它们大多局限于没有集成电路的问题。在控制和/或状态上添加ic通常会增加问题的难度，因为拉格朗日乘法器的存在具有比解决方案更低的规律性。近年来，针对这类问题的优化算法有了明显的进展，然而，期望通过优化过程中所需的线性代数技术的改进可以进一步获得显著的效率。在这个项目中，PI的具体目标是为线性系统开发最优阶多水平预调节器，这些系统出现在由控制和/或状态上具有ic的线性和半线性椭圆或抛物型pdesc约束的优化问题的内点法和半光滑牛顿法求解过程中。对于更困难的状态ic问题，将考虑Lavrentiev和Moreau-Yosida正则化。长期目标是为流体流动的大规模控制问题（Stokes和navier -Stokes系统）开发有效的多层算法。该项目的结果有望使软件的最终用户——工程师、应用科学家——能够以与执行单个模拟相当的（小倍）成本解决高分辨率、相关的优化问题。长期目标应用包括天气预报和空气污染建模的数据同化。例如，高分辨率模型的快速数据同化将能够及时获得对飓风周围大气当前状态的更好的定量理解，从而潜在地提高当前的预测能力。从教育的角度来看，这个成功的项目将有助于PI在马里兰大学巴尔的摩县（UMBC）推广这一研究领域的努力，它将允许研究生和本科生在马里兰大学巴尔的摩县（UMBC）的研究领域获得战略利益的经验，这可能会增加他们在研究型大学或实验室找到一个好职位的机会。

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.